Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

Converse theorems for GL n

Converse theorems for GL n by J. w. GOGDELL* and I. I. PIATETSKI-SHAPIRO** The objective of this article is to prove a criterion for a given irreducible represen- tation II of GL~(A) to be automorphic. This criterion traditionally is called a Converse Theorem, after Hecke's celebrated Converse Theorem [17, 18]. The converse theorems of this paper have an application to the problem of Langlands' lifting of automorphic representations from classical groups to GL~. This application will be considered in a future joint publication with S. Gelbart, D. Ginzburg, S. Rallis, and D. Soudry. The first converse theorem was actually proved by Hamburger in 1921 [5]. This theorem states that any Dirichlet series satisfying the functional equation of the Riemann zeta function ~(s) and suitable regularity conditions must be a multiple of ~(s). The generalization to L-functions corresponding to holomorphic modular forms was done by Hecke in 1936 [17]. The leading idea of Hecke was the connection of L-functions which satisfy a certain functional equation with modular forms. However Hecke was able to prove this connection only for holomorphic modular forms with respect to the full modular group, In 1944 Maass extended Hecke's method to his non-holomorphic forms, but still only for the full modular group [35]. The next very important step was made by Well in 1967 [42]. Well showed how to work with Dirichlet series corresponding to holomorphic modular forms with respect to congruence subgroups of the full modular group. Weil proved that if a Dirichlet series together with a sufficient number of twists satisfy nice functional equations with suitable regularity then it comes from a holomorphic modular form with respect to a congruence subgroup. The work of Well marks the beginning of the modern era in the study of the connection between L-functions and automorphic forms. In 1970 a remarkable new book came out: "Automorphic Forms on GL(2)" by Jacquet and Langlands [21]. In this book, instead of automorphic forms, a new object came into this scheme: automorphic representations. The basic result of Jacquet and Langlands was the following. They attached to each automorphic representation of GL(2) an L-function and proved that the nice properties of this L-function, i.e., holo- * The first author was supported in part by NSA grant MDA904-91-H-0040. ** The second author was supported in part by NSF grant DMS-8807336. 158 J. w. COGDELL AND I. I. PIATETSKI-SHAPIRO morphic continuation and nice functional equations, are equivalent to the representation being automorphic. In the following we will use the language of automorphic representations rather than the classical language of automorphic forms. However, for the applications to the problem of lifting of automorphic representations, we need results which are more similar to Well's theorem rather than that of Jacquet and Langlands. In order to get this result we have to use Well's idea, but disguised in the language of automorphic represen- tations. A preliminary version of these results over Q was given in [39]. We would like to thank Herv6 Jacquet and the referee for pointing out some mistakes in an earlier version of this work. 1. Some basic definitions and notation Throughout we will take k to be a global field. Let o denote its ring of integers. For each place vofk we will let k~ denote the completion ofk at v. At the non-archimedean places we will let o v denote the ring of integers of kv, p, the unique prime ideal of o,, m, a choice of generator of p, and we will normalize the absolute value so that ] m, Iv ----- q~-a where q, = [ o,[p, [. We will use either o~ x or u~ for the group of local units. The symbol A will denote the ring of adeles of k and A � its group of ideles. Thus, A is the restricted pro- duct II',k, of the completions of k with respect to the compact subrings 09. If S is a finite set of places of k we will let k s = II,~ s k, and A s = II',~ s k, so that A = k s A s. We will use a similar notation for ideles. For each finite set of places S of k containing all archimedean places, the ring of S-integers is os = k n k s I1~r s o~. We may view o s as a discrete subgroup of k s through the embedding of k into k s. Let u s = [[~s u~C (A� s. The class number hs of os, called the S class number of k, is the cardinality of the S-class group = � � x ~s k \A /k s u s We fix a non-trivial normalized additive character ~ of A which is trivial on k. Fix a basis { e, } of k" with respect to which the matrix structure of GL, is defined. Let B. denote the Borel subgroup of upper triangular matrices, A~ its Levi subgroup consisting of all diagonal matrices, and N, its maximal unipotent subgroup. Let P'~ denote the standard parabolic subgroup of GL~ associated to the partition (n- 1, 1) of n. Let P,~ C P'~ be the mirabolic subgroup consisting of those matrices in P', whose last row is (0, ..., 0, 1). Let P, C P~ denote the opposite mirabolic and parabolic. So ~ = ,(p,)-l. By Z,, we denote the center of GL,. For each non-archimedean place v we will let K, = GL,(0,) be a maximal compact subgroup. We will always consider admissible representations II, of GL,(k~) on a complex vector space VII v in the usual sense [6, 8, 9]. As is common, we will not distinguish between admissible representations of GL.(k,) and of its Hecke algebra [6]. We will 159 call an admissible representation unramified if the space of vectors fixed by K, is one- dimensional. At an archimedean place v, we select as maximal compact subgroup K, either O(n) or U(n) defined with respect to the basis above. At an archimedean place v of k by an admissible representation H~ of GL,(k~) we will mean a smooth representation of GL,(k,) on a complete Frechet space Vnv whose subspace of K,-finite vectors is an admissible representation of its Hecke algebra [6] and such that (II,, Vuv ) is a canonical smooth model of moderate growth (in the sense of Casselman and Wallach) of the underlying representation of its Hecke algebra [10, 27]. Let v be any place of k and let +, be any non-trivial additive character of k,. Then +9 defines a character of N,(k,), which by abuse of notation we again denote by +,, by +,(n) = +,(nl. ~ + n~. 3 + ... + n,_l,,) where n = (n~,j) eN,(k~) relative to the basis above. Let (II,, Vn~) be a finitely generated admissible representation of GL,(k,). We let V$o denote the space of +,-Whittaker functionals on Vn,, i.e., the space of continuous linear functionals k, on Vno such that )~,(1-I~(n)~)= +,(n)),~(~,) for all n e N.(k,) and all ~, ~ Vno. A representation H, of GL.(k,) is of Whittaker type if H. is finitely generated, admissible, and dim(VS,) = 1. In this case we have a non-zero intertwining map from Vno to the Whittaker space given by ~, ~-~ W~,(g) = X,(1-I,(g) 4,) where ~, e V~o is a non-zero Whittaker functional. We will call the space of functions zCU(II,, +,) = { W~(g) ] ~, e H, } the Whittaker model of H, (even though it is a model for the Whittaker quotient of H, unless the Whittaker map above is injective) and it is unique. An irreducible admissible representation of Whittaker type is called generic. For our purposes, we will only need consider representations of Whittaker type of a certain nature. Let Q be the parabolic subgroup of GL, associated to the partition (rx, ..., r,~) ofn. For each i let ~ be a quasi-square integrable representation of GLn(k,) (i.e., an irreducible admissible representation whose matrix coefficients become square- integrable modulo the center after twisting by a suitable character of GL,i(k,)). Then the (unitarily) induced representations are of Whittaker type [4, 27]. Throughout this paper, by an induced representation of Whittaker type (or, more succinctly, an induced of Whittaker type) we will always mean one of these induced representations. From this it is clear that induced representations of Whittaker type have well-defined central characters. Also, the subspace of K,-fixed vectors is at most one-dimensional. In particular, let Q. be the parabolic subgroup of GL, associated to the partition (rl, ..., %) of n. For each i, let P~,o be a tempered representation of GLn(k,). Let Ul > u~ > ... > u,, be a sequence of real numbers. Set T.a,qGYun(kz~) r / ~ . 9 . =, = L 0 .0 I I | | (p.,0| [')] ...~s.~,,~;..n~r.~*v~r,v,~wj~ J. w. COGDELL AND I. I. PIATETSKI-SHAPIRO (unitary induction). We call such a representation an induced representation of Langlands type. Then ~v is an induced representation of Whittaker type [19, 26, 27]. If II, is an irreducible generic representation of GL,,(kv) then II v is necessarily an irreducible induced representation of Langlands type [26, 27]. The Langlands classification for GL,(kv) says that every such ~v has a unique irreducible quotient II v and every irreducible admissible II v occurs uniquely as the quotient of some E, [7]. Consider a representation II of GL,(A) on a space V = Vn. This representation is called factorizable if there are local representations 1-[~ of GL,(k,) on spaces V, such that II, is unramified for almost all places v and (II, V) is the restricted tensor product of the (IIv, Vv) as in Flath [12]. We will denote this simply by II = @ II v. We will always consider admissible representations of GL,(A) in the sense of [6] or [12]. If II -~ (~)1-I, is factorizable and admissible then each II, is admissible, and conversely. If S is a finite set of places of k we will let G s = GL,(ks) =IIve s GL,(k,) and GS= GL,(A s) = 1-I'v~ s GL,(k,). Similarly, for II = (~II v factorizable we shall let 1-Is = (~,Es IIv be the associated representation of G s and II s = @,~s 1-Iv be the asso- ciated representation of G s, so that II = 1-I s | 1-I s. If fl = @ II, is an admissible factorizable representation of GL,(A) we will say that II is of Whittaker type, induced of Whittaker type, or generic if each FIv is and, in addition, at the places v where II v is unramified the space of K:fixed vectors is not in the kernel of the map to the Whittaker quotient. (This last condition is automatic if II v is generic or induced of Langlands type since in these cases the map to the Whittaker model is an isomorphism [26].) In these cases there is a unique global Whittaker func- tional k (up to scalars) given by the product of the local Whittaker functionals ?'v suitably normalized. At the places v where II, is unramified, there is a distinguished unramified vector ~0 with respect to which the restricted tensor product is taken. At these places we always normalize the Whittaker functionals kv so that kv(~ ~ = 1. In terms of the local Whittaker models, this implies that W~o(I,) = 1. If~CU(fl,, +~) are the local Whit- taker models, then r +) = (~zCV(IIv, ~). It is again clear that global induced representations of Whittaker type have central characters. If H = (~ l-I v and II' = (~)II~ are two factorizable admissible representations of GL,(A) then we will say that they are quasi-isomorphic if 1-I, ~_ FI', for all non-archi- medean v for which both 1-I, and 1-I~ are unramified. By an automorphic representation of GL,(A) we will mean an admissible sub- quotient representation of the space of automorphic forms d(GL,(k)\GL,(A)) [6]. By a proper automorphic representation we will mean an admissible subrepresentation of the space of automorphic forms d(GL,(k)\GL,(A)). By a cuspidal automorphic represen- tation we will always mean an irredudbh cuspidal automorphic representation. These are of course always proper. 161 2. Basic converse theorems Let II = @ FI, be an admissible factorizable representation of GL,(A) such that each II, is either irreducible or induced of Whittaker type. Let v denote a factorizable automorphic representation of GL,.(A) for some m with 1 ~< m ~< n -- 1 such that each % is irreducible or induced of Whittaker type. Then from the local theory of L-functions for GL,(k,) [24, 27] for each place v we have a local L-function L(II, x %, s) and local c-factor ~(II, � %, s, +~) attached to II and "r. We may then formally define a global L-function L(H X v, s) = I-IL(H, x %, s) and a global ~-factor ~(n � ~, s, +) = 1-I ~(no x ~,, s, +o). To see that these are actually well-defined we need the following elementary lemmas. Lemma 2.1. -- The a-factor ~(H X ~, s, +) is absolutely convergent and if the central character ~o n of II is invariant under k � then ~(n x ,~, ~) = II ~(n, x .~,, s, +.) is independent of +. Proof. -- For almost all v, II,, % and +~ will be unramified and so ~(1J~ � ~, s, +~) - 1 for these places. Thus r � v, s, +) is convergent. To prove that the product is independent of the choice of additive character we must consider how the local c-factor changes when we change our additive character +(x) to t~X(x) = +(kx) with X e k � Recall that the local ~-factor is defined by the local func- tional equation [24, 27] uZ(W, W'; s) CF(p(w,,,~) ~r, T~r,; 1 -- s) r � %, s, +~) o~v (- 1) "-1 = L(n~ � ~o, s) L(~, � %, 1 -- s) Where W e W~(II,, +~), W' e W'(%, +~-1), ,~(g) = W(w, ,g-l) e ~r +~-1), and ~r,(g) = W(w~ tg-1) e ~(~,, ~,). The Weyl elements involved are w~ = (1 1), 21 162 J. W. COGDELL AND I. I. PIATETSKI-SHAPIRO the longest Weyl element of GL,, and o) ?-O n _ whereas ~ denotes right translation in the Whittaker model. The integrals themselves are given by .c*,~\ot.c~) In- and, setting k = n -- m -- 1, ~r(w,w';s) W'(g) [ det(g)I '-''-"'t2 dx dg. rd~d\GLm{kv) 'rnr 0 To change the Whittaker model from those with respect to d?, to those with respect to +x, note that if we set r -- 1 ~t ~ 2 ~,(x) = ~B,(k), then for W ~r162 +,) we have/(an(X)) W(g) = W(an(X ) g) eqF'(II,, d~x). Note that the local L-functions are independent of the choice of d?,. Hence r � v~, s, +x) is defined by the local functional equation 'r(t(~n(X)) w,t(,.(x)) w';s) ~(II. � ~~ ~, +~) ~.o(- 1) n-' L(II, � "%, s) _~P(p(w.,.) (t(~.(x))w) ~, (t(~.(x))w')~; 1 -s) L(H. � ~'., 1 -- s) Now a straightforward computation gives 'r(t(a.(x)) w,t(a.(x)) w'; s) =lxly+',%(x)"-"'t~(p( I" W, W'; s) a._.(x)) 163 with A = m(m -- n) -- ~ m(m -- 1) 1 (m 1 ) 1 1) m(m + 1). B =~(m--n) (m--n)---~m(m-- 1) +~(m-- An equally straightforward calculation gives ~v(p(w.,.) (t(~.(x)) w) ~, (t(a.(x)) w')~; 1 - s) = I x IC'l-" + v ~no(X)" %(x)" re (0(o.,.) (p ( I" ,W'; 1-- a n_ m(X with C =--m ~+~m(m- 1) 1 (1 ) D = -~ m(n -- m-- 1) (n--m--2) +-~(n--m) m S-~m(m- l) +~(m-- 1) m(m+ 1). Then using the definition of the respective local e-factors, we find 9 (n~ x ~, s, +~) = ~0(x)" o,0(x) ~ I x 17 '-~ ~(n~ � ~0, s, +0) with d :nm -- -~ m(m + 1). Taking the product over all places of k and using the product formula we find ~(n � ~, s, +~) = on(X) ~ o.(x)" ~(n � ~, s, +). Since v is automorphic its central character is invariant under k � Hence if Olt is invariant under k � we see that the product ,(II � % s, q) = r X % s) will be independent of the choice of d/. [] Lemma 2.2. -- Suppose L(II, s) : II~ L(II~, s) is absolutely convergent in some half- plane. Then for any automorphic representation v : (~) % of GL,,(A) which is either irreducible or induced of Whittaker type the Euler product for L(II � % s) is also absolutely convergent in some half-plane. Proof. -- Let T be a finite set of places of k containing all archimedean places such that II~ is unramified for v r T. Then the local L-factor for the places v r T will be of the form L(n., s) = II (1 -- a,., q~-,)-l. ~=1 164 j. W. COGDELL AND I. I. PIATETSKI-SHAPIRO Globally, let us write L(II, s) = Lr(rl, s) LT(n, s), where ~(n, s) = II L(H~, s) v~T is a finite product, and hence always absolutely convergent, and LT(n, s) = H L(rL, s) = II II (1 -- a,,. q~-*)-a. Then if the Euler product for L(H, s) is absolutely convergent for Re(s) > Co, we have the estimate [ a,, ~ I ~ q~ for all v r T, with the implied constant independent of i and v. Let v = | % be an automorphic representation of GLm(A) which is irreducible or induced of Whittaker type. Then we know that the Euler product for L(':, s) converges absolutely in some half-plane, say Re(s) > q. Enlarging T if necessary, we may assume that % is also unrarnified for v r T. Then, as above, we have L(%,s) = fi (1 -- b j,~ ~,,-t } $=1 with the estimate [bj,~ [ ~ q~. The Euler factor for L(II~ � %, s) for v r T is given by ~m L(II~ � %,s) = fi fi (1--a~,~b~..,q~-')-a= II (1--ck, ~q~-~)-'. i=1 j=l k=l Since we have the estimate I ck,. 1= ]a,,~ I I bj,~ I ~ q~+ =1 for v r T, we see that the Euler product for L(H � % s) is absolutely convergent for Re(s) > co + ct + 1. [] Let g' denote the outer automorphism g ~ g' = ~g-1 of GL,. For any repre- sentation = of GL, over a local or global field, let =~(g) = =(g'). If II, is an induced of Whittaker type, then so is W~. If II~ is irreducible, then so is H~, and in fact H i _~ H,, the contragredient representation. Lemma 2.3. -- Suppose L(II, s) converges in some half-plane and tkat the central character of H = | II, is invariant under k � Then the Euler product for L(W, s) also converges absolutely in some half-plane, as do the L(YP � v ~, s) for any automorphic representation v of GL,,(A) whiek is irreducible or induced of Whittaker type. Proof. -- We may assume that the central character O~rl of II is unitary. For ff it is not, we have [ on(a)[ = [ a I d for some d =~ 0. If we let o~_a!,(a ) = ] a [-al. and set II' = II | o~x, then II' has a unitary central character. Since L(II', s) = L(II, s -- din) we see that L(H, s) is absolutely convergent in some half-plane ff and only if L(II', s) is. 165 For v r T, with T as in Lemma 2.2, II~ will be unramified and we have unramified characters ~1,~, ..., ~,,~ of GLI(k,) such that II~ is the unramified constituent of Ind~.~'(~l,.| ... | The local factor L(II,,s) is then rI(1- a,,,q:') -1 with ai, ~ = ~t,,,(%). The central character of rl. is tor~. = r[~,,.. Since this is unitary, we have 1--I ton~ l = I] I ~,.~(~o)I = ~I la,~l. i=! ' Still for v r T, if II, is as above, then II', will be an unramified constituent of "'~.~k.~ ~m,. "- ~,.). Its local factor will then be L(HL~, s) ~ l-I(l -- b~,. q~-i)-i with b~,. = ~z~,.(%) -1 = a~,..-1 Now assume that L(II, s) converges absolutely for Re(s) > c, so that we have the estimate [ a~,~ I ~ q~. Then for [ b~, ~ [ we have Ib,.~l = II Ib,,~l -~= II la,,,[ ~ q~-l, 0. Hence the Euler product for L(IP, s) converges absolutely for Re(s) > (n -- 1) e + I. The rest of the lemma now follows form Lemma 2.2 applied to IP. [] Definition. -- Let II : | II~ be a factorizable admissible representation of GL,(A) such that each local component II~ is either irreducible or induced of Whittaker type and such that its central character to n is invariant under k � and its L-function L(II, s) is absolutely convergent in some half-plane. Let .c be an automorpkic representation of GLm(A) which is either irreducible or induced of Whittaker type. We will say that L(II � % s) is nice/fL(II � v, s) and L(17 ~ � r s) have an analytic continuation to entire functions of s which are bounded in vertical strips and satisfy the functional equation L(n x., s) = ~_(n x -~, s) L(II ~ x r 1 -- s). A converse theorem for GL~ is a criterion in terms of the L(II x v, s) for deter- mining when II is actually an automorphic representation. Our first converse theorem, modeled on that of Jacquet and Langlands, is one of the end products of years of colla- boration of the second author with H. Jacquet and J. Shalika (for example [22-24]). In the function field case, this theorem was proven in the 1970's by the second author [38]. The same method of proof works in the number field case now that the local archimedean theory has been completed by Jacquet and Shalika [27]. Theorem 1. -- Let II be an irreducible admissible representation of GL~(A) whose central character ton is invariant under k � and whose L-function L(1-I, s) is absolutely convergent in some half-plane. Suppose that L(II � v, s) is nice for every cuspidal automorphic representation -e of GLm(&) for all m with 1 <~ m <~ n -- 1. Then II is a cuspidal automorphic representation of GL.(A). This theorem yields maximal information about rI, namely that it is actually cuspidal automorphic, but it requires nice behavior of the L-functions under twists J. w. COGDELL AND I. I. PIATETSKI-SHAPIRO by all cuspidal automorphic representations on all smaller GL,~'s. We will give variants of this theorem where we require the L-functions to be nice under a smaller set of twists. For this we must use the ideas of Well. The most typical converse theorem of this type is the following. Fix a finite set of places S ofk containing all archimedean places. For each integer m, let f~s(m) = { r: : 7r is an irreducible generic automorphic representation of GL~(A), unramified at all v r S }. Similarly, let f~(m) be the set of cuspidal elements of f~s(m). Theorem 3. -- Let n >1 3. Let II be an irreducible admissible representation of GL,(&) whose central character co n is invariant under k � and whose L-function L(II, s) is absolutely convergent in some half-plane. Let S be a non-empty finite set of places of k containing all archi- medean places such that the ring os of S-integers has class number one. Suppose that for every m with 1 <~ m <~ n -- I and every 9 E s the L-function L(II � v, s) is nice. Then there exists an irreducible automorphic representation II' Of GL,(&) such that 1-I'~ = II,for all non-archimedean places v where II, is unramified. This will be proved in Section 11. We will also give a version of this theorem where we put the extra hypothesis that II be generic. In this case we can draw slightly stronger conclusions. These are stated as Theorem 2 and its corollaries, which can be found in Section 7. We believe that it is not necessary to have control of so many twists to be able to draw conclusions about the automorphic nature of II. Twists by characters of GL x might be enough. We state this in the following conjecture. Conjecture. -- Let II = | YI, be an irreducible admissible representation of GL,(A) whose central character O)r~ is invariant under k � and whose L-function L(fl, s) is absolutely convergent in some half-plane. Assume that L(II | co, s) is nice for all characters co of k� � Then there exists an automorphic representation II' of GL,(&) which is quasi-isomorphic to II and such that L(II | ~o, s) ----- L(fl' | o~, s) and ~(H | co, s) = r | co, s). The validity of this conjecture would have very fundamental applications to the problem of Langlands lifting. This conjecture is known to be true for n = 2 [21] and n = 3 [22] and we actually have II = II'. The first example where II 4= II' was constructed in [38] for n = 4 and the construction provides examples for all n I> 4. 3. Outline of the proof of Theorem 1 Let us first outline the proof Theorem 1 under the more restrictive hypothesis that II = | II, is generic, i.e., each II~ is generic. Let us begin with an arbitrary ~ E VII. Our goal is to embed Vn in CONVERSE THEOREMS FOR GL n 167 ~r such that the actions of GL~(A) are intertwined. Since Vn is linearly spanned by decomposable vectors we may assume that ~ is decomposable, i.e., ~ = | ~, with ~ e Vnv. As a first step let us associate to ~ some function on GL~(A). This is where the assumption that II is generic comes into play. Each II, has a unique Whittaker model ~r ~b,) and to each ~, is associated a function W~(g,)er162 qb~). For almost all v, II, will be unramified and there is a distinguished unramified vector ~o with respect to which the restricted tensor product is taken. At these places we normalize the Whittaker model so that W~o(I,) = 1. Now to ~ e VII associate the global function W~(g) = 1-[, W~v(g,). Since for almost all v, ~, is the distinguished unramified vector ~0 in Vn~ and g~ ~ GL,(o,), this product converges absolutely to a continuous function on GL, (A). We first attempt to make an automorphic function from ~ by averaging as much as possible over GL,(k). First note that W~(g) is left invariant under both N,(k) and Z,~(k). To get further invariance, consider the sum U~(g) ----- ~1 W~(yg) = ~] W~ g . Y ~ ~,n(k)\Pn(lc) y' ~ Nn - I(/O\GLn_ l(k) ((0 ~ This sum converges absolutely and uniformly on compact subsets to a continuous function on GL,(A) which is cuspidal along the unipotent radical of any maximal parabolic subgroup of GL. containing B.. As a function on GL,(A), Us(g ) is left invariant with respect to P,(k) and Z,(k) and hence with respect to the full parabolic subgroup P'.(k) associated to the partition (n -- 1, 1) of n. We next construct a second function V~(g) associated to ~ which will be related to U~(g) via the functional equation of the L-function. Put - (i1 11) where (11) is the longest Weyl element of GL,. Then, if we consider W~(~, g), this is left invariant under ,1 /i* ~-1 N.(k) a. = 0 1 o oo ~t which we will denote by N'.(k). Note that N'.(k) C P.(k) where P. is the mirabolic opposite to Pn" 168 J. W. COGDELL AND I. I. PIATETSKI-SHAPIRO Then let us set Vg(g) = Z Wg(0% vg) = Z Wg ~. g . ( (; V ~(k)\Pn(k) T' ~ l(k) To deduce the properties of Vg from those above for Ug, let us recall that if we set "~/~(g) = W~(w. ,g-l), then "W~ is in the Whittaker model for H, the contragredient representation [24, 25, 27]. Then we have V~(g) = X We, g' 7' ~ Nn-t(k) \OI~-l(k) ((o ~ where ,=(o. 11), x(w. 1 l) and oi) . We may conclude that Vg(g) converges absolutely and uniformly on compact subsets to a continuous function an GL.(A). It is left invariant with respect to P'.(k) = ~P'.(k) -1. To any ~ we have attached two functions on GL.(A), one invariant under P'.(k) and the other under P'.(k). Note that together P'.(k) and P'.(k) generate all of GL.(k). Our strategy will be to use the global functional equation to prove that U~(g) = V~(g), which will show that this function is in fact invariant under GL.(k) and hence automorphic. To relate U~ and V~ to the L-function we consider the following integrals. If we restrict U~(g) or V~(g) to GL._a(A)C GL.(A) embedded in the standard way, then u~(h 0 ~)and v~(h 0 01)are rapidly decreasing automorphic forms on GL,_~(A). Let v be an irreducible proper automorphic representation of GLn_x(A ) and let ~ be an automorphic form in the space of v. Set I(~, 9; s) = U~ 1) ~?(h) ] det(h)["-(a/~' dh. ;0 (;0, L n _ l(k) \GLn_ I(A) The integral I(~, q~;s) converges absolutely for Re(s)>> O. If we unfold the series defining u~(h 0 01), we find ~) W~(h) det(h)I '-`1/2' dh i(~,~;s) = f~ w~(h ~ n_ I (A) \GLn_ I (A) = T(W~, W~; s) i"~'n-l(k)\GI.,~,~- CONVERSE THEOREMS FOR GL a 169 where n - 1 (k) \lqn - 1 (A) i.e., W,(h) eSF(% +-~). Similarly, for Vr we may define the integral I(~, ~; s) -- V~ ~) q~(h) I det(h)I '-am dh. L n _ irk) \GL._ I(A) This will converge for Re(s) ~ O. If we unfold it, we find ~r (h 0 O1) ~r l det(h) l'a-"-'a/~' dh = W(Wr W~;1- s), where, as before, we set W~(h) = W~(W._ a 'h--l), ~rr = Wr ,g-a) Both of these families will have an analytic continuation to entire functions of s, bounded in vertical strips. To see this we must relate these global integrals to the global L-function. Up to this point, nothing is used other than general properties of Whittaker func- tions. To prove the continuation of these integrals and relate them, we must use our assumptions on the L-functions. The integrals are related to the global L-functions through their expressions as Whittaker integrals. In fact, we have I(~, ~; s) -- 't'(W~, W~; s) = L(n � ~, s) Eft) ]'(r ~; s) = ~v(~, ~; 1 - s) = L(~ � % 1 -- s) E(s) where E(s) and E(s) are entire functions of s. The analytic continuation of the global L-functions then implies that I(~, ~; s) and I(~, c?; s) both have continuation to entire functions of s which are bounded in vertical strips. The global functional equation for L(H � % s) will allow us to relate I(~, ~; s) and I(~, ~; s) and hence Ur and Vr From the local functional equation we have ,v(W~o, w~o; ,) 'v(~'~o, ,~'~; 1 - s) L(n~ x ~, s) ~(n~ x ~o, s, +~) %(- 1) "-a = ~ L(Ho � %, 1 - s) Using this, the global functional equation will imply that upon taking products we have "F(W~, W~; s) = ~F(W~, W~; 1 -- s) or I(~, $; s) =: I(~, $; s) for all s. 22 j. w. COGDELL AND I. I. PIATETSKI-SHAPIRO Ifwe set Fr = U~- V~ then F~(h 0 ~) is rapidly decreasing on GL._ ~ (k)\GL,_ ~ (A). If we restrict to SLn_l(A) then F~(k 0 ~) will be in L~(SL._I(k)\SL~_I(A)) andif we interpret the above equality in terms of F~, we see that ;SLn_l(k)\ 8Ln_I(A F ~ (h 0 :)~(h)dh=O for all ~ occurring in irreducible automorphic subrepresentations of SL._I(A ). If we then apply the weak form of Langlands' spectral theory we may conclude that Since F~(h 0 ~)--O, we have that Ut o(h O1)=V~(: 01)for all k eSL,~_I(A ) and in particular U~(1) = V~(1). Since this is true for all ~, then U~(g) = Un,g, 4(1) ---- Vn(o)r = V~(g) for all g ~ GL.(A). We now have that Ur is invariant under P.(k), Pn(k), and Zn(k ). Since these generate GL.(k) we see that U~ ~d(GL.(k)\GLn(A)). Thus the map ~-~ U~(g) embeds II into M(GL.(k)\GLn(A)). Hence II is an automorphic subrepre- sentadon. To see that II is cuspidal, since U~(g) is given by the convergent " Fourier expansion " Y ~ Nn_ l(kl\GLn- l(/c) without constant term, we observe that for any parabolic Q., the constant term of U~ along O is 0. Hence U~ ~ ~r176 i.e., U~ is cuspidal and hence II is cuspidal. This is the conclusion of Theorem 1. 4. Preliminary considerations on Whittaker models Before we turn to the rigorous proof of Theorem 1, we would like to gather together some known results which we will need. We begin with the following local and global estimates for Whittaker functions. If v is a place of k then by a gauge on GL.(k.) is meant a function ~, which is left invariant under N.(k~), invariant on the right under Kn,. and which on A.(kv) has the form ~,(a) = l ala2 ... a,-11-' qb(al, a~, ..., an_i) , CONVERSE THEOREMS FOR GL n 171 where al a2 . . . a n ] a~ . .. a n a~ An(k~ a n t is real and non-negative, and q) is a Schwartz-Bruhat function on k~- ~ [22, 27]. A global gauge on GLn(A ) is defined analogously [22]. Then the standard estimates on Whittaker functions are the following. Lemma 4.1. -- a) Let II~ be a generic representation or an induced representation of Whittaker type of GLn(k,) with con, its central character. Let I cony(X)[ = [ x [~. Then for any W, ~ ~//'(II,, +,) there is a gauge 6, such that [ W,(go) l ~< ~,(g,) I det(g,)[a/n. b) Let II be a generic representation or an induced representation of Whittaker type OfGLn(A ) whose central character con is invariant under k x and whose L-function L(II, s) is absolutely convergent in some half-plane. Let [ con(X) [ = [ x [a. Then for any ~ ~ II there exist a global gauge ~ such that I Wg(g) [ ~< ~(g) [ det(g)[a/,. Proof. -- a) When v is a non-archimedean place of k and H~ is generic this is just Proposition 2.3.6 of [22]. As noted in Remark 2.2.5 of [22], the proof is still valid if II~ is induced of Whittaker type. For v an archimedean place this follows from Proposi- tion 2.1 of [27] and the comments following it. b) Let H' = H | co_d/, as in the proof of Lemma 2.3, so that H' has a unitary central character. The product L(H', s) also converges absolutely in some half-plane. Let T be a finite set of places ofk such that II~, and hence II',, is unramified outside T. As in the proof of Lemma 2.3, for v r T we have that II: is the unramified constituent of Ind~ | | ~,, ~). If L(II', s) converges absolutely for Re(s) > e 0 then we have seen that we have the uniform estimate [ ~q,,(~) [ ~ q~. Since II' has a unitary central character, we also have a uniform lower estimate of q~-tn-1)~ ~ [~h,,(t%)[. Hence there exists a uniform do, independent of v, such that q~-ao < ] ~,,,(t%)[ < q~ for all v r T. Now applying Proposition 2.4.1 of [22] we see that we can choose a compatible family of local gauges { ~, } for the II, as in a) such that ~ = 1V[ ~ is a global gauge and gives the estimate in b). [] 5. Prellmln~ry considerations on Langlands' spectral theory For the proof of Theorem 1 we will need the weak form of Langlands' spectral theory for SL n_ x. We recall here what we will need, specialized to SL._ x. For details, see [33], [14] or [36]. j. w. COGDELL AND I. I. PIATETSKI-SHAPIRO Ifwe set Fr = U~- V~ then F~(h 0 ~) is rapidly decreasing on GL._ ~ (k)\GL,_ ~ (A). If we restrict to SLn_l(A) then F~(k 0 ~) will be in L~(SL._I(k)\SL~_I(A)) andif we interpret the above equality in terms of F~, we see that ;SLn_l(k)\ 8Ln_I(A F ~ (h 0 :)~(h)dh=O for all ~ occurring in irreducible automorphic subrepresentations of SL._I(A ). If we then apply the weak form of Langlands' spectral theory we may conclude that Since F~(h 0 ~)--O, we have that Ut o(h O1)=V~(: 01)for all k eSL,~_I(A ) and in particular U~(1) = V~(1). Since this is true for all ~, then U~(g) = Un,g, 4(1) ---- Vn(o)r = V~(g) for all g ~ GL.(A). We now have that Ur is invariant under P.(k), Pn(k), and Zn(k ). Since these generate GL.(k) we see that U~ ~d(GL.(k)\GLn(A)). Thus the map ~-~ U~(g) embeds II into M(GL.(k)\GLn(A)). Hence II is an automorphic subrepre- sentadon. To see that II is cuspidal, since U~(g) is given by the convergent " Fourier expansion " Y ~ Nn_ l(kl\GLn- l(/c) without constant term, we observe that for any parabolic Q., the constant term of U~ along O is 0. Hence U~ ~ ~r176 i.e., U~ is cuspidal and hence II is cuspidal. This is the conclusion of Theorem 1. 4. Preliminary considerations on Whittaker models Before we turn to the rigorous proof of Theorem 1, we would like to gather together some known results which we will need. We begin with the following local and global estimates for Whittaker functions. If v is a place of k then by a gauge on GL.(k.) is meant a function ~, which is left invariant under N.(k~), invariant on the right under Kn,. and which on A.(kv) has the form ~,(a) = l ala2 ... a,-11-' qb(al, a~, ..., an_i) , CONVERSE THEOREMS FOR GL n 173 Now let ~ be an irreducible admissible cuspidal representation of M(A). Then we may form the induced representation Indg~s which we view as the space of functions ~ : SL,_I(A ) ~ C such that for all g ~ SL,_I(A), the function m r-~ ~(mg) is in ~ | 8~/2. Let I(~) denote the subspace of admissible vectors of this induced repre- sentation. So q~ ~ I(~) if it is smooth, K-finite, and satisfies the previous condition. If g e SL,_I(A), then g will have an Iwasawa decomposition g = umk relative to P, where u e U(A) the unipotent radical of P(A), m e M(A), and k ~ K = I-I, K~. Write m = re(g). This is not unique, but its image ,~(m(g)) in M(A)/MI(A) is Uniquely defined. If r ~ I(a) then the function ~z~:g ~ ~(g) t m(g) I ~ is in I(~ | X~). We are now ready to define the Eisenstein series we will use. If M is a Levi subgroup of a parabolic subgroup P of SL,_x, cra unitary cuspidal representation of M, q0 e I(a), ands~C'-I set E~(g; s) -- ~] ~(Yg) I m(yg)18 7 6 P(k)\ SL.- l(k) whenever this converges. The facts we will need about the Eisenstein series are contained in the following theorem. Theorem $1. --The series defining E~(g; s) converges absolutely and uniformly on compact subsets for all s in the positive cone X + ={seX MlRe(s,)-Re(s,+l)> 1} (set s, = 0). In this region, E~(g; s) is a holomorphic function of s and is of moderate growth on SL,_I(k)\SL,_I(A ). Moreover for s ~ X + E~(g; s) e ,~/(SL._I(k)\SL._I(A)). For generic s e X + , I(a| Zs) is irreducible and the map ~ ~-~ E,(g; s) defines an embedding of I(~| as an automorphic subrepresentation of ~r Besides the Eisenstein series we need another family of functions which seem to go by many names (incomplete theta series, pseudo Eisenstein series, etc.). Let us introduce them through the Paley-Wiener functions on X~. If k is a number field, so X~, __ C '-1, then P(XM), the space of Paley-Wiener functions on XM, is the space of holomorphic functions f: X~, -+ C which satisfy an estimate of the following type. For each f e P(XM) there exists a real number r and for each n ~ N there exists a constant C, such that lf(s)l c. + II s I1)-". 174 J. W. COGDELL AND I. I. PIATETSKI-SHAPIRO Ifk is a function field, then X M ~ C'-1/2hi Z,_ 1 and P(XM) is the set of functions /log q which are given by polynomials in q,1, ..-, q,,-x and their inverses. If we define the Fourier transform on functions on X M by 3~(m) = ~.~,o,f(s)[m I' ds, thenaf(m ) is a function on V~ = M(A)/MI(A) and the space of Paley-Wiener functions on X~ has the equivalent characterization by f~P(X~) if and only iff~C~(VM). For ~ ~I(~) andf~P(X~) we define O~./(g) = E r(yg)f(m(Tg)). 7 ~ P(~)\ 8Ln-l(k) Then first basic result on these functions is the following. Theorem $2. -- The sum O~,/(g) converges absolutely and uniformly on compact sets to a rapidly decreasing function on SL._x(k)\SL._I(A ). It has an expansion in terms of Eisenstein series by 0~'t(g) = :m,,, = m,*o, E~(g; s) f(s) Us for any s o ~ X + . To state what we have called the weak spectral theorem, let us recall the convention that for M = SL._ 1 itself, both the Eisenstein series E~(g; s) and the series 0~,t(g ) reduce to just the cusp forms ~ in the cuspidal representation a of SL._ ~(A). Then by weak spectral theory we mean the following result [36, Theorem II. 1.12]. Theorem $3. -- The collection of all functions of the form 0~, f(g) obtained as M runs over all Levi subgroups of SL._I, ~ all unitary cuspidal representations of M (A), ~ E I(~), and f ~ P(XM) are dense in L2(SL._a(k)\SL._I(A)). We will use this in the form of the following standard corollary. We repeat the proof for the convenience of the reader. Corollary. -- Let F(g) be a smooth function of rapid decay on SL._I(k)\SL._I(&). Suppose that :s~_~,~,\S~_x,A, F(g) E~(g; s) dg = 0 for all Einstein series E~(g; s) as M runs over all Levi subgroups ofSL._I, a all unitary cuspidal representations of M(A), ~ e I(a), and all s in a Zariski open subset of X + . Then F(g) - 0. Proof. --Since F(g) is smooth and of rapid decay it lies in L~(SL._ a(k)\SL._ I(A)), and hence by Theorem $3 it suffices to show that I(%f) = :SL._I~k,~L._I,A, F(g) 0~, :(g) dg = 0 CONVERSE THEOREMS FOR GL n 175 for all 0., 1(g) as in the statement of that theorem. If we replace 0., t(g) by its expansion in terms of Eisenstein series from Theorem $2, we have I(%f) = fsL,_x,k,\sI~_~,A)f~*,~,= ~,,o, F(g) E,(g; s)f(s) ds dg. Since F(g) is of rapid decay, ] E~(g; s) [ satisfies a moderate growth estimate depending only on Re(s), and f is Paley-Wiener, we may interchange the order of integration to obtain I(%f) = f~,.,=~,.,, (fsT.._,,k,XST.._,,., F(g) g.(g; s) dg) f(s) ds, By our assumption, fsN_l,k,\sN_x~*, F(g) E~(g; s) dg = 0 except possibly on a set of measure zero in the set Re(s) = Re(s0). Hence I@,f) = .Is,.._l,k,\s~._l,A,. F(g) 0~,1(g ) dg = 0 and we are done. [] 6. Proof of Theorem 1 Let H = | II~ be an irreducible, admissible, not necessarily generic representation of GL.(A) whose central character o H is invariant under k � and whose L-function L(H, s) is absolutely convergent in some half-plane. By the Langlands classification for GL.(k~) at each place v there is an admissible induced representation ~.. of Langlands type such that II~ is the unique irreducible quotient of E, [7]. The representation ~ is induced of Whittaker type and ~., = H, only if H~ is generic [26]. The induced representation E, has a well-defined central character ~os~ and this will be the central character of any constituent of E,. In par- ticular ~, and H, will have the same central character. The point of introducing the ~ is that for non-generic representations like II~ their local L-function is defined through the L-functions of the ~ where an integral representation via Whittaker models can be used [22]. More specifically, from the definition of the local L-functions [24, 27], for every irreducible admissible representation % of GL,(k,) we have L(II~ X %, s) = L(E, x %, s) with a similar equality for the ~-factors. Therefore if we consider the admissible repre- sentation E = | E~, this representation will have the same central character as H and its L-function will be nice for all twists by every cuspidal automorphic representation -r of GL~(A) for all m with 1 ~< m ~< n -- 1. It has the extra advantage that it is induced of Whittaker type. If H was generic to begin with, then H = E. 176 J. W. COGDELL AND I. I. PIATETSKI-SHAPIRO For the convenience of the reader, let us recall how the local L-functions L(~, � %, s) are defined through their Whittaker models. The representations E~ and % are both induced representations of Whittaker type and thus have Whittaker models. For each pair of smooth functions W~(g) e ~f'(~, d?,) and W',(g) e#'(-%, +;-1) there is associated an integral ~(W,,W',;s) =fN W~(0 0 )W',(g) 'det(g)]"-'"-~'/~dg which is absolutely convergent for Re(s) >~ 0 by the estimates in Section 4. Ifk~ is non-archimedean then by Theorem 2.7 of [24] we know the following. The integrals ~F(W~, W'~; s) define rational functions of q~-". As the functions W~ and W'~ run over their respective Whittaker spaces, this family of integrals form a C[q~", q~-"]-fractional ideal in C(q;-'). The local L-factor L(~, � %, s) is the generator of this ideal of the form L(~, � %, s) = P(q~-')-I with P(X) e C[X] a polynomial having P(0) = 1. Moreover, these integrals satisfy a local functional equation of the form 9 (W~, W~;' s) CF(p(w.,,~) W., W.,~'" 1 -- s) L(no x .~, s) r x .., s, +.) o~.o(- 1) ~ = L(H~ x ~%, 1 -- s) In this functional equation, the function UF(Wv, W'~; s) is defined by the integral N t ,r(wo, w,; s) W;(g) I det(g)] "-C~-~'/2 dx dg, m(kv) \GLmlkv) ' m(kv) 0 where k =n--m-- 1. (Note that if re=n-- 1 then CF(W~,W',;s) =tF(W,, W',;s).) The Whittaker functions involved are ~r =W,(w ~g -1) e CC(~,t~-l), and W'(g) = W'~(w,, ,g-l) a~(v~, ~,), where ~%~' is the representation of GL,(k,) on the same space as .~, but with action E',(g) =- E~(tg -1) and similarly for v',. The Weyl elements involved are 1 1), W r the longest Weyl element of GL r and (i ~ 0) As before, p denotes right translation in the Whittaker model. These integrals have the same analytic properties as the u~(W~, W'~; s). The s-factor is of the form CONVERSE THEOREMS FOR GL n 177 r � %, s, +~)= Aq; -z* tbr appropriate constants A and B. The local functional equation is also written as = W~; 1 --s) 'V(W~, W;;s) 7(Zo X "~, s, +o) ~('Wo, ~' where y(E~ X %, s, d?,) ---- t%(-- 1) '~ s(E, X %, s, +~) L(E; x "r~, 1 -- s) L(E~ X %, s) If the local field k~ is archimedean, then the integrals tF(W~, W~; s) extend to meromorphic functions of s. For the L-function L(~ � %,s) and the s-factor s(~, � %, s, +~) we may take the L-function and e-factor of the nm-dimensional repre- sentation of the local WeLl group associated to the pair (E~, %) by the archimedean local Langlands correspondence as in [5, 27, 29]. The ratio W(W~, W~; ' s)/L(~ "~ � ,., s) is again entire and satisfies the same functional equation as in the non-archimedean case. These results are all due to Jacquet and Shalika and the details can be found in [27]. To prove Theorem 1, let us begin with an arbitrary ~ s V=. Since V z is linearly spanned by decomposable vectors we may assume that ~ is decomposable, i,e., ~ -= | ~, with ~ e Vz. Each E~ has a unique Whittaker model W'(E~, d~) and to each ~ is associated a function W~(g~) eqC/'(E,, +~). Now to ~ e V z associate the global function W~(g) = II, W~,(g~). Since ~. is the distinguished unramified vector ~0 in Vz, for almost all v and g, e GL,(o,) for almost all v, this product converges absolutely to a continuous function on GL,(A). The function W~(g) is left invariant under both N.(k) and Z,(k). Consider the sum U~(g) = ]~ W~(vg ) = ~ W~ g . ((; 7 E Nn(k)\Pn(k) V' ~ Nn-l(k)\OLn-t(k) From the global gauge estimate of Lemma 4.1 we may estimate U~(g) and find the following. Lemma 6.1. -- The sum Ur converges absolutely and uniformly on compact subsets to a continuous function on GL,(A). Moreover it is cuspidal along the unipotent radical of any maximal parabolic subgroup of GL, containing B,. If k is a number field, f~ a compact subset of GL,(A) and c > 0 there exists t o suck that if t >t t o then there is a constant c' with property that I W~(ao)] .<. c' II l ada,+t I -"+''"-x-*' I det(a)l a/" i=l for o~ e f~ and a (al a) satisfying [ a,/a,+ 1 I>1 c for 1<<. i <<. n- 2, where d is suck tkat l co=(a) I = l a I a. Proof. -- This is just Propositions 12.2 and 12,3 of [22]. 1:3 23 178 J. w. COGDELL AND I. I. PIATETSKI-SHAPIRO Lemma6.2.--Lel]tEGL._l(A) andcon$idert]lefunctionU~(h 0 0l).TkozU~(h 0 ~) is a ra?idly decreasing automorphic function on GL._z(& ) and furthermore it satisfies the estimate IUr 0 ~) I ~< c, [ det(k) [ -' for suffdently large t > O. m That Ur (h 0 ) is automorphic follows from the formula Proof. "Y' E Nn_ l(kl\GT'n- l(k) First, assume that k is a number field. Then, by reduction theory for GL._I(A), we may write h = yaco where to eft, a compact subset of GL~_I(A), y ~GL._I(k ) and a= with [a~[a~+ zl >c for some c and i= 1,2,...,n--2. a,_ 1 Then from Lemma 6.1 we have (setting a, = 1) the estimate g~ <~ c~ II [ ai/ai+ 1 [-ti+it,-1-il i det(a)f." 4=1 Since the ratios aJa~+ z for 1 <~ i ~< n -- 2 are the simple roots of GL~_a, this shows that Ur (h 0 01) is rapidly decreasing on GI,,-z(A). n--1 On the other hand, since Idet(a) l = II l ada,+l 1' we see that i=1 II [ aJa, +1 1--t, = ] det(a) I-' i=1 [II aJa,+l I '"-1-' c I det(a)I '"-1' 4=1 and therefore ]U~(hO 10)<~c;[det(a)[ -'+''-1'+'~"' ~< c;, I det(a)1-" for t' > 0. Since k = yao~ with [ det(y) [ = 1 and [ det(~) [ bounded for ~ e f~, this gives the estimate when k is a number field. 179 CONVERSE THEOREMS FOR GI n Now assume that k is a function field having a finite field of q elements as its field of constants. It is easy to see from the transformation property defining the Whittaker function that there is a sequence of constants c ----- { r } with c~ = 0 for almost all v such that ff W~(g) 4= 0 with g =: nak, where n e N.(A), k e K = II GL~(oJ, and then [aJ.+l] <..qCvfor l<..i~n--l. Taldnga,~-~l weseethatWg(h 0 ~)vanishes identically for [det(k)[ sufficiently large and so the same will be true for U~(h 0 ~). This establishes the estimate for [ det(h) I large. On a set { h e GL._.I(A) [ I det(h) I = q" } of matrices with fixed determinant, the function U~ (h 0 ~} is compactly supported mod GL._I(k), and hence is rapidly decreasing as an automorphic form on GL._I(& ). To see this, recall that by the reduction theory for GL._I(A ) [16] there exists a set of constants X = { X, } with Xo = 0 for almost all v and a compact subset ~ C N._I(& ) such that, if we set ~(X, a) = { k = nake GL._I(A) [ a = diag(at, ..., a._~) e A._t(A), neff, kel-IGL._l(o.) with for l~<i~<n--2}, then GL._t(/k ) = GL._x(k) ~(~., f~). Hence it suffices to prove that U~ (h 0 ~) has compact support in ~,(~,, f~) = { h e ~(~,, f~) [ I det(h) l = q' }. On such a set if suffices to prove that l al l is bounded if Ur (h 0 :)4:0. IfU~(k 0 01)*0 then there must be a y e GL._I(k ) such that W~ 4= 0. Write y = (y~, 5) as a (,: 01) matrix. First we assume that Y.-1,1 4= 0. Write yh = ynak. It is easy to see that the (n--1, 1) entry of "r~ is y._l, la~. Write yna =bk' with k' eK and b EB._I(& ). Then we have 1Y.-1,1 al I~ ~< [ b._1,._1 [. ~< q~v for all places v. Hence laxl = I Y.- ,laa I = IIl < IIq~ where [ c[ = ]~ c~. The general case proceeds in the same way using the first y~,t such that %,1 4= 0 and u = 0 for all k> ~, giving [ a~ I ~< qlel where co = c~ . 180 J. w. COGDELL AND I. I. PIATETSKI-SHAPIRO Finally, the polynomial estimate for small determinant is now a consequence of the reduction theory and the gauge estimate of Lemma 4.1 as in the number field case. [] Before we proceed, let us note the following. Lemma O. 3. -- The function U~(g) /s not identically O. Proof. -- If we compute the +-Fourier coefficient of U~(g) we find IN U~(ng) +-l(n) dn = I~ ~ W~(yng) +-l(n) dn n(k)\ ~Tn(A) ntk)\ NntA) = 5] W~ ng +-l(n) dn. (k) \ Nn(A ) We now proceed by induction. Let N" C N, be the unipotent radical of P~, i.e., N" = In-1 . 0 ... 0 Then since N" is normal in N. we may integrate over it first, and the formula for an individual term in the b-Fourier coefficient becomes fN"(k)\ N"(A) Now, GL,_I normalizes N" so +((0 But +-l(n") an" = o v' r P._l(k). 1 Y' e P,~_x(k) Hence this term vanishes unless y' c P._l(k). We now proceed by induction in this way and finally conclude f U~(ng) +-l(n) dn = W~(g). Since W~(g) is not identically 0, because ~ ~ W~(g) is injective, this shows that U~(g) cannot be identically 0. [] fNn_l(k)\Nn_l(A) Nn-l(k)\GLn-l(k) Nn(k)kPn(~) CONVERSE THEOREMS FOR. GL n As a function on GL.(A), U~(g) is left invariant with respect to P.(k) and Z.(k) and hence with respect to the full parabolic subgroup P'.(k) associated to the partition (n-- 1, 1) ofn. Consider now a second function V~(g) associated to ~. Let us set ((; :)) Y E.-" N~(k)\ Pn(k) y' ~ lff n_ l(k)\ OL n _ x(k) where the notation is as in Section 3. To deduce the analytic properties of V~ from those above for Ur let us recall that if we set W~(g) = W~(w. tg-1) then ,~r is in the Whittaker model for ~'. Then we have where 1(w._11) ~'=~ ~. -101) and We may conclude that V~(g) converges absolutely and uniformly on compact subsets to a continuous function an GL,(A). It is left invariant with respect to P~(k) = tP',(k)-1. The function V~(g) does not vanish identically. Furthermore, if we consider the function v~(h 0 ~) on GL.- l(&) it is a rapidly decreasing automorphic functi~ ~ GL,- l(&)- The only difference is that our determinant estimate becomes for t>O. These facts follow from Lemma 6.2. From ~ we have produced two functions on GL.(A), one invariant under P'~(k) and the other under P',(k). Note that together P'~(k) and P~(k) generate all of GL,(k). To relate U~ and V~ to the L-function consider the following integrals. Let x be an irreducible proper automorphic subrepresentation of GL,_I(A ) and let 9 ~ Vs. Set I~n - 1 (k)\ GL n _ 1(A) 182 J.W. COGDELL AND I. I. PIATETSKI-SHAPIRO As a function on GL~_I(&), ~0(h) is of moderate growth and transforms via a central character o~. On the other hand, U~ (h 0 01) is rapidly decreasing on GL,_ I(k)\GL~_ ~(A) and, in terms of the determinant, satisfies lUg(: ~)l,<c, ldet(h)l-'forevery t>O. Hence I(~, ~; s) converges absolutely for Re(s) >> O. On the other hand, if we unfold the series defining U~ (h 0 ~) we find I(~, q~; s)= Io Ut (: 01)~o(h):1 det(h)I'-a/2'dh IJn_ l(k) \O]',n_ l(A) 1,A detlh ,s ,1,2 = ~(W~, W~; s), where WAh) = / ~(nh) +(n) an n- l{k)\ Nn- 1{ A) i.e., W~(h) eYCz(% qb-1). Hence we have: Lemma 6.4. -- For any % irreducible automorphic subrepresentation of GL~_I(A), and o eV, the integral I(~,~0;s) converges for Re(s)>>0. Moreover, in this range, I(~, ~; s) = ~F(W~, W~; s). Hence 1(4, ~; s) = 0 if\ is not generic. Similarly, for V~ we may define the integral ]~(~. q~; s) = fo 'v~(h 001)~(k)[det(h)'8-a'2'dh Ln_l(k}\GLn_ 1( } which will converge for Re(s) ~ 0. If we unfold this, then we find T(~, ~; s)= f~._.,A,\o._~,. ~r~ (; Ol)'W~(h)'det(h)[n-"-n'~'dh CONVERSE THEOREMS FOR GL n where, as before, we set W~(g) = W~(w, ,g-X), ~,(h) = W,(w._, 'h-'). Hence we have proven: Lemma 6.5. -- For v any irreducible automorpkic representation of GL,_~(A) the integral I(~, ~; s) converges for Re(s) < 0. Moreover, in this range, I(~, ~?; s) = W(~V~, ~; 1 -- s). Hence I(~, ~; s) = 0 if .~ is not generic. Both of these families will have an analytic continuation to entire functions of s, bounded in vertical strips. To see this we must relate these global integrals to the global L-function. We will work with I(~, ~; s) in detail, then I(~, q~; s) proceeds in the same way. Proposition 6.1. -- The integral I(~, ~ ; s) has an analytic continuation to an entire function ofs. Proof. -- We will consider two cases, although this is not really necessary. First, assume that 9 is cuspidal. We take I(~, q~; s) = ~F(W~, W~; s). Assume that and q~ are decomposable. (This is possible since the decomposable vectors span V= and V,.) Then we have I(L ~; s) = II ~'(w~, w~o; s). Now, from the local theory of L-functions ~F(W%, W~o; s) = E~(s) L(E~ x ~, s) is an entire function of s. If v is non-archimedean, E,(s) eC[q~, q~-'] and if both ~ and % are the distinguished unramified vectors, which is true for almost all v, E.(s) -- 1 [24]. If v is archimedean, then E~(s) is an entire function of s [27]. Hence, setting E(s) = l-i v E,(s), we find I(L ~; s) = 11 L(~ � ~o, s) E,(s) = L(Z � ~, s) E(s). So if v is cuspidal, then by our hypothesis on L( ~ � v, s) we have that L(E � v, s) is an entire function. The same holds true of E(s) and hence for 1(4, ~; s). Now suppose v is not necessarily cuspidal. Since I(~, q~; s) - 0 unless 9 is generic, we may assume that v is an irreducible generic automorphic subrepresentadon of GL,_ I(A). Then by the work of Langlands [34] there exists a partition (q, ..., r~) of n -- 1 and irreducible cuspidal representations % of GL,i(A) such that x is a subrepresentation of T = I,,~OL,-lCAIt_ C~ | %), where Q is the standard parabolic associated to this partition. The theorem as stated in [34] only gives ~ as a subquotient of T. But if one begins with v an automorphic subrepresentation, the proof presents v as a subrepresen- 184 J. W. COGDELL AND I. I. PIATETSKI-SHAPIRO tation of the induced Y. Let us sketch Langlands' proof, referring to [34] for more details. Langlands begins with the realization of the automorphic representation ~ on V/U where V is a space of automorphic forms generated by a single form q~ and U is a subspace of V. Since we are assuming that T is an irreducible subrepresentation of the space of automorphic forms, we may take V irreducible and U ---- { 0 }. Langlands then realizes in the space of constant terms q0p ~ Vp of the forms q0 ~ V along a suitably chosen para- bolic P with Levi M. Since our V is irreducible, we get a realization of'r on a subspace Vp of these constant terms and Up : { 0 }. Again, using the fact that Up : { 0 ) for this realization of r the argument of Lemma 6 in [34] produces a generator ~r of Vp such that r ---- x(a) 9v(g) with ~( a character of the center of M(A) for all g e G(A) and a in the center of M(A). Langlands then projects each ~0 v e Vp to a function ~v in the space of constant terms transforming by an irreducible cuspidal representation ~ of M(A) having the central character ~(. Since Vp is still irreducible as a G(A) representation, this mapping is an injection and realizes Vp as a subspace V~ of these functions, i.e., a subspace of the induced representation from this cuspidal representation ~ of M(A) to G(A). Taking G = GL,_ 1 and P ---- Q we obtain the conclusion stated above. Now, locally for each place v of k, % will be a generic irreducible subrepresentation of Y, ~--- .XtUQ(kv )]'-'~G:Sn-x(~v)t-kul, ~ ~ ... | Since each ~r~ is cuspidal and hence generic, the local components ~, ~ must also be generic. Then the results of Rodier [40] and Jacquet [19] imply that each T~ is of Whittaker type, that is, has a one-dimensional space of Whittaker functionals. Hence it has at least one generic constituent. If k, is non-archimedean, the results of Bernstein and Zelevinsky [4] imply that there is a unique generic constituent and so T, must be it. If k. is archimedean, then using the Casselman subrepresentation theorem for each %, and the transitivity of induction we can embed Y, into a representation Y'~ which is induced off the Borel subgroup. Now the results of Kostant [31] imply that T'~ has a unique generic constituent. Since Y, is a subrepre- sentation of Y~ we have that Y~ can have at most one generic constituent. Since we already have seen that it has one generic constituent, namely "r,,, we have that % must be the unique generic constituent of Y~ in the archimedean case as well. % cannot lie in the kernel of the map from Y~ to its Whittaker model, since if it did this would imply that Y~ would have at least two generic constituents. Hence, the Whittaker model of % will be a subspace of the Whittaker model of Y,. In particular, the family of integrals defining L(E~ � %, s) will be a subspace of those defining L(E~ � T~, s). At those places where % is unramified, these families agree. Hence from the computation of the local L-functions in [24] we see that at all non-archimedean places we have and if E, and % are unramified then CONVERSE THEOREMS FOR GL n 185 Hence for v non-archimedean we have L(~o � 7,, ~) = ~I L(ao � ~,,., s) E:(~), where E'~(s) is endre, bounded in strips, and identically one for almost all v. If o is archi- medean then from [27] we have L(z~ � 7. ~) = fl L(=. � ~,,,~, s). Hence, globally we have ,m L(~ x 7, ~) = E'(~) 11 L(~ x ~,, ~). By our hypothesis, each L(E � ,~, s) is entire and bounded in strips since each ,~ is cuspidal. The same is true of E'(s) = I-i v E~(s). Hence it is true of L(E X 7, s). (We will use the boundedness in strips in the proof of Proposition 6.3.) If we now write I(~, ~; s) = L(~. � ~, s) F.(s) as above we see that I(~, q~;s) is entire as desired. [] Proposition 6.2. -- The integral I(~, ~ ; s) has an analytic continuation to an entire function ofs. Proof. ~ In this case we write Y(~, ~; s) = "I'(~r ~; 1 s). By the local theory of L-functions, we may relate this integral to the global L-function L(Et x 7~; 1 -- s) and proceed as before. [] We next relate these integrals, again using the properties of the global L-function --this time the functional equation. th/s Proposition 6.3. -- As entire functions of s, I(~, ,; s) = I(~, ~; s). Moreover, function is bounded in vertical strips. Proof.-- If we write these integrals as Euler products I(~, ~; s) := 11 ,v(w~o, w~0; s) I'(~, ~?; s) ---- ~ ~F(l~rcv , l~r; 1 -- s) J. W. COGDELL AND I. I. PIATETSKI-SHAPIRO then the local factors are related by the local functional equation [24, 27] v(W~o, w~v; s) V(~'~o, X~',po; 1 -- s) r x %, s, +,3 co.~,,(-- 1)"-: = L(~ x .~,l-s) L(E, X %, s) or 9 (W~v, W,v; s) y(g, x %, s, ~,) = ~(X~r~, X~r,,; 1 -- s). Therefore, taking the product over v, we have 9 I'(W~, w~; s) v(~ x % s) = 'I'(~, "~; 1 - s). Since L(E x % s) is assumed to satisfy the global functional equation, T(g X % s) --- 1 if-~ is cuspidal and hence I(~, $; s) = I(~, 9; s) in the cuspidal case. If'~ is not cuspidal, then we may assume it is generic (since otherwise both sides are identically zero) and hence, as in the proof of Proposition 6.1, is a subrepresentation of an induced repre- sentation where each o, is cuspidal and generic on GL,I(A ). Now, from the local theory, i=1 and hence globally ~(~ x % ~) = ~I ~(-: x ~,, ~). i=1 Now, by assumption, since each ~, is cuspidal the global y(g x ~, s) = 1. Hence in the case of non-cuspidal -~ we still have I(~, ,; s) = I'(~, 9; s). We need to show that this function is bounded in vertical strips. Note that from the integral representations, I(~, 9; s) is bounded in vertical strips in its half-plane of absolute convergence Re(s) >> 0 and T(~, 9; s) is bounded in strips in its half-plane of absolute convergence Re(s) < 0. To verify that it is bounded in any vertical strip we just need to see that it grows sufficiently slowly that the Phr~igmen-Lindel6ff principle applies. From the proof of Proposition 6.1, we have I(~., ,p;s) = L(~ x %s) IIS~(s). t~ The factor L(E � % s) is bounded in any vertical strip as in the proof of Proposition 6.1. The factor E~(s) is identically 1 for almost all places. At the remaining non-archimedean places E~(s) belongs to C[q*, q-'] and is thus bounded in any vertical strip. If v is archi- medean, then E,(s) = 'I'(W~o, W~o; s) L(~ x "~, s) CONVERSE THEOREMS FOR GL n From the local archimedean theory [27] the numerator decreases like 1 over a polynomial in s at infinity in vertical strips while the denominator is a linear exponential factor times a product of P-functions. Then Stirling's formula applied to this product of U-functions gives a bound on I L(E, � v,, s)[-1 of the form Ce a~ at infinity in any vertical strip, where we have written s = a 4- it as usual. Hence Phr~tgmen-Lindel/Sff applies to I(~, ~; s) and we may conclude that it is indeed bounded in any vertical strip. [] This concludes our use of the L-function. We now maneuver ourselves into a position where we can apply the weak form of Langlands' spectral theory for auto- morphic representations. For each idele a let us set and similarly for I1(~ , ~; a). The integrals I1(~ , @; a) and I1(~ , @; a) are continuous functions on kX\A x. Note that if we replace x by x | co for a (unitary) character co then I1(~, ~.co; a) = co(a) I1(~, ~; a) and similarly for Ix(~, ~; a). Hence we may write I(~,~.co;s) =f Ii(~,~;a) co(a)la[~-(1/2'd� for Re(s) >>0 Jk X\A� I(~,q~.co;s) =9 Ii(~,@;a ) co(a)la['-(lmd � for Re(s) <0. J~ x\A� We may now apply the following elementary lemma of Jacquet-Langlands [21, Lemma 11.3.1]. Lemma. -- Let fl and f~ be two continuous functions on kX\A x . Assume there is a constant c so that for all (unitary) characters co of k� X the integral fk fx(a) co (a) I a l" dX a x\ A� is absolutely convergent for Re(s) > c and the integral co(a) I a I' dX a is absolutely convergent for Re(s) < -- c. Assume that the functions represented by these integrals can be analytically continued to the same entire function and that this entire function is bounded in strips. Then fl and f2 are equal. 188 J. w. COGDELL AND I. I. PIATETSKI-SHAPIRO Therefore we may conclude that I1(~, ~; a) = I'x(~, ~; a) for all a e A � and all ~. In particular, for a = 1, we have U~ (h 0 ~)v(h)dh= Vr (h 0 01)~(h)dh. Note that since U~ and V~ are rapidly decreasing on GL._~(k)\GL._~(A) they are also on SL._I(k)\SL._I(A ) and hence Ur162 ~L*(SL._I(k)\SL._I(A)). Let F~(g) = U~(g)-- V~(g). Then F~(h 0 ~)israpidlydecreasing and (h 0 ~) ~(h) dh ~- 0 for all ~ occurring in irreducible automorphlc subrepresentations of GL~_I(A). Proposition6.4.--Wehave F~(h 0 01) =0. Proof. -- We wish to apply the weak form of Langlands spectral theory for SL,_ 1 as formulated in Section 5. Before doing so we must relate automorphic representations of SL,_I(A ) to the restrictions of automorphic representations of GL,_I(A ). By a result of Labesse-Schwermer [32] (see also Lemme 5.6 of Clozel [11]) given any irreducible cuspidal representation -r 1 of SL,_I(A ) there exists an irreducible unitary cuspidal representation -r of GL,_a(A) whose restriction to SL,_I(A) contains -q. The argument in [32] extends to yield that any irreducible cuspidal representation -r 1 of M 1 = S(GL,I(A ) � ... � GL,,(A)) extends in this way to an irreducible unitary cuspidal representation v of M = GL,I(A) � ... x GL.,(A). If we apply this fact in the construction of Eisenstein series, we find that for any partition (nl,..., n,) of n- 1 and any irreducible cuspidal representation a� of the Levi subgroup M1 = S(GL,I(A) � 9 .. � GL.~(A)) the space of Eisenstein series E,(h; s) for ~ c I(~i) and s ~ X+ is obtained by the restriction of Eisenstein series on GL._I(A ) formed with the extention a of ai to M = GL.I(A ) � ... � GL,,(A). In the realm of absolute convergence for these GL._ I(A) Eisenstein series, the induced representations are irreducible for all parameters in a Zariski open subset, and hence for these values of the parameter the Eisenstein series generate irreducible automorphic subrepresen- tations. Hence for all s in a Zariski dense subset of X+ the Eisenstein series E,(h; s) are obtained from restriction of irreducible automorphic subrepresentations of GL,_I(i ). As a consequence, we see that fSLn_x(k)\SLn_liA)F~ fs~_~(~)\sr~._~(A~ Is~_~(~\s~._lCA) CONVERSE THEOREMS FOR GL n for all Eisenstein series E,(h; s) as M 1 runs over all Levi subgroups of SL,_ 1, al all unitary cuspidal representations of MI(A), ~ e I(al) , and all s in a Zariski open subset ofX+~. Hence by the Corollary to Theorem S3, F~(h0 ~)-0. D Since F~(h 0 ~)--0, we have that u~(h 0 ~)= v~(h 0 ~)for all heSL,_~(A) and in particular U~(1) = "V~(1). Since this is true for all ~, U~(g) = Umo,~(1 ) ----- Vr~,g,~(1) = V~(g) for all g e GL,(A). We now have that U~(g) is invariant under P,(k), P,(k), and Z,(k). Since these generate GL,(k) we see that U~ e ~r Thus the map ~-*U~(g) embeds "z into d(GL,(k)\GL,(A)). Hence E is an automorphic sub- representation. In fact the map ~ ~-~ U~(g) embeds E in the space of cusp forms. To see this, we must show that for any parabolic Q, with unlpotent radical NQ, the constant term of U~ along Q is 0, i.e., f~ U~(ng) dn = O. q(k)\ Nq(A) Since U~(g) is left invariant under GL~(k) and all k-rational Borel subgroups of GL. are conjugate under GL.(k) it suffices to compute the constant term along the unipotent radicals of standard parabolic subgroups O.~D B., so that N-Q C N.. If Q' is a maximal parabolic subgroup such that Q' D Q.D B., then N a, is a normal subgroup of NQ and in computing the constant term along Na we can integrate along NQ, first. Hence to show that U~(g) is cuspidal it suffices to show that it is cuspidal along the unipotent radical of any standard maximal parabolic subgroup. But this is guaranteed by Lemma 6.1. Hence U~ e ~r176 i.e., U~ is cuspidal for every ~ and hence ~ is cuspidal. As a constituent of ~., H will then be cuspidal automorphic as well. However, we can say a little more. Since H is cuspidal, it is generic. Thus each local component II, is generic. But as we have pointed out, when II, is generic, II, = ~, ~. Hence 1I ~ [] 7. A second converse theorem Theorem 1 is a generalization of results of Jacquet and Langlands for GL(2). It gives the most information about H, namely that it is not only automorphic but also cuspidal. However Theorem 1 requires information about L(II � v, s) which is usually not available. More precisely, in Theorem 1 we assume that L(H � % s) is entire for twists by all euspidal automorphic forms on all GL~(A) with m < n. It is very difficult to obtain such information. Andr6 Weft, even before Jacquet-Langlands, suggested a 190 J. w. GOGDELL AND I. I. PIATETSKI-SHAPIRO different method of proving this type of theorem, which will allow us to obtain a result suitable for applications. In the method of Weil the first step is the construction of some periodic holomorphic function which is supposed to be an automorphic form. From given information about the functional equations satisfied by the associated Dirichlet series and their twists, Well derived the conclusion that this function was an automorphic form with respect to some congruence subgroup. In the following Theorems 2 and 3 we will follow the method of Well disguised in the language ofautomorphic representations. For each finite set of places S of k containing all archimedean places and for each integer m, let ~s(m) = { n : zc is an irreducible generic automorphic representation of GL,,(A), unramified at all v r S }. Similarly, let ~~ be the set of cuspidal elements of f~s(m). Theorem 2. -- Let n >1 3. Let II be an irreducible admissible generic representation of GL,(A) whose central character o~ n is invariant under k x and whose L-function L(I-I, s) is absolutely convergent in some half-plane. Fix a non-empty finite set of places S of k containing all archimedean places such that the ring o s of S-integers of k has class number one. Suppose that for every m with 1 <<, m <~ n -- 1 and every ,r E f~~ ) the L-function L(II � v, s) is nice. Then there exists an irreducible automorphic representation II' of GL,(A) such that II, ~_ II'~ for all v ~ S and for all non-archimedean v such that I-I, is unramified. In order to prove Theorem 2, we will first use the framework of Theorem 1 to construct an embedding ofII s in the space of smooth functions on rs\G s for a congruence subgroup r s of G s with respect to an appropriate Hecke algebra. Let us recall that according to the general Duality Theorem [ 13], it is known that " classical " automorphic forms with respect to a group r are in duality with embeddings of given irreducible representations of GLz(R) into the space L2(I'\GL2(R)). In the case n I> 3 there is a simplification compared with Well's theory, which in fact says that the set of assumptions (i.e. necessary twists) does not depend on the conductor of the representation II. The reason for this simplification is that the congruence subgroup theorem is true for SL, for n/> 3. There are two extensions of this which follow after some extra arguments. Currently they are separate statements, but we hope that they will eventually coalesce. Corollary 1. -- With the hypotheses of Theorem 2, there exists a proper automorphic repre- sentation 17" with II'~' ~_ l-I, for all non-archimedean v for which II, is unramified. The next Corollary is the one which is most useful for the application to Langlands' lifting. Corollary 2. -- With the hypotheses of Theorem 2, there is a unique irreducible generic auto- morphic representation II" such that II'~' ~ II,for all v ~ S and all non-archimedean v for which I-I, is unramified. GONVERSE THEOREMS FOR GL n 191 8. The conductor of a representation Let II = @ H, be an irreducible admissible generic representation of GL,(A). Let S be a finite set of places ofk containing all archimedean places. For almost all places ~ S, the representation II~ is unrarnified, that is, II~ contains a vector which is fixed by the maximal compact subgroup K~ = GL,(o~). This vector is unique up to scalar multiples. Let T denote the smallest finite set of places containing S such that II~ is unramitied for v r T and let T' = T\S. So T' is the set of places not in S for which II, is ramified. For those places v ~ T', it is known from [23] that there is a unique integer m, > 0 such that if we set Kl,,(p~) := g e GL.(o,) : g =- (mod p~) 0 ... 0 then the dimension of the space of Kl,~(p~o)-fixed vectors in YI~ is one. Set m~=0 for veT. We will call the compact subring n= II p~'~CA s the S-conductor of II. If S is precisely the set of archimedean places, hence is empty in the function field case, we will call rt the conductor of II. It determines (and is determined by) an ideal of o s by n s = k c3 k s rt C o s . To simplify notation we will denote rt s simply by u, since they can be distinguished by context. Note that Ds[u -~ 1-I~s odP~ ~. If we set Kx(u) = g e,~s GL.(o~) : g = (mod u) 9 0 ... 0 = II Kl,,(pr 0) c G S, then the dimension of the space of K1(u)-fixed vectors in II s is exactly one. We may similarly define (mod p~) K0,,(p~) = g ~ GL.(o~) : g - 0 ... 0 1 E 192 j. W. COGDELL AND I. I. PIATETSKI-SHAPIRO and (mod n) Ko(n ) = ge II GLn(o.):g--- ,$8 0 ... 0 = II Ko, o(p~ ) C G s. The group K~, o(p~ .) will then be a normal subgroup of K0, o(p~ ~) with abelian quotient given by Ko, o(p~)/Ki, o(p~ .) _ (oo/p~) � and Ki(rt ) is a normal subgroup of Ko(rt ) with quotient I-[**s(oJp~o) � ~ (osflt)� Then the action of Ko(rt ) will preserve the one-dimensional space of Kl(rt)-fixed vectors and act on it by a character of K0(rt ) trivial on Ki(~t). It is easy to compute the action of K0(n ) on the space of Ki(rt)-fixed vectors. Let C ~ be a non-trivial Kl, o(p~,)-fixed vector in H o for v ~ S. Then the tensor product C ~ = (~ C ~ is a non-trivial Ki(n)-vector in II s. If v r T then Kx, o(p~ ) = K0,,(p~o) = GLn(o,) and so for go e Ko, o(p~ o) we have Ho(go) ~o ___ ~o. If v e T' and g. = (g~, s) e Ko,.(p~) then from the congruence condition we have [ gn. j [~ < 1 for 1 ~< j < n. Since go e GL.(oo) we must have max{[g.,~[o}= 1 and hence [g.,nlo= 1 and so gn, ne0~ x. Then we may write go = (g,, n In) g~, o with g~, ~ e Kx, o(p~). Then iio(g~, ) ~o = iio(g,, ' n In) ~o = O~,,(gn. n) C ~ where % is the central character of H,. So we may define a character Z = @ X, of K0(rt ) by xo(g~) = 1 if v ~ T and xo(go) = c~ n) if v e T'. This is guaranteed to be a character by construction. If we wish to emphasize the dependence on the central character co of II we will write X = Xo,. We have IIS(g) ~0 = X,o(g) ~o for g e K0(rt ). There is another useful construction of X,~. Consider the central character co of H. If v r T then for any local unit u, ~ o~ x we have u, In ~ GL~(oo) = Kl.o(P~V) and so c%(uo) C ~ = IIo(u o I,) C ~ = ~o so that %(u,) = 1. Similarly, ff v e T' and u o is a local unit of the form 1 + p~v then o~,(uo) = 1. So co o is unramified at v r T and has conductor at least p~v at the places v e T'. Since (Os/rt) � _ IIo(oo/p~ v) x, the character o~ defines a character Xo of (Os/n) � via this isomorphism by X~ ----Iioes %. Then, through the isomorphism K0(rt)/Ki(n ) -(os/rt) � this character X,o defines a character of K0(rt) trivial on Kx(rt ) which is easily seen to be the same character as defined above. Hence we could write zo(g) ---- xo(g,, ,) for g = (&, j) ~ K0(n). CONVERSE THEOREMS FOR GL n 193 9. Generation of congruence subgroups Let n >/ 3. Let S denote a non-empty finite set of places of k containing all archi- medean places, Let os denote the S-integers of k. Since OL.(os) = GL.(k) n Gs HsGL.(o,) , we may view GL,(os) as a subgroup of GL,(k) embedded in G s. Then GL,(os) is a discrete subgroup of G S. For the proof of Theorem 2 we will need a preliminary result on the generation of certain congruence subgroups of GL~(os). The heart of this proof is Lemma 9.1 which is extracted from the proof of Theorem 4.2 of Bass [1]. This result from the stable algebra of GL, plays a role in the solution of the congruence subgroup problem for SL, [2]. This is the place where the restriction n >/ 3 comes from, as in the congruence subgroup theorem. Let T' be a finite set of places disjoint from S and let T = S w T'. For each v e T' let m~ be a positive integer and for v ~ T set m~ = 0. Let n = I-I~ $ s P~~ C k s. As in Section 8, n defines an ideal, again denoted 1t, in a S. The congruence subgroups of GL,(os) we are interested in are (mod 1I) Pl(rO = 7eGL.(os):7- 0 I and P0(n ) ---- y~GL.(os):7 = 9 (mod 11) I" /o 0 If we define Ki(n ) C G s for i = 0, 1 as in Section 8 then we can also characterize F,(rt) by P~(n) = GL~(k) c3 Gs.K~.(n ). Consider the following subgroups of F,(rt). Set P;(os) = P'.(k)n P'.(k)n G.K S, where as usual we have set K S = 1I,$ S K,. This is the set of all matrices in GL~(os) whose last row is of the form (0, ..., 0, ,) ifi= 0 or (0, ..., 0, 1) ifi = 1. It is inde- pendent of 11. Set P~(rt) = P'~(k) c3 Gs.K,(rt ). This is the subgroup of F,(n) consisting of those matrices whose last column is t(0, ..., 0, ,). There is a congruence condition on the last row of these matrices. 25 194 J.W. COGDELL AND I. I. PIATETSKI-SHAPIRO Proposition 9.1. -- The groups P'(os) and P'(rt) together generate the congruence sub- group P,(rt) fir i = 0, 1. --t For now, let A,(n) denote the subgroup of GLs(os) generated by P~(os) and P~ (n). Note that Al(n ) C A0(rt ). Lemma 9.1. -- Let (a 1, ...,as)~o~ be a unimodular sequence suck that (al, ..., a,)- (0,..., 0, d)modrt. Then there exists an element 7 e Aa(tt) such that (hi, ..., as) "~ = (0, ..., 0, d). Proof. -- The sequence (al, ..., a,) is unimodular in the sense that there exist cl, 9 9 c. in os such that 1 = ~c~ a,. Therefore al = ]~a 1 c, a t = a 1 cl a~ q- 1~=.~ a~ ci a~. If we substitute this expression for a 1 into 1 = Y~c, a~, and let q = a 1 c~, we find 1 =c lqa l-k- ~ c~(q+ 1) a~. S=2 Since a 1 e 1t, we have al ca = q ~ 1t, and we see that the sequence (qal, as, ..., as) is again unimodular. Since 0s is a Dedeldnd domain, n = 2 defines a stable range for D s in the sense of [1]. (Note that there is a shift of one in the definition of stable range between [1] and [2].) This implies that there exist a'~ = a~ § b~ qal with b~ s Ds such t ! that the sequence (a2, ..., a,) is unimodular. Let t ! Then (a,, ..., as) v a = (al, as, ..., as). Note that we still have t t (al, a=, ..., a,) - (0, ..., 0, d) (mod n). r t p t Since (a~,...,a,) is unimodular, we may write 1 =Z"~=2qa~. Write a' s=d+q. t t t with qs e 1t. Then we have q, -- ale 1t and by the unimodularity of (a~, ..., a's) we may write this element as q', -- al = Y~,"=2 di a" with d i e ft. Now let T~ = 9 kd, 0 ... 1 ' ~,) ,~ (q',, a~,.., Then v 2 e P[( n) since d s elt. So (al, a~, .. ., = ' . a's) . Now set (11) CONVERSE THEOREMS FOR GLn 195 t i e i l so that (q,, a~, ..., a,) = =-- (q',, a=, ..., a,_l, d). Note that we still have t t , a t q,,a~, . .~ ,-1 err. So if we set 7 8 ~--- 9 . 9 -- ~._~ - - r then % ~P;(u) and (q',, ~, ..., a',_l, 1) % = (0, ..., O, d). Therefore (al, ..., a,) 71 7 2 ~7 s = (0, ..., 0, d). Since 71, ~ ~ Pl(os) and 72, % e P~(u) we see that 71 7~ aTs E AI(U ). [] Proof of the proposition.- Since P;(os), P;(u) C F~(u), it is clear that A,(u) C P,(u). Now let y e F~(n), so y - (mod u) 0 ... 0 with d = 1 if i = 1. Let u = det(y). This is a unit in Os and the diagonal matrix diag(u, 1, ..., 1) is in P~(Os). Then diag(u -1, 1, ..., 1) Y has determinant 1. Hence its last row is unimodular in the sense of the lemma and we still have y - (mod u). 1 0 ... 0 Now, by our lemma, there exists Y1 ~ AI(u) such that YYI = = P" 1 0 ... 0 But then p e P~(Os). Hence Y = diag(u, 1, ..., 1) py~ -1 e P~(Os) Al(u) C A,(u). [] 196 J. W. COGDELL AND I. I. PIATETSKI-SHAPIRO 10. Proofs of Theorem 9, and its corollaries Let u denote the S-conductor of H and co = to n its central character. Let t ~ = @~s t ~ be a non-zero Kl(rt)-fixed vector in II s as in Section 8. So t ~ transforms by the character Z~ of Ko(rt ) as in Section 8. Proof of Theorem 2. -- For each ~s ~IIs consider the functions U~| ) and V~| to(g) associated to the vector ~ = ~s | ~0 ~ II. The function Ut~| is left invariant under P,(k) and V~| to is left invariant under P.(k). Both are invariant under Z,(k). In addition, U~| and V~| are right invariant under Kl(n ) for all ~s ~ IIs. Now, if we restrict these functions to GL,_I(A ) we find: Lemma 10.1. -- In addition to the properties from Lemma 6.1 and Lemma 6.2, the t x # (: )are a, ,e .re #naion k- invariant under KS ,-1 = II,~s GL,-I(~ Proof. -- For v such that II, is unramified this is clear since t ~ is fixed by GL,(o,) D GL,_x(o,). At the remaining places t ~ is fixed by K~,~(p~,)DGL,_~(o,). [] Now consider the integrals I(~ s | t ~ q~; s) and ]'(~s | t ~ q~; s) for q~ lying in a proper automorphic representation v of GL,_I(A ) as defined in Section 6. Since U~| to (h 0 ~) is unramified, we find that I(~s | ~~ ~; s) = f~ Ut~| to (h 0 ~)*?(h)ldet(h)] *-nmdh L._ l(k)\OL. _ llA) = fl%_ gn)\Gi~_ltn)Wts @ t~ (h 0 ~)W~ (h)[det(h)''-(lmdh K,_ a. The same is true for Hence I(~ s | ~o, q~; s) - 0 unless v contains a vector fixed by s ~(~s | ~ Hence, if vr 1), then I(~ s| ~ =0=I(~s |176 On the other hand if-c e~~ 1), then by our assumption on the L-function we have as in the proof of Theorem 1 that I(~ s | t ~ c?; s) -- I(~s | t ~ c?; s). If v ~ fls(n -- 1) but is non-cuspidal, then v must still be generic for the integrals to be non-zero and as before we have that v is a subrepresentation of a representation induced from cuspidals. Since 9 is unramified for v 6 S and generic, these cuspidal representations must also be unramified for v $ S and generic. Then arguing as before, we find that, in this case as well, I(~ s | t ~ q~; s) ----- "I(~s | t ~ c?; s). Hence we have the following result. CONVERSE THEOREMS FOR GL n 197 Proposition 10.1. -- For all proper automorphic representations -r of GL,_I(A ) we have I(~s | ~o, r s) = I(~s | ~o, ~; s) for all ~s e 1-I s . From here, applying the weak form of Langlands' spectral theory as in the proof of Theorem 1, we find Proposition 10.9.. -- The equality U~| ) = V~s| holds for all ~s ells. Since ~s was arbitrary in I/s and ~ transforms by the character X~ of K0(u ) as in Section 8, we find that in fact U~| to(g) = Vt~ | to(g) for all g e G s . K0(u ) C GL,(A) and all ~s e H s . However, since we have fixed the vector ~0 at places v r S, we cannot conclude equality for all g e GL,(A). For this reason we are unable to embed II as a subrepresentation of d(GL,(k)\GL,(A)). We will only be able to embed 1-I s as a subrepresentation of a space of classical modular forms on G s trans- forming by the Nebentypus character X~ ~ of F0(rt ). To simplify notation, let us introduce the functions (I)~(gs) = Ut~| IS)) = V~| lS)), where 1 s = YI~ s 1~ e G s and (gs, lS) e G s G s : GL,(A). This associates to each ~s e H s a function on G s. Let Po(os) and Po(rt) be the discrete subgroups of G s defined in Section 9. These are both subgroups of Fo(rt ). Since Fl(rt ) is a normal subgroup of F0(rt ) with abelian quotient (Os/rt) � the central character ~ of II induces a character Z~ of F0(n ) through the character Z,~ of (os/rt) � defined in Section 8. Lemma 10.2. -- The function dp~ is left invariant under Pl(os) and PI('It) and transforms by the character X~ 1 under Po(os) and Po(rt). Proof. --This is the standard argument. Write an element g ~ GL,(A) as g = (gs, gS) with gs ~ Gs, gS ~ G s. Then for y e Po(as) we have @~(Ygs) = U~|176 IS)) 9 Since U~s| is left invariant under P',(k) this is dP~s(ygs) = U~| 7-1)) = U~| to((gs, IS)) 9 But now y -1 e K0(rt ). Since U~s| transforms by Zo under K0(rt ) we have r = X~I(Y)@~(gs). The argument for P'o(rt) is the same, but using V~| Since Xo is trivial on the subgroups P'~(Os) and P~(n) we obtain the invariance of ~ under these groups. [] 198 J.W. COGDELL AND I. I. PIATETSKI-SHAPIRO By Proposition 9.1, the groups P~ (Os) and P'(rt) generate the congruence subgroup F~(rt) fi GL,(os) for i = 0, 1. Hence we may conclude that for every ~s e 1-I s the function r is left invariant under Fl(rt ) and transforms by the character X= 1 under P0(rt). Let d(r0(n)\Gs; cos, Z= -~) be the set of automorphic forms 9 on G s in the sense of [6] which also satisfy (1) r = X='(V) O(gs) for T e Fo(rt) (2) r gs) = r r for z s e Z.(ks) N k~, where r s is the central character of 1-I s. The character X= x is referred to as the Neben- typus character [18]. We then have the following. Proposition 10.3. -- The map ~s ~ (I)r embeds H s as an irreducible subrepresentation of d(ro(n)\Os; o,s, From Section 1 of the appendix, we know that d(Fo(n)\Gs; cos, X= 1) is naturally isomorphic to the space d(GL,(k)\GL,(/k); co)x,~m of Kl(rt)-invariant functions in the space of automorphic forms transforming by the character co under the center. To relate irreducible subrepresentations of ~r cos, Z~ x) to automorphic repre- sentations of GL.(A) occurring in ~r co) we need to know that the representation consists of Hecke eigenforms for an appropriate Hecke algebra. In the appendix we explain this relationship and the Hecke algebras involved when the S-class number is equal to one. We refer the reader to the appendix for the notation to be used. Let T be the smallest finite set of places containing S and such that II~ is unramified at all v r T. Let T' ---- T~S. So T' consists of those places dividing the S-conductor ft. Then rIT is an irreducible unrami_fied representation of G T and hence corresponds to a character A of the Hecke algebra ~(G T, K T) of compactly supported KT-bi-invariant functions on G T. Since ~0 is the unique KI(n ) D KZ-fixed vector in [I s we see that for an g g(G K 1-i~(~) ~o __ A(O) ~o. There is a natural Hecke algebra, which we will denote by ~',(rt), acting on the space d(I~o(rt)\Gs; cos, X~I) 9 To describe ~c(rt), let M = GL,(k) c~ (II~e ~, K0,.(p~)) G T. This M consists of those rational matrices y ~ GL~(k) such that for all v e T' the v-compo- nent y~ lies in K0,~(p~ ). Then Fx(rt) C M. Let ~,~,(rt) depote the C-span of the double cosets Fdn)\M/Pl(n ). The algebra ~,(n) is related to the following adelic Hecke algebra. Let GS(rt) ----- (II~ T, K0,~(p~)) G T. Then GS(rt) D Kx(rt ) and we may from the associated Hecke algebra ~(GS(rt), Kx(rt)) of compactly supported Kl(n)-bi-invarlant func- tions on GS(rt). From the appendix we know that this algebra is isomorphic to GONVERSE THEOREMS FOR GL n 199 C[(os/rt) � | ~, K T) and so contains o~~ ~, K T) as a subalgebra. Then there is a natural isomorphism ~ :o*'c(tt ) -+og~ which takes the double coset Fx(n)tI'l(n ) to the normalized characteristic function Ot of the double coset Kl(n ) tKa(rt ). The algebra structure on o~~ is the pull back of that of ~(tt) via ~. In particular, 3$~ has a subalgebra o~ corresponding to o~ff(G T, K ~) via ~. If Fx(rt ) tI'l(rt ) e d~'o(n) then the associated Hecke operator $'~ acting on d(I'0(rt)\Gs; cos, X= a) is defined as follows. For f~ ~r ~0s, X~ ~) and Pl(rt ) tPl(rt ) = II aj Pl(tt ) the action is (g',f) (gs) = ~. f(a; 1 gs)" The algebra ~(rt) acts on ~r o) x~la) by convolution (0 * 9) (g) = forum ~"b(h) eo(gh) ah for 9 eg~~ and ~? e d(GL,(k)\GL,(&) ; c0) xa~m. These facts can be found in Section 3 of the appendix. Proposition 10.4. -- For each ~s ~ IIs the function cb~ is a Hecke eigenform for #f'~ with eigencharacter A, i.e., 8" t q)~ = A(O,) O~ for each I'l(rt) tI'l(rt) e ~o~. Proof. ~ Let Fl(rt) tI'l(tt ) be a double coset in ~,o~ and $'~ the associated Hecke operator. Write Pl(u ) tPl(rt) = H a~ El(u) with a~ = p~ g~ e P~(k) P0(rt ). This choice of coset representatives is possible by Lemma A.2 of the appendix. Then since Or transforms by the Nebentypus character Z= ~ we have = Z Xo(Y~) U~|162 1)). Since U~| is left invariant under P'.(k) and t ~ transforms by Xo under Ko(rt ) we have X,~(V~) U~| ~o((p[ ~ gs, 1)) = Zo(Vj) U~. ~o((gs, P~)) = U~| ~0((gs, lS)). Thus (~', qSCs) (gs) = U~s| r lS)). As noted above, ~o is an eigenfunction for oY'(G T, K T) with eigencharacter A. Thus (g', Or.) (gs) --= A(O,) U~| IS)) t~a = A(O,) *v.(gs). [] J. W. COGDELL AND I. I. PIATETSKI-SHAPIRO We now have that l-I s is an irreducible subrepresentafion of d(F 0 (u)\Gs; ~s, ~(~ 1) which consists of Hecke eigenvectors for the Hecke algebra 3r with eigencharacter A. We may then apply Theorem A of the appendix to conclude that there exists an irre- ducible automorphic representation 1-I' of GL,(A) such that ri~ _~ rI s (since I-I s is irreducible) and such that II '~ is the unique irreducible representation of G ~ with eigencharacter A for ~#(G T, K~). Thus II '~ ___ II ~ and II' satisfies the conclusions of Theorem 2. [] Proof of Corollary 1. -- We begin with the representation II' from Theorem 2. II' is an automorphic representation with the desired properties, but it may be only a subquotient of the space of automorphic forms. The fact that our original representation II is generic will allow us to pass from II' to a proper automorphie representation, that is, a subrepresentation of the space of automorphic forms. Since our original representa- tion II was generic, then II' is quasi-generic in the sense that II' = ~) II~ is irreducible and for almost all v, II~ is generic. Hence to complete the corollary as stated, it is enough to prove the following result. Proposition 10.5. -- Let II' be an irreducible automorphic quasi-generic representation of GL,(A). Then there exists an irreducible proper automorphic representation II" which is quasi- isomorphic to II'. Moreover, II', N II',' for all non-archimedean places v where II'~ is both generic and unramified. To prove this we will use the following well-known fact. Lemma 10.3. -- Let II, be an irreducible admissible unramified generic representation 0fGL,(kv) over a non-archimedean local field k~. Then there exist unramified characters Xl, ~, 9 9 X,. of GLI(k~) = k~ such that T-,~aL,(ko)t . @ 9 | X,,~). I~v ~ ~lXUBn(k v) kLl, v " 9 Proof. -- By the theory of spherical functions [8] we know that there are unramified characters 11,,, 9 9 Z,,, of GLI(k,) such that II, is the unique unramified constituent of Indg~'(Zl,~| ... | Without loss of generality we may write each Z,,v(x) = Ix ]~" with the u i E C and assume Re(u1)/>... /> Re(u,). Following Jac- quet [20], if we group the characters into families with Re(u~) equal and induce these up to the appropriate GL 8 we get a sequence of quasi-tempered representations vl, ~, ..., vr,,. Since these induced representations v~,~ are irreducible [19] we may use induction in stages to get i.,Aa~,(ko)t~, | | Z,,,) ---- Ind~"ck~)(va, | | v,,,) XXUBn(kv) kill, v " " " , " " " for an appropriate parabolic Q_~. Then this induced representation is actually an induced representation of Langlands type. As Jaequet observed in [20] II~ is in fact the Langlands CONVERSE THEOREMS FOR GL n 201 quotient of this representation. Since H, is generic we know by Jacquet and Shalika [26] that this induced representation of Langlands type must actually be irreducible and hence H~ -= Ind~Ln(ko)('rl ~ | | % ~) = T,,,tGI~%)f,, | | X,, ~)" [] , " " " , ~'~'-tBn(k v) kA,1, v " " " Proof of the Proposition. -- Since II' is automorphic, then by Langlands [34] there exists a partition (rl, ..., r,,) of n and irreducible cuspidal representations a, of GLri(A ) - ~ -~Gr,.(A)~ | | tr,~), where Q, is the standard such that II' is a subquotient of .~ = lnuQ(A) tal ... parabolic subgroup associated to the partition. Let v be a non-archimedean place where II', is both generic and unramified. By Lemma 10.3, there exist unramified characters ZI,,, ..., Z,,, of GL~(k,) such that Ilvt = ~"~Bn(k v | " 9 @Z,,,)" On the other hand, H', is a generic unramified ~ ~ox~(k~/_ | | ~,~ ~). By [3, Lemma 2.24] and Rodier [40], constituent of ~ = lnua,~ )tox, ~ . .. , each %~ must be generic and unramified. By Lemma 10.3, each ~,,, must be fully induced from unramified characters of Bri(k,) and, by transitivity of unitary induction, there are unramified characters ~q,,, ..., ~t,,, of GL,(k,) such that E~ = Ind~)(~x,~ | | ~,, ~). Since II; and E~ are both fully induced off the Borel and have a common constituent, namely II~, by [4] they have the same Jordan-HSlder constituents. But II', is irreducible. Hence, so is ~.~ and E. =-- II',. Since ~ is now irre- ducible at almost all places and has a finite composition series at the remaining finite number of places, we see that the global representation ~ will have a finite composition series and each composition factor will be admissible. Using the theory of Einstein series, at least one constituent of ~ embeds into the space ,~(GL,(k)\GL,(A)) as a proper automorphic representation. In fact, Lemma 7 of Langiands [34] gives a non-zero intertwining of a subrepresentation of E to the space of automorphic forms. Taking any irreducible submodule of the image gives a consti- tuent of E embedded as a proper automorphic representation. Let II" be this component. At all v where E~ is irreducible, we must have II~' ----- E~. In particular II~' = H~ at all non-archimedean v where II" is generic and unramified. [] This completes the proof of Corollary 1. [] Proof of Corollary 2. --Take II' from the conclusion of Theorem 2. As in the proof of Proposition 10.5 we have that there exists a partition (rl, ..., r,,) of n and irreducible cuspidal representations ~i of GL,i(A) such that II' is a subquotient of -- T~s Q Q t~m). By [25] the components II', __ 1-I~ for non-archimedean v where II, is unramified completely determine the partition and the a~, so this data is completely determined by II. Furthermore, as in the proof of Proposition 10.5, at the places where II~ is unra- miffed E, is irreducible and II, __ II" _~ E,. Set II~' = E, at these places. Now consider any other non-archimedean place v. Since the a~ are cuspidal, v)TnrlGLn(kv)[~t\/,.1, 202 j. w. COGDELL AND I. I. PIATETSKI-SHAPIRO they are genetic and the same is true of their local components. Hence at any finite place, .~ has a unique generic constituent. Let II" be this constituent. At those places v e S, let II~'= II~ "~ I/~. This is a genetic constituent of ~,. Let 1-I" = | II~'. Then 17" is the unique generic constituent of ~ subject to lq~' = 1-I, for v e S. By Langlands' result [34] II" is automorphic. This is the desired representation. [3 11. A third converse theorem In the next version of the converse theorem we relax the condition that/I be genetic. The cost is that we can no longer guarantee that the automorphic representation II' we produce agrees with II at the places v ~ S. We now repeat the statement, already given in Section 2, of the precise result: Theorem 8. -- Let n >>. 3. Let 17 be an irreducible admissible representation of GL,(A) whose central character co n is invariant under k � and whose L-function L(II, s) is absolutely convergent in some half-plane. Let S be a non-empty finite set of places of k, containing all archi- me&an places, such that the S-class number of k is one. Suppose that for every m with 1 <~ m <~ n -- 1 and every v ~ ~~ ) the L-function L(I-I � % s) is nice. Then there exists an irreducible auto- morphic representation II' of GL,(A) such that II', ~_ II, for all non-arehimedean places v where II~ is unramified. This is only a mild modification of Theorem 2. Proof. -- For each v let E~ be the representation of Langlands type having II, as its unique irreducible quotient. Each ~ is of the form E,, = Ind,~'k"'(p,,,, I I',~ | | Pmo,, I[ "m''') where Q,, is a standard parabolic subgroup associated to a partition (rl,,, ..., r~.,) of n, p,,, is an irreducible tempered representation of GLuey(k,) and the u~., are real numbers satisfying ul,, > ... > u~,,. As we noted in the proof of Theorem 1, each E, has the same central character as II, and each E, is an induced of Whittaker type and hence injects into its Whittaker model. By the local theory of L-functions for non-genetic representations [24, 27] we have by definition L(n. x .~, s) = L(S. x .., s) ~(n,, x ,~,, s, +,,) = ~(~,, x ~,~, s, +,,) for all irreducible admissible % of GL,~(k~) with 1 ~< m ~< n -- 1. Now if we form the representation E -----| ~, then ~ is a global induced repre- sentation of GL~(A) of Whittaker type having an automorphic central character and such that the L-function L( .~. � "5 s) is nice for every -~ ~ ~(m) with 1 ~< m ~< n -- 1. CONVERSE THEOREMS FOR GL n 203 To proceed as in Theorem 2 and embed =s into a space of classical automorphic forms we need to choose a standard vector in each F,, for all v r S. For each v r S for which II, is unramified, E, must also be and it must have a unique K, = GL,(o,)-fixed vector t ~ which projects to the distinguished K~-fixed vector of II,. Since l-I, is the unique irreducible quotient, Eo must be cyclic and generated by ~o. Since t ~ is the unique K,-fixed vector in E~, it must transform by a character A, under the local Hecke algebra o~~ K,) of compactly supported K,-bi-invariant functions on GL,(k,). Since the quotient map E, ~ II, is intertwining, the image of t ~ in II, will also transform by this character and II, is the unique irreducible unramified representation of GL,(k~) associated to this character. For the places v not in S where II, is not unramified, if we let ~g'(E,, +) be the Whittaker model of E, then by Jacquet and Shalika [26] the restriction of the functions in ~r d?,) to the mirabolic P,,, contains all smooth functions on P,,, which are left quasi-invariant under N,, ~, i.e., the space of Ind~,,~(~b,). Choose a function W', which is fixed by K, n P,,,. The corresponding function W [ in~/.~(E,, +~) will have a stabilizer containing Kx(p~ ~ for some m, >/ 0. We take the corresponding vector ~0 as our standard vector at this place. If we let ~o = @~s ~, then t ~ e =s and t~ fixed by KI(U ) where u = II,e ~, p~~ The argument of Section 8 still gives that t ~ transforms by the character X~, under Ko(rt ) even though =. is not irreducible since E has central character ~ ---- ~n. We now proceed as in Theorem 2. For each ts ~ '='s we form the functions U~| ~o(g) and Vv~| ~o(g). From the methods of Theorem 1 and 2, U~s| lS)) = V~s| lS)) = q)~(gs) for gs ~ Gs and the map ts ~"* ~s(gs) embeds E s into d(l"0(rt)\Gs; o~s, )(gl). Since E s has II s as its unique irreducible quotient, if we take a vector ~s ~ Es which has a non- zero projection to II s then ts must be a cyclic generator for E s. Hence the image of Es in d(P0(rt)\Gs; COs, )~1) is cyclic with a generator f0. As noted before, for all places v r T (as before T is the smallest set of places contai- ning S outside of which l-I,, is unramified) t ~ is a Heeke eigenvector. Hence our standard vector ~0 is an eigenvector for o~g'(G T, K ~) with eigencharacter A = @,~TA,. Then Proposition 10.4 shows that for every ~s ~ =s the function ~ is a Hecke eigenfunction for o~ with eigencharacter A. We now have that "='s is a cyclic subrepresentation of d(P0(n)\Gs; COs, Zg 1) which consists of Hecke eigenvectors for the Hecke algebra oct~ with eigencharacter A. Applying Theorem A of the appendix, we conclude that there exists an irreducible automorphic representation II' of GL,(A) such that II~ is a constituent of E s and II '~ is the unique representation of G ~ with eigencharacter A. But as we have seen above, IF is also the unique representation of G T with eigencharacter A. Therefore II" _ II, for all non-archimedean v where II, is unramified. [] J. W. COGDELL AND I. I. PIATETSKI-SHAPIRO 12. Our final converse theorem Theorems 2 and 3 have the drawback that the automorphic representation II' associated to 11 need not be cuspidal nor unique. However it is possible to associate to II a unique collection of cuspidal representations ~, on general linear groups GL,I(A ) with (rl, ..., r~) a partition of n. Theorem 4. -- Let II be an irreducible admissible representation of GL,(A) satisfying the hypotheses of Theorem 3. Then there exists a partition (q, ..., r,~) of n and irreducible cuspidal representations a~ 0fGL,i(A) such that for all non-archimedean places where H, is unramified we have that a~,, is unramified and L(II,, s) = Il~ L(a~,,, s). Moreover, the sequence (~, ..., a,,) is unique up to permutation. Proof. -- By either Theorem 2 or Theorem 3 we have associated to 11 an automorphic representation 11' such that II,_ 11', for all non-archimedean places v where II, is unramified. By Langlands [34] there is a partition (rz, ..., rm) of n and irreducible cuspidal representations a~ of GL,~(A) such that 11' is a constituent of ~- = ,naQ(A ~- | ... | am) where O is the standard parabolic associated to the parti- tion. By Jacquet and Shalika [25] this sequence of cuspidal representations is unique up to ordering. Moreover, the sequence is uniquely determined by the unramiffed constituent of ~, at those places where ~, is unramiffed. But this unramified constituent is exactly II,. Hence the sequence (az, ..., am) is uniquely determined by 11. In computing the local L-factors, at the places where a representation is unramified, the local L-function can be computed from the unramified vector using Hecke theory. At those places where .~., is unramified, each a,., must also be and ~, has a unique unra- miffed vector which in turn projects to the unramified vector in its unramified quotient, namely II" Thus when 11" is unramiffed L(H~, s) -- L(~,, s) -= II L(a,,,, s). But when 11, is unramified, 11, ~_ II~. Thus : L(II~, s) = IlL(a,,,, s) as desired. ! t t t Next, suppose that (rl, ...,r,) is another partition of n and ai, ..., a, are cuspidal automorphic forms on the GL,~(A) such that for all v where 1I, is unramified we have L(FI,, s) = IIL(,~,,, s) = IlL(a,,,, s). Then let -- TnAOL"r ~ | a~) ~.~t = .... Q'(A) \~ ~ " " " ~OT,,(AI,~al CONVERSE THEOREMS FOR GL n 205 where Q' is the standard parabolic subgroup associated to the partition (r~, ..., rs At the places v where II, is unramified we then have L(.'~.,, s) = L(~.;, s). For GL,, the L-function of an unramified representation % completely determines the Satake parameter t., e GL,(C) of % since L(II~, s) = det(I, -- t~ q;-,)-l. Thus we see that for these places E~ and E~ must have the same unramified constituent. Hence again Jacquet and Shalika [25] let us conclude that n = m, r~ = r~, and (h -~ (h after reordering. [] APPENDIX We retain the notation and conventions of Sections 7-10. In particular, k is a global field, S is a non-empty finite set of places of k containing all archimedean places, and Os is the ring of S-integers of k. Let o~ be a character of A � which is trivial on k � Let d(GL.(k)\GL.(A); co) denote the space of automorphic forms on GL.(A) which transform under the center Z.(A) by the character o~, i.e., f(zg) = co(z)f(g) for g c GL.(A) and z c Z~(A). The purpose of this appendix is to explain the connection between a space of classical auto- morphic forms with Nebentypus d(r0(rt)\Gs; o~s, X~ 1) and the subspace of the adelic automorphic forms d(GL,,(k)\GL.(A); o~) which are fixed by Kl(rt ). The automorphic forms in d(Fo(n)\Gs; COs, =1) are analogous to the functions in ~qC(F0(N)\SL,(R); Z) which are obtained by lifting classical modular forms on the upper half-plane ~ with respect to Po(N ) and Nebentypus character Z to functions on the group SL2(R). For this reason, we will refer to the functions in ~qC(F0(rt)kGs; COs, ~(~x) as " classical " automorphic forms. The functions in ~r ; co) we will refer to as " adelic" automorphic forms. In the case of class number one fields, and forms without Neben- typus, this is explained in [6]. For the convenience of the reader, we recall the extensions to the S-arithmetic case, still assuming the S-class number is one. 1. Relation between automorphic forms Assume that the S-class number of k is one. One consequence of this is that A � = k � k~ u s. As a consequence of strong approximation for SL. [30] and the fact that det(Kl(rt)) = u s we have that GL.(A) may be decomposed as (A.1) GL.(A) = GL.(h) G s K~(rt) as in [6]. Since Kl(rt ) C K0(n ) we also have (A.2) GL.(A) ----- GL.(k) G s Ko(rt ). From the decomposition (A. 1) we have d(GL.(k)\GL.(A))K~("' ___ d(r,(n)\G,) (A.3) where the isomorphism associates to each Kl(n)-invariant automorphic formfon GL.(A) the classical form fc given by f0(gs) =f((gs, lS)) 9 CONVERSE THEOREMS FOR GL n 207 For our purposes we need to keep track of the central character. Let us suppose that ~(GL,(k)\GL,(A) ; co) xl~"~ is non-empty. WHte n = H,~ s p~ with m~/> 0 and m~ =0 for almost all v. Let T'={vlm, 4=0}and T=S~3 T'. Sore,=0 for vr Let f(g) be a non-zero function in this space. If v r T then for any local unit u~ e o~ we have u~ I N e GL,(o,)----Kl.,(p~) which is naturally embedded in Kx(rt ) and so %(u,)f(g) -----f(u, I,g) =f(g) so that %(u,) = 1. Similarly, if v ~T' and u, is a local unit of the form 1 + p~", then %(u,) = 1. So % is unramified at v ~ T and has conductor at least p~"o at the places v e T'. Since (Os/rt) � _ II,es(OJp~)� co defines a character X~ of (Os/n) � via this isomorphism by X~----II,~s c%. The central character co allows us to define a character Z~ ---- IIX, of K0(rt ) as in Section 8. The construction there was not dependent on the space of Kl(rt)-fixed vectors being one, just on the existence of K~(rt)-fixed vectors and the central character. Since the second construction of this character in Section 8 is through the character Z,o of (Os/rt) � we see that Z,~ also defines a character of Fo(rt ) through the quotient map ro(n)/r~(n) ~ (os/n) � as in Section 10. Now let ~r c%, ~1) be the space of classical automorphic forms f~ on G s satisfying (I) f~(Ygs) = X~'(Y)f~(gs) for y e ro(rt ) C O s (2) f~(Zs gs) = r for z s e Z,(ks) ___ k~. Then from the decomposition (A. 2) we have (A. 4) ~r ; o~) ~''' _~ d(ro(n)\G s; '~s, z~') where the isomorphism associates to every Kl(rt ) invariant automorphic form f on GL,(A) the classical form f, on G s given by f,(gs)=f((gs, IS)) and to a classical form f, on G s with Nebentypus character Z~ 1 the adelic form given by f(Ygsko) =f~(gs)X,o(k0) where y e GL,,(A) and k 0 e K0(n ) as in the decomposition in (A.2). 2. Comparison of Heeke algebras Both the spaces in (A.3) and (A.4) have natural Hecke algebras which act on them. We will describe these algebras and compare their actions. If G is any locally compact totally disconnected topological group and K is an open compact subgroup of G we will let .~(G, K) denote the space of K-bi-invariant compactly supported functions on G. This space is an algebra under convolution: the Hecke algebra of G with respect to K. The space d(GL,(k)\GL,(A)) xll"~ is most naturally a module for the Hecke algebra 9ff(G s, Kl(rt)) = (~,$s ~~ KI,,(P~V)) acting by fight convolution. Since Kl(rt ) is not the maximal compact subgroup of G s, the algebra 9ff(G s, Kl(n)) is not 208 J. w. COGDELL AND I. I. PIATETSKI-SHAPIRO necessarily commutative. However ~(G T, K T) = @~ ~ 3r K~) is commutative since K T = 1-I~r K~ and K~ is the maximal compact subgroup at all places v r T. The algebra Yg(G T, K ~) is naturally a subalgebra of 3/QG s, Kl(n)) by the embedding (P ~-~ (Px~, | where ~K~, is the normalized characteristic function of K~, = II,e r, Kl(p~ "~) C Gr,, i.e., the Characteristic function of K T, divided by the volume of KT,. Then 3r T, K T) is the subalgebra of functions whose support lies in K. r, G r. For any g e G s let (P~ denote the characteristic function of the double coset K,(n) gK~(n) divided by the volume of K~(U). The commutative algebra s/g(G T, K ~) acts naturally on ~'(GL~(k)\GL.(A)) Klm~ by right convolution (r 9 ~) (g) = fG~r ~(gh) dh for ap eaiP(G T, K T) and ~ e auf(GL,(k)\GL,,(A)) K*''~. When keeping track of the actions of K0(n ) and P0(rt) it is most convenient to use an intermediate algebra between ~(G s, KI(n)) and 3/~ T, KT). Let us set GS(n) = ( [I K0,,(p~")) G ~. rUT' Then G s DGS(rt)DG T and GS(n)DKo(n)DK,(rt). Since K0(n) normalizes Kl(n ) and has as quotient Ko(n)/Kl(n ) ,~ II,e T, Ko(pD)/Kl(p~ "~ ~ [I,em,(odpD) � - (os/rt) � we see that the double coset algebra aC~(GS(n), Ka(n)) is naturally isomorphic to C[(os/n) � |162 T, K r) and is therefore again commutative and contains ~(G r, KT). Let us set #g(n) = ~(GS(n), K~(n)). As before, the commutative algebra ~(rt) acts on aC(GL,(k)kGL,(A); (o) xllm by right convolution dh (r 9 (g) = r ~(gh) for @ eaCg(n) and q~ ~ aC(GL.(k)\GL.(A); co) xl"'. There is a corresponding classical Hecke algebra, which we will denote by Wc(rt), which acts on d(l~0(n)\Gs; COs, Z~I). To describe afro(n), let M = GL.(k) n G s GS(n). The group M consists of those rational matrices "r e GL.(k) such that for all v e T' the v-component y. lies in K0,.(p~ ). We may view M as a subgroup of GL.(k) and hence of both O s and G s. Then r,(n) = GL.(k) G s Ks(n)C M. Lemma A.I. -- The map ~: rl(n)\M/Vl(n)-+Ki(n)kGS/K~(n) which is given by ra(rt)tPx(n)~-*Kx(n)tKl(rt) is injective with image Kl(n)kGS(n)/Kx(n). Further- more, if we have the decomposition into right cosets Fl(n)tFa(rt)= lla~ Pl(rt) then also K,(n) tKl(rt) = H aj K,(n). Proof. ~ This argument is modeled after that of Shimura [41], who proved a similar result for GL2. CONVERSE THEOREMS FOR GL n 209 Let V be the n-dimensional vector space on which GL,(k) acts having basis { el, 9 9 e, } with respect to which K and Kx(n) are defined. IfA is the space of os-lattices in V then there is a natural action of both GL,(k) and G s on A [37]. Let L o = o s e 1 + ... + o s e, be the free os-lattice such that GL,(o s) is the stabilizer of L o in G s and GL,(os) is the stabilizer of Lo in GL.(k). Set L1 = Os e~ + ... + Os e~_ ~ + rte., so that Ko(n ) is the set ofg e G s such that gL 0 = Lo and gL1 = L1, and K~(n) is the subgroup of elements g which act trivially on L0/L1. We define F~(rt) in GL~(k) by the same conditions. Let u e Kx(rt) and t e M. We first claim that there exists 7 e F~(n) and u~ e K~(rt) such that ut = 7tu~. To see this, consider the lattices L~ = tL o and L3 = utLo. After scaling by an element of Z,(k) if necessary, which will not effect our conclusion, we may assume L,, L~C L 0. By the theory of invariant factors, Theorem 81.11 of [37], there exists a basis x~, ..., x, of V and os-ideals at, 9 .., a. and fractional ideals b~, 9 .., b~ such that Lo=aiXl+ ... +a.x. L~=alblxl+... +%b.x. and L3 = uL,. Consider a place v ~T'. Then u, ~Kl, v(~0~v ) and t, ~Ko,~(p~v ). Therefore 1 lt~, Lo,, = L2,, -- L3, ~. Hence u~ 1 L2 ' , = Ls ' , with u~ = Consider a place v C T. Then Ls, ~ = u~L.,,~ = u~(]~(a~), (b~)~xi). Now write u~ = u~ d, where det(u~) = 1 and d~ is the diagonal matrix diag(det(u~), 1, ..., 1) with respect to the basis {xl,...,x,}. Then d~L0, ~=Lo, ~ so that d veK~ and hence 1 La, ~ Hence 1 Le,, Ls,,. u~ 1 e K~. Also u~ L2, ~ = u~ 14(Z(a,L x,) = . uo = Let u I = II u,. t Then u 1 e Kl(rt ) SL,(A s) is such that u IL~ = L3. Let c c k � be such that L1, Lg., L3 D cL o. Then by strong approximation for SL, there exists 7 eSL,(k) such that 7 -- ul (modcos). Then yL2 = L3, 7Lo = Lo, and since y,- 1 (modco,) for v ~T', y, LI,, = Lx., and Y acts trivially on L0/L1. Hence 7 ~ SL,(k) c~ G s Kx(u ) C Ial(n), We now have 7tLo = 7L, = L3 = utLo. Hence there exists u 1 in the stabilizer of L 0 in G s, namely K s, such that ut = 7tul. Since t, u, and 7 are all in Ko(u), we must have u 1 e Ko(u ) as well. However, since u 1 = t -1 7 -1 ut we see that u 1 acts trivially on Lo/L x and so u 1 s Kx(rt ). Thus ut = 7rut with Y s Px(u) and ut ~ Ka(rt). We are now ready to prove injecfivity. Let tl, t, e M be such that Kt(n) tl Kl(n) = Kt(rt) t~ Kt(tt). Then there exist ul, u s c K10t ) such that u 1 t 1 = t 2 u,. Write u 1 t 1 = Y1 tl ua with Y1 e FI(lt ) and u s e Kt(rt ). Then Y1 tl us = ts u,. Hence t~ -1 "#1 c GL.(k) c~ G s KI(n ) ----- Ft(n ). Thus Yt tt ----- t, y, and rl(rt) tt Pt(n) = Pt(n) ta Pt(rt). Thus the map ~ is injecfive. The fact that the image is Kt(rt)\GS(rt)/Kl(rt) is clear. 27 210 J. W. COGDELL AND I. I. PIATETSKI-SHAPIRO Now suppose that for t e M we have Px(U) tPx(u) = Hs~l a s Pi(u). Multiplying by Ki(u) on the right we have Pl(u) tKi(u) = Ua~ Ki(u) and it is easy to see that a~ Ki(u ) = a s Kl(u) implies that a~ Pi(rt) = a s Pl(u) so that the right hand side is a disjoint union. But as we have seen above, any ut with u e Kx(rt) and t e M can be written as ut = ytu i with y e Pi(u) and ui e Kl(u ). Hence Kl(u ) tK~(u) = P~(u) tKi(u). This finishes the proof. [] We will also need the following result on the choice of coset representatives. Lemma A.2. -- For t ~ M there exists a decomposition Pi(n ) tPl(u ) ----Ha s Pi(rt) with a s e P'.(k) Po(n). Proof. -- This lemma is a consequence of the class number one assumption. To better illustrate this, let us first consider the case where there is no level, so u = o s and P = Pi(n ) = Po(U) = GL,(os), and remove the class number assumption for the moment. Then M = GL,(k). We claim that ]P',(k)\GL,(k)/P] = h s. To prove this, let us first recall some facts about the classification of lattices over the Dedekind domain os [28]. If L is a os-lattice of rank n then L has the form L=aiXl+ ... +a.x. with a~ fractional Os-ideals. The group GL,(k) acts on these lattices and this action has a complete invariant, namely the Steinitz invariant St(L) = cl(al ... a.) where cl(b) represents the ideal class of the fractional ideal b. So, given two rank n lattices L1 and L, there exists an element y e GL,(k) such that yLi = L, if and only if St(L1)= St(Lz) [28, Theorem 10.14]. Since P = GL,(os) is the stabilizer in GL,(k) of the standard lattice Lo---:Osei+ -.. +Ose, then the set GL,(k)/P is in one-to-one correspondence with the set A 0 of all rank n os-lattices with trivial Steinitz invariant. Now consider the action of P',(k) on the space A 0. Geometrically P',(k) is the subgroup of GL,(k) which preserves the subspace ( el, ..., e,_ i ) spanned by the first n- 1 of the standard basis vectors. It has the structure of a semi-direct product of GL~_I(k ) � GLI(k) acting on k "-1. If L 1 eA 0 then we may associate to L i the rank n- 1 sublattice L~ = L 1 c~ (el,..., e,_ 1 ). We claim that the Steinitz invariant of L~, i.e., St(L 1 rn ( el, ..., e,_l )) is a complete invarlant of the action of P',(k) on A o. Suppose that L1, L~ e A o and L1----pL~ with p e P',(k). Let L~ = L~ n <el, ..., e,_l >. CONVERSE THEOREMS FOR GL. 211 Then by Theorem 81.3 of O'Meara [37] there exists y, ---- ~a,, ~ e~ with a,,. 4= 0 and fractional ideals o~ such that (A.5) L, = L~ + o~y,. Let the action of p on (el,..., e._l ) be given by the element A ~ GL._I(k) then from Lx = pLz we find L~ + alYl = AL~ + a2PY2. Since neither Yx nor PY2 lie in (ex, ..., e~_t) we find pL, n (el, ..., e,_l) = Lt n (e~, ..., e,_x) = L[ = AL~ and hence St(L, n ( el, ..., e,_x )) = St(l_~) = St(AL~) = St(pL n ( el, ..., e,_ 1 )). Hence St(L n ( el, ..., e,_l )) is a P',(k) orbit invariant. Now suppose that L1, Lz ~ A 0 are such that St(L1 n (el, ..., e,_l )) : St(L2 n (el, ..., e,_l)). Let L~ ----- L, n ( el, ..., e,_ ~ >. Then there exists A e GL,_i(k) such that L~ = AI~. Write each I~ as L, : L~ + 0~y, as in (A. 5). Since St(L1) ---- St(Lz) and St(L~) = St(I~) we see that ai and r are in the same ideal class. So, modifying yz by a non-zero scalar if" necessary, we may assume L, = L~' + ay,. Since each y, = ]~a,, ~ea with ai,, 4= 0 we may solve the equation (with A e GL._I(k ) as above) for the (n -- 1) � 1 vector b and the non-zero scalar d. Thenp = ( A ~)EP',(k)andpL,=Ll, so that Ll and L~lie in the same P',(k)-orbit. Hence St(L n (el, ..., e,_ 1 )) is a complete invariant for the action of P,(k) on A 0 _ GL,(k)/P. Since this invariant can take on any ideal class as a value, we see that [ P',(k)\GL,(k)/P [ = h s as desired. In the case n = 2 this is the usual proof that the number of cusps for the full Hilbert modular group is equal to the class number of the underlying field. Now let us return to the class number one case, i.e., we again assume h s = 1. Then the above argument gives that GL,(k) = P',(k) GL,(os) which implies the lemma when there is no level 1t. In the case of level, we claim that M = (P'~(k) n M) ro(11 ) from which the lemma follows. Of course, we have M D (P~(k) c~ M) I'0(1t ) so we need only prove the opposite inclusion. Let m e M. Since h s = I we may write m = py with p e P'~(k) and y e GL,(as). View GL~(os) as GL,(k) n G s K s. Then for v e T' we have m~ =p, y, or p~ = m~ 1 y,. Since m, e K0,,(p~ ) and T~ e GL,(o,) we see that p~ ~ GL,(o~) at these places. But P',(k~) n GL,(o~) C K0. ~(p~'o). Hence p, e Ko,,(p~) for v e T' and p e P'.(k) n M. Now consider T" Since y e GL.(os) = GL.(k) t~ G s K s we know that for all v r S we have T. e K.. Now if v ~ T' we have Y. = P~- 1 m. e K0, .(p.~.). J. w. COGDELL AND I. I. PIATETSKI-SHAPIRO Hence y ~ GL,(k) n G s K0(rt ) ---= F0(n ). Thus M =- (P',(k) c~ M) I'o(n ). This then proves the lemma for h s =- 1. [] Let o~(rt) denote the C-span of the double cosets Fi(rt)kM/Fi(tt ). Then the map ~ induces a C-linear bijection ~ :#f0(rt)-+Yg(n) which takes the double coset Fx(rt) tPi(n ) to the normalized characteristic function qS,. The algebra structure on 3r is the pull back of that of 3f'(n)via ~. If I'i(rt ) tFi(rt ) e ~r then the classical Hecke operator ~ acting on ~/(Fo(n)kGs; o~,, Z~ ~) is defined as follows. For fe J~'(Fo(rt)\Gs; r Z~ ~) and Fi(n ) tFi(n ) = IJaj Fi(rt ) the action is (~f) (gs) = ~. f(a-[ ~ gs). If we recall that o~(n) acts on d(GL,(k)\GL,(A); o~) Kl(n) by convolution (0 * (g) = fG,,., .(h) (gh) dh for q) eW(rt) and q~ e d(GL,(k)\GL,(A); ~)~:1~"), then we have the following result. Proposition A. 1. -- The bijection d(GL,(k)\GL.(A) ; co) xl'"' --% ~(F0(n)\G s; co s, Z~ 1) given in (A. 4) is an isomorphism of I-Iecke modules under the identification of algebras given by 3. Comparison of automorphic representations We would now like to compare certain automorphic subrepresentations of d(Fo(rt)\Gs; COs, ?(~1) which consist of Hecke eigenfunctions for the subalgebra oct~ of ~(n) which corresponds via ~ with the subalgebra 3r T, K T) of ~(rt), with the representations they generate in d(GL,(k)\GL,(A); o~). For the sake of envisioned applications we work in the context of cyclic representations rather than irreducible ones. Theorem A. -- Let 1-1 s be a cyclic automorphic subrepresentation of d(D0(n)\Gs; co, Z~ 1) which cons#ts of Itecke eigenvectors for 3~~ ~_ ~,vt~ K T) with eigencharacter A. Then there exists an irreducible automorphic representation II' of GL,(A) such that II' s is a constituent of l-I s and H '~ is the unique irreducible representation of G T with eigencharacter A. Proof. -- Using the isomorphism of (A.4) we may embed l'I s as a Gs-invariant subspace of d(GL,(k)\GL,(A); co) consisting of Ki(n)-fixed vectors. Let f0 be a cyclic generator of II s in d(GL,(k)\GL,(A); ~). Let (Hi, V1) be the GL,(A) subrepresen- tation generated by 1-I s. Then H 1 will also be cyclic, generated by f0- Let II ~ be the unique irreducible admissible G%module associated to the cha- racter A of W(G T, K T) [3]. CONVERSE THEOREMS FOR GL n 213 Since f0 is a Hecke eigenfunction for o~ with eigenfunctional A, then as an element of d(GL,(k)\GL,(A) ; ~) it is an eigenfunction for W(G T, K ~) as well. Let U be a maximal GL,(A)-invariant subspace of V1 not containing f0 (such a U exists by Zorn's lemma). Then V~/U is a non-zero irreducible subquotient of the space of automorphic forms and hence admissible by [6], paragraphs 4.5 and 4.6. Call the representation of GL,(A) on this quotient II'. Then II' is an irreducible automorphic representation and II' = | II~. Since II' is irreducible and contains a K ~ fixed vector with eigenfunctional A, namely the image off0, we see that II '~ ~ IF. Now consider II~. Since the map V~ -+ II' is intertwining, we see that II~ is an irreducible quotient of V 1. Since II 1 was generated by the G s module IIs, II~ must be isomorphic to an irreducible constituent of II s. [] REFERENCES [1] H. BAss, K-theory and stable algebra, Publ. Math. IHES, 29. (1964), 5-60. [2] H. BAss, J. MILOR, and J.-P. S~.RRE, Solution of the congruence subgroup problem for SLn(n ~> 3) and SP2n(n >1 2), Publ. Math. IHES, 33 (1967), 59-137. [3] J. N. BEm~STEm and A. V. ZELEVINSKY, Representations of the group GL(n, F) where F is a non-archimedean local field, Russian Math. Surveys, 31"3 (1976), 1-68. [4] J. N. BERNSTEIN and A. V. ZELEVINSKY, Induced representations of reductive p-adic groups, I, Ann. scient. Ec. Norm. Sup., 4 e s~rie, 10 (1977), 441-472. [5] A. BO~RL, Automorphic L-functlons, Proc. Symp. Pure Math., 33, part 2 (1979), 27-61. [6] A. BOR~L and H. JACQUET, Automorphic forms and representations, Proc. Symp. Pure Math., 83, part 1 (1979), 189-202. [7] A. BOREL and N. WALLACH, Continuous Cohomology, Discrete Subgroups, and Representations of Reductive groups, Annals of Mathematics Study, No. 94, Princeton, Princeton University Press, 1980. [8] P. CARTIER, Representations of p-adic groups: a survey, Proc. Syrup. Pure math., 88~ part 1 (1979), 111-155. [9] W. CASS~.LMA~r, Introduction to the theory of admissible representations of p-adic reductive groups, manuscript, 1975, [10] W. CASSELMAN, Canonical extensions of Harish Chandra modules, Can. J. Math., XLI (1989), 385-438. [11] L. CLOZEL, Representations galoisiennes associ6es aux repr6sentations automorphes autoduales de GL(n), Publ. Math. IHES, 73 (1991), 97-145. [12] D. FLATH, Decomposition of representations into tensor products, Proc. Syrup. Pure Math., 88~ part 1 (1979)o 179-183. [13] I. M. GELFAN9, M. I. Gga~.V and I. I. PIATETSla-SrIAPmO, Representation Theory and Autornorphic Functions, Boston, Academic Press, 1990. [14] R. GODEraEWr, Introduction ~t la th6orie de Langlands, S~minaire Bourbaki, 1966-1967, No. 321. [15] H. HAMBUROm% l~lber die Riemannsche Funktionalgleichung der ~-Ftmktion, Math. Zeit., 10 (1921), 240-254; 11 (1922), 224-245; 13 (1922), 283-311. [16] G. HARDER, Minkowskische Reductionstheorie tiber Funktionenk6pern, Inv. Math., 7 (1969), 33-54. [17] E. HEctor, l~Iber die Bestimung Dirichletscher Reihen durch ihre Funktionalgleichung, Math. Ann., 119. (1936), 664-699. [18] E. HECKS, Mathematische Werke, G/Sttingen, Vandenhoeck und Ruprecht, 1959. [19] H. JACQUET, Generic representations, in Non-commutative Harmonic Analysis, Lecture Notes in Mathematics No. 587, Berlin-Heidelberg-New York, Springer Verlag, 1977, p. 91-101. [20] H. JACQIJET, Principal L-functions of the linear group, Proc. Syrup. Pure Math., 88, part 1 (1979), 111-155. [21] H.JAcQm~T and R. P. LAnOLANDS, Automorphic Forms on GL(2), Lecture Notes in Mathematics No. 114, Berlin- Heidelberg-New York, Springer Verlag, 1970. [22] H. JACQUET, I. I. PIATSTSI~I-SHAI'mO and J. SrlALIKA, Automorphic forms on GL(3), I & II, Ann. of Math., 109 (1979), 169-258. J. W. COGDELL AND I. I, PIATETSKI-SHAPIRO [23] H. JACUET, I. I. and J. SHALIKA, Conducteur des repr6sentations du groupe lin6aire, Math. Ann., 256 (1981), 199-214. [24] H. JACQUET, I. I. PL~TETSKI-SHAPmO and J. SHALIKA, Rankin-Selberg Convolutions, Am. J. Math., 105 (1983), 367-464. [25] H. JACkeT and J. SHALXX~, On Euler products and the classification of automorphic representations, I, Am. J. Math., 108 (1981), 449-558; II, 103 (1981), 777-815. [26] H. JAC~UET and J. SHALIKA, The Whittaker models of induced representations, Pacific J. Math., 109 (1983), 107-120. [27] H. JACOtmT and J. SHALtr,.A, Rankin-Selberg Convolutions: Archimedean Theory, Festschrift in Honor of L I. Piatetski-Shapiro, part I, Rehovot, Weizmann Science Press, 1990, p. 125-207. [28] N. JAcoBso~, Basic Algebra II, San Francisco, W. H. Freeman and Co., 1980. [29] A. KNAPP, Local Langlands correspondence: Archimedean case, Proc. Syrup. Pure Math., 55, part 2 (1994), 393-410. [30] M. KNESER, Strong approximation, Proc. Syrup. Pure Math., 9 (1966), 187-196. [31] B. KosT^~vr, On Whittaker vectors and representation theory, Inv. Math., 48 (1978), 101-184. [32] J.-P. L~RssE andJ. SCn'~WR~Ea, On liftings and cusp cohomology of arithmetic groups, Inv. Math., 83 (1986), 383-401. [33] R. P. LANOLANDS, On the Functional Equation Satisfied by Eisenstein Series, Lecture Notes in Mathematics No. 544, New York, Springer Verlag, 1976. [34] R. P. L~OLANDS, On the notion of an automorphic representation, Proc. Syrup. Pure Math., 33, part 1 (1979), 203-207. [35] H. MAASS, l~ber eine neue Art yon nichtanalytischen automorphen Funktionen und die Bestimmung Dirich- letscher Reihen durch Funktionalgleichungen, Math. Ann., 121 (1944), 141-183. [36] C. MOEGLIN and J.-L. WALDSPU~:OER, Ddcomposition spectrale et Sdries d'Einstein. Une paraphrase de l'dcrlture, Boston, Birkh~iuser, 1994. [37] O. T. O'ME~.A, Introduction to Quadratic Forms, Berlin-G6ttingen-Heidelberg, Springer Verlag, 1963. [38] I. I. PmTETSFa-St-mPmo, Zeta-functions of GL(n), Preprint, University of Maryland, 1976. [39] I. I. PIATETSKX-S~emO, The converse theorem for GL(n), Festschrift in Honor ofl. L Piatetski-Shaplro, part II, Rehovot, Weizmann Science Press, 1990, p. 185-195. [40] F. RODmR, Whittaker models for admissible representations of reduetive p-adic split groups, Proc. Symp. Pure Math., 26 (1973), 425-430. [41] G. SmMURA, On Dirichlet series and Abelian varieties attached to automorphic forms, Ann. Math., 76 (1962), 237-294. [42] A. WEIL, Ober die Bestimung Dirichletscher Reihen durch Funktionalgleichungen, Math. Ann., 168 (1967), 149-156. J. w. G. Department of Mathematics, Oklahoma State University, Stillwater, OK 74078 I. I. P.-S. Department of Mathematics, Yale University, New Haven, GT 06520 Manuscrit re(u le 11 mai 1992. Rgvisg le 21 octobre 1993. PIATETSKI-SHAPIRO, http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Publications mathématiques de l'IHÉS Springer Journals

Converse theorems for GL n

Loading next page...
 
/lp/springer-journals/converse-theorems-for-gl-n-a7Hlzo6crB
Publisher
Springer Journals
Copyright
Copyright © 1994 by Publications Mathématiques de L’I.H.É.S.
Subject
Mathematics; Mathematics, general; Algebra; Analysis; Geometry; Number Theory
ISSN
0073-8301
eISSN
1618-1913
DOI
10.1007/BF02698889
Publisher site
See Article on Publisher Site

Abstract

by J. w. GOGDELL* and I. I. PIATETSKI-SHAPIRO** The objective of this article is to prove a criterion for a given irreducible represen- tation II of GL~(A) to be automorphic. This criterion traditionally is called a Converse Theorem, after Hecke's celebrated Converse Theorem [17, 18]. The converse theorems of this paper have an application to the problem of Langlands' lifting of automorphic representations from classical groups to GL~. This application will be considered in a future joint publication with S. Gelbart, D. Ginzburg, S. Rallis, and D. Soudry. The first converse theorem was actually proved by Hamburger in 1921 [5]. This theorem states that any Dirichlet series satisfying the functional equation of the Riemann zeta function ~(s) and suitable regularity conditions must be a multiple of ~(s). The generalization to L-functions corresponding to holomorphic modular forms was done by Hecke in 1936 [17]. The leading idea of Hecke was the connection of L-functions which satisfy a certain functional equation with modular forms. However Hecke was able to prove this connection only for holomorphic modular forms with respect to the full modular group, In 1944 Maass extended Hecke's method to his non-holomorphic forms, but still only for the full modular group [35]. The next very important step was made by Well in 1967 [42]. Well showed how to work with Dirichlet series corresponding to holomorphic modular forms with respect to congruence subgroups of the full modular group. Weil proved that if a Dirichlet series together with a sufficient number of twists satisfy nice functional equations with suitable regularity then it comes from a holomorphic modular form with respect to a congruence subgroup. The work of Well marks the beginning of the modern era in the study of the connection between L-functions and automorphic forms. In 1970 a remarkable new book came out: "Automorphic Forms on GL(2)" by Jacquet and Langlands [21]. In this book, instead of automorphic forms, a new object came into this scheme: automorphic representations. The basic result of Jacquet and Langlands was the following. They attached to each automorphic representation of GL(2) an L-function and proved that the nice properties of this L-function, i.e., holo- * The first author was supported in part by NSA grant MDA904-91-H-0040. ** The second author was supported in part by NSF grant DMS-8807336. 158 J. w. COGDELL AND I. I. PIATETSKI-SHAPIRO morphic continuation and nice functional equations, are equivalent to the representation being automorphic. In the following we will use the language of automorphic representations rather than the classical language of automorphic forms. However, for the applications to the problem of lifting of automorphic representations, we need results which are more similar to Well's theorem rather than that of Jacquet and Langlands. In order to get this result we have to use Well's idea, but disguised in the language of automorphic represen- tations. A preliminary version of these results over Q was given in [39]. We would like to thank Herv6 Jacquet and the referee for pointing out some mistakes in an earlier version of this work. 1. Some basic definitions and notation Throughout we will take k to be a global field. Let o denote its ring of integers. For each place vofk we will let k~ denote the completion ofk at v. At the non-archimedean places we will let o v denote the ring of integers of kv, p, the unique prime ideal of o,, m, a choice of generator of p, and we will normalize the absolute value so that ] m, Iv ----- q~-a where q, = [ o,[p, [. We will use either o~ x or u~ for the group of local units. The symbol A will denote the ring of adeles of k and A � its group of ideles. Thus, A is the restricted pro- duct II',k, of the completions of k with respect to the compact subrings 09. If S is a finite set of places of k we will let k s = II,~ s k, and A s = II',~ s k, so that A = k s A s. We will use a similar notation for ideles. For each finite set of places S of k containing all archimedean places, the ring of S-integers is os = k n k s I1~r s o~. We may view o s as a discrete subgroup of k s through the embedding of k into k s. Let u s = [[~s u~C (A� s. The class number hs of os, called the S class number of k, is the cardinality of the S-class group = � � x ~s k \A /k s u s We fix a non-trivial normalized additive character ~ of A which is trivial on k. Fix a basis { e, } of k" with respect to which the matrix structure of GL, is defined. Let B. denote the Borel subgroup of upper triangular matrices, A~ its Levi subgroup consisting of all diagonal matrices, and N, its maximal unipotent subgroup. Let P'~ denote the standard parabolic subgroup of GL~ associated to the partition (n- 1, 1) of n. Let P,~ C P'~ be the mirabolic subgroup consisting of those matrices in P', whose last row is (0, ..., 0, 1). Let P, C P~ denote the opposite mirabolic and parabolic. So ~ = ,(p,)-l. By Z,, we denote the center of GL,. For each non-archimedean place v we will let K, = GL,(0,) be a maximal compact subgroup. We will always consider admissible representations II, of GL,(k~) on a complex vector space VII v in the usual sense [6, 8, 9]. As is common, we will not distinguish between admissible representations of GL.(k,) and of its Hecke algebra [6]. We will 159 call an admissible representation unramified if the space of vectors fixed by K, is one- dimensional. At an archimedean place v, we select as maximal compact subgroup K, either O(n) or U(n) defined with respect to the basis above. At an archimedean place v of k by an admissible representation H~ of GL,(k~) we will mean a smooth representation of GL,(k,) on a complete Frechet space Vnv whose subspace of K,-finite vectors is an admissible representation of its Hecke algebra [6] and such that (II,, Vuv ) is a canonical smooth model of moderate growth (in the sense of Casselman and Wallach) of the underlying representation of its Hecke algebra [10, 27]. Let v be any place of k and let +, be any non-trivial additive character of k,. Then +9 defines a character of N,(k,), which by abuse of notation we again denote by +,, by +,(n) = +,(nl. ~ + n~. 3 + ... + n,_l,,) where n = (n~,j) eN,(k~) relative to the basis above. Let (II,, Vn~) be a finitely generated admissible representation of GL,(k,). We let V$o denote the space of +,-Whittaker functionals on Vn,, i.e., the space of continuous linear functionals k, on Vno such that )~,(1-I~(n)~)= +,(n)),~(~,) for all n e N.(k,) and all ~, ~ Vno. A representation H, of GL.(k,) is of Whittaker type if H. is finitely generated, admissible, and dim(VS,) = 1. In this case we have a non-zero intertwining map from Vno to the Whittaker space given by ~, ~-~ W~,(g) = X,(1-I,(g) 4,) where ~, e V~o is a non-zero Whittaker functional. We will call the space of functions zCU(II,, +,) = { W~(g) ] ~, e H, } the Whittaker model of H, (even though it is a model for the Whittaker quotient of H, unless the Whittaker map above is injective) and it is unique. An irreducible admissible representation of Whittaker type is called generic. For our purposes, we will only need consider representations of Whittaker type of a certain nature. Let Q be the parabolic subgroup of GL, associated to the partition (rx, ..., r,~) ofn. For each i let ~ be a quasi-square integrable representation of GLn(k,) (i.e., an irreducible admissible representation whose matrix coefficients become square- integrable modulo the center after twisting by a suitable character of GL,i(k,)). Then the (unitarily) induced representations are of Whittaker type [4, 27]. Throughout this paper, by an induced representation of Whittaker type (or, more succinctly, an induced of Whittaker type) we will always mean one of these induced representations. From this it is clear that induced representations of Whittaker type have well-defined central characters. Also, the subspace of K,-fixed vectors is at most one-dimensional. In particular, let Q. be the parabolic subgroup of GL, associated to the partition (rl, ..., %) of n. For each i, let P~,o be a tempered representation of GLn(k,). Let Ul > u~ > ... > u,, be a sequence of real numbers. Set T.a,qGYun(kz~) r / ~ . 9 . =, = L 0 .0 I I | | (p.,0| [')] ...~s.~,,~;..n~r.~*v~r,v,~wj~ J. w. COGDELL AND I. I. PIATETSKI-SHAPIRO (unitary induction). We call such a representation an induced representation of Langlands type. Then ~v is an induced representation of Whittaker type [19, 26, 27]. If II, is an irreducible generic representation of GL,,(kv) then II v is necessarily an irreducible induced representation of Langlands type [26, 27]. The Langlands classification for GL,(kv) says that every such ~v has a unique irreducible quotient II v and every irreducible admissible II v occurs uniquely as the quotient of some E, [7]. Consider a representation II of GL,(A) on a space V = Vn. This representation is called factorizable if there are local representations 1-[~ of GL,(k,) on spaces V, such that II, is unramified for almost all places v and (II, V) is the restricted tensor product of the (IIv, Vv) as in Flath [12]. We will denote this simply by II = @ II v. We will always consider admissible representations of GL,(A) in the sense of [6] or [12]. If II -~ (~)1-I, is factorizable and admissible then each II, is admissible, and conversely. If S is a finite set of places of k we will let G s = GL,(ks) =IIve s GL,(k,) and GS= GL,(A s) = 1-I'v~ s GL,(k,). Similarly, for II = (~II v factorizable we shall let 1-Is = (~,Es IIv be the associated representation of G s and II s = @,~s 1-Iv be the asso- ciated representation of G s, so that II = 1-I s | 1-I s. If fl = @ II, is an admissible factorizable representation of GL,(A) we will say that II is of Whittaker type, induced of Whittaker type, or generic if each FIv is and, in addition, at the places v where II v is unramified the space of K:fixed vectors is not in the kernel of the map to the Whittaker quotient. (This last condition is automatic if II v is generic or induced of Langlands type since in these cases the map to the Whittaker model is an isomorphism [26].) In these cases there is a unique global Whittaker func- tional k (up to scalars) given by the product of the local Whittaker functionals ?'v suitably normalized. At the places v where II, is unramified, there is a distinguished unramified vector ~0 with respect to which the restricted tensor product is taken. At these places we always normalize the Whittaker functionals kv so that kv(~ ~ = 1. In terms of the local Whittaker models, this implies that W~o(I,) = 1. If~CU(fl,, +~) are the local Whit- taker models, then r +) = (~zCV(IIv, ~). It is again clear that global induced representations of Whittaker type have central characters. If H = (~ l-I v and II' = (~)II~ are two factorizable admissible representations of GL,(A) then we will say that they are quasi-isomorphic if 1-I, ~_ FI', for all non-archi- medean v for which both 1-I, and 1-I~ are unramified. By an automorphic representation of GL,(A) we will mean an admissible sub- quotient representation of the space of automorphic forms d(GL,(k)\GL,(A)) [6]. By a proper automorphic representation we will mean an admissible subrepresentation of the space of automorphic forms d(GL,(k)\GL,(A)). By a cuspidal automorphic represen- tation we will always mean an irredudbh cuspidal automorphic representation. These are of course always proper. 161 2. Basic converse theorems Let II = @ FI, be an admissible factorizable representation of GL,(A) such that each II, is either irreducible or induced of Whittaker type. Let v denote a factorizable automorphic representation of GL,.(A) for some m with 1 ~< m ~< n -- 1 such that each % is irreducible or induced of Whittaker type. Then from the local theory of L-functions for GL,(k,) [24, 27] for each place v we have a local L-function L(II, x %, s) and local c-factor ~(II, � %, s, +~) attached to II and "r. We may then formally define a global L-function L(H X v, s) = I-IL(H, x %, s) and a global ~-factor ~(n � ~, s, +) = 1-I ~(no x ~,, s, +o). To see that these are actually well-defined we need the following elementary lemmas. Lemma 2.1. -- The a-factor ~(H X ~, s, +) is absolutely convergent and if the central character ~o n of II is invariant under k � then ~(n x ,~, ~) = II ~(n, x .~,, s, +.) is independent of +. Proof. -- For almost all v, II,, % and +~ will be unramified and so ~(1J~ � ~, s, +~) - 1 for these places. Thus r � v, s, +) is convergent. To prove that the product is independent of the choice of additive character we must consider how the local c-factor changes when we change our additive character +(x) to t~X(x) = +(kx) with X e k � Recall that the local ~-factor is defined by the local func- tional equation [24, 27] uZ(W, W'; s) CF(p(w,,,~) ~r, T~r,; 1 -- s) r � %, s, +~) o~v (- 1) "-1 = L(n~ � ~o, s) L(~, � %, 1 -- s) Where W e W~(II,, +~), W' e W'(%, +~-1), ,~(g) = W(w, ,g-l) e ~r +~-1), and ~r,(g) = W(w~ tg-1) e ~(~,, ~,). The Weyl elements involved are w~ = (1 1), 21 162 J. W. COGDELL AND I. I. PIATETSKI-SHAPIRO the longest Weyl element of GL,, and o) ?-O n _ whereas ~ denotes right translation in the Whittaker model. The integrals themselves are given by .c*,~\ot.c~) In- and, setting k = n -- m -- 1, ~r(w,w';s) W'(g) [ det(g)I '-''-"'t2 dx dg. rd~d\GLm{kv) 'rnr 0 To change the Whittaker model from those with respect to d?, to those with respect to +x, note that if we set r -- 1 ~t ~ 2 ~,(x) = ~B,(k), then for W ~r162 +,) we have/(an(X)) W(g) = W(an(X ) g) eqF'(II,, d~x). Note that the local L-functions are independent of the choice of d?,. Hence r � v~, s, +x) is defined by the local functional equation 'r(t(~n(X)) w,t(,.(x)) w';s) ~(II. � ~~ ~, +~) ~.o(- 1) n-' L(II, � "%, s) _~P(p(w.,.) (t(~.(x))w) ~, (t(~.(x))w')~; 1 -s) L(H. � ~'., 1 -- s) Now a straightforward computation gives 'r(t(a.(x)) w,t(a.(x)) w'; s) =lxly+',%(x)"-"'t~(p( I" W, W'; s) a._.(x)) 163 with A = m(m -- n) -- ~ m(m -- 1) 1 (m 1 ) 1 1) m(m + 1). B =~(m--n) (m--n)---~m(m-- 1) +~(m-- An equally straightforward calculation gives ~v(p(w.,.) (t(~.(x)) w) ~, (t(a.(x)) w')~; 1 - s) = I x IC'l-" + v ~no(X)" %(x)" re (0(o.,.) (p ( I" ,W'; 1-- a n_ m(X with C =--m ~+~m(m- 1) 1 (1 ) D = -~ m(n -- m-- 1) (n--m--2) +-~(n--m) m S-~m(m- l) +~(m-- 1) m(m+ 1). Then using the definition of the respective local e-factors, we find 9 (n~ x ~, s, +~) = ~0(x)" o,0(x) ~ I x 17 '-~ ~(n~ � ~0, s, +0) with d :nm -- -~ m(m + 1). Taking the product over all places of k and using the product formula we find ~(n � ~, s, +~) = on(X) ~ o.(x)" ~(n � ~, s, +). Since v is automorphic its central character is invariant under k � Hence if Olt is invariant under k � we see that the product ,(II � % s, q) = r X % s) will be independent of the choice of d/. [] Lemma 2.2. -- Suppose L(II, s) : II~ L(II~, s) is absolutely convergent in some half- plane. Then for any automorphic representation v : (~) % of GL,,(A) which is either irreducible or induced of Whittaker type the Euler product for L(II � % s) is also absolutely convergent in some half-plane. Proof. -- Let T be a finite set of places of k containing all archimedean places such that II~ is unramified for v r T. Then the local L-factor for the places v r T will be of the form L(n., s) = II (1 -- a,., q~-,)-l. ~=1 164 j. W. COGDELL AND I. I. PIATETSKI-SHAPIRO Globally, let us write L(II, s) = Lr(rl, s) LT(n, s), where ~(n, s) = II L(H~, s) v~T is a finite product, and hence always absolutely convergent, and LT(n, s) = H L(rL, s) = II II (1 -- a,,. q~-*)-a. Then if the Euler product for L(H, s) is absolutely convergent for Re(s) > Co, we have the estimate [ a,, ~ I ~ q~ for all v r T, with the implied constant independent of i and v. Let v = | % be an automorphic representation of GLm(A) which is irreducible or induced of Whittaker type. Then we know that the Euler product for L(':, s) converges absolutely in some half-plane, say Re(s) > q. Enlarging T if necessary, we may assume that % is also unrarnified for v r T. Then, as above, we have L(%,s) = fi (1 -- b j,~ ~,,-t } $=1 with the estimate [bj,~ [ ~ q~. The Euler factor for L(II~ � %, s) for v r T is given by ~m L(II~ � %,s) = fi fi (1--a~,~b~..,q~-')-a= II (1--ck, ~q~-~)-'. i=1 j=l k=l Since we have the estimate I ck,. 1= ]a,,~ I I bj,~ I ~ q~+ =1 for v r T, we see that the Euler product for L(H � % s) is absolutely convergent for Re(s) > co + ct + 1. [] Let g' denote the outer automorphism g ~ g' = ~g-1 of GL,. For any repre- sentation = of GL, over a local or global field, let =~(g) = =(g'). If II, is an induced of Whittaker type, then so is W~. If II~ is irreducible, then so is H~, and in fact H i _~ H,, the contragredient representation. Lemma 2.3. -- Suppose L(II, s) converges in some half-plane and tkat the central character of H = | II, is invariant under k � Then the Euler product for L(W, s) also converges absolutely in some half-plane, as do the L(YP � v ~, s) for any automorphic representation v of GL,,(A) whiek is irreducible or induced of Whittaker type. Proof. -- We may assume that the central character O~rl of II is unitary. For ff it is not, we have [ on(a)[ = [ a I d for some d =~ 0. If we let o~_a!,(a ) = ] a [-al. and set II' = II | o~x, then II' has a unitary central character. Since L(II', s) = L(II, s -- din) we see that L(H, s) is absolutely convergent in some half-plane ff and only if L(II', s) is. 165 For v r T, with T as in Lemma 2.2, II~ will be unramified and we have unramified characters ~1,~, ..., ~,,~ of GLI(k,) such that II~ is the unramified constituent of Ind~.~'(~l,.| ... | The local factor L(II,,s) is then rI(1- a,,,q:') -1 with ai, ~ = ~t,,,(%). The central character of rl. is tor~. = r[~,,.. Since this is unitary, we have 1--I ton~ l = I] I ~,.~(~o)I = ~I la,~l. i=! ' Still for v r T, if II, is as above, then II', will be an unramified constituent of "'~.~k.~ ~m,. "- ~,.). Its local factor will then be L(HL~, s) ~ l-I(l -- b~,. q~-i)-i with b~,. = ~z~,.(%) -1 = a~,..-1 Now assume that L(II, s) converges absolutely for Re(s) > c, so that we have the estimate [ a~,~ I ~ q~. Then for [ b~, ~ [ we have Ib,.~l = II Ib,,~l -~= II la,,,[ ~ q~-l, 0. Hence the Euler product for L(IP, s) converges absolutely for Re(s) > (n -- 1) e + I. The rest of the lemma now follows form Lemma 2.2 applied to IP. [] Definition. -- Let II : | II~ be a factorizable admissible representation of GL,(A) such that each local component II~ is either irreducible or induced of Whittaker type and such that its central character to n is invariant under k � and its L-function L(II, s) is absolutely convergent in some half-plane. Let .c be an automorpkic representation of GLm(A) which is either irreducible or induced of Whittaker type. We will say that L(II � % s) is nice/fL(II � v, s) and L(17 ~ � r s) have an analytic continuation to entire functions of s which are bounded in vertical strips and satisfy the functional equation L(n x., s) = ~_(n x -~, s) L(II ~ x r 1 -- s). A converse theorem for GL~ is a criterion in terms of the L(II x v, s) for deter- mining when II is actually an automorphic representation. Our first converse theorem, modeled on that of Jacquet and Langlands, is one of the end products of years of colla- boration of the second author with H. Jacquet and J. Shalika (for example [22-24]). In the function field case, this theorem was proven in the 1970's by the second author [38]. The same method of proof works in the number field case now that the local archimedean theory has been completed by Jacquet and Shalika [27]. Theorem 1. -- Let II be an irreducible admissible representation of GL~(A) whose central character ton is invariant under k � and whose L-function L(1-I, s) is absolutely convergent in some half-plane. Suppose that L(II � v, s) is nice for every cuspidal automorphic representation -e of GLm(&) for all m with 1 <~ m <~ n -- 1. Then II is a cuspidal automorphic representation of GL.(A). This theorem yields maximal information about rI, namely that it is actually cuspidal automorphic, but it requires nice behavior of the L-functions under twists J. w. COGDELL AND I. I. PIATETSKI-SHAPIRO by all cuspidal automorphic representations on all smaller GL,~'s. We will give variants of this theorem where we require the L-functions to be nice under a smaller set of twists. For this we must use the ideas of Well. The most typical converse theorem of this type is the following. Fix a finite set of places S ofk containing all archimedean places. For each integer m, let f~s(m) = { r: : 7r is an irreducible generic automorphic representation of GL~(A), unramified at all v r S }. Similarly, let f~(m) be the set of cuspidal elements of f~s(m). Theorem 3. -- Let n >1 3. Let II be an irreducible admissible representation of GL,(&) whose central character co n is invariant under k � and whose L-function L(II, s) is absolutely convergent in some half-plane. Let S be a non-empty finite set of places of k containing all archi- medean places such that the ring os of S-integers has class number one. Suppose that for every m with 1 <~ m <~ n -- I and every 9 E s the L-function L(II � v, s) is nice. Then there exists an irreducible automorphic representation II' Of GL,(&) such that 1-I'~ = II,for all non-archimedean places v where II, is unramified. This will be proved in Section 11. We will also give a version of this theorem where we put the extra hypothesis that II be generic. In this case we can draw slightly stronger conclusions. These are stated as Theorem 2 and its corollaries, which can be found in Section 7. We believe that it is not necessary to have control of so many twists to be able to draw conclusions about the automorphic nature of II. Twists by characters of GL x might be enough. We state this in the following conjecture. Conjecture. -- Let II = | YI, be an irreducible admissible representation of GL,(A) whose central character O)r~ is invariant under k � and whose L-function L(fl, s) is absolutely convergent in some half-plane. Assume that L(II | co, s) is nice for all characters co of k� � Then there exists an automorphic representation II' of GL,(&) which is quasi-isomorphic to II and such that L(II | ~o, s) ----- L(fl' | o~, s) and ~(H | co, s) = r | co, s). The validity of this conjecture would have very fundamental applications to the problem of Langlands lifting. This conjecture is known to be true for n = 2 [21] and n = 3 [22] and we actually have II = II'. The first example where II 4= II' was constructed in [38] for n = 4 and the construction provides examples for all n I> 4. 3. Outline of the proof of Theorem 1 Let us first outline the proof Theorem 1 under the more restrictive hypothesis that II = | II, is generic, i.e., each II~ is generic. Let us begin with an arbitrary ~ E VII. Our goal is to embed Vn in CONVERSE THEOREMS FOR GL n 167 ~r such that the actions of GL~(A) are intertwined. Since Vn is linearly spanned by decomposable vectors we may assume that ~ is decomposable, i.e., ~ = | ~, with ~ e Vnv. As a first step let us associate to ~ some function on GL~(A). This is where the assumption that II is generic comes into play. Each II, has a unique Whittaker model ~r ~b,) and to each ~, is associated a function W~(g,)er162 qb~). For almost all v, II, will be unramified and there is a distinguished unramified vector ~o with respect to which the restricted tensor product is taken. At these places we normalize the Whittaker model so that W~o(I,) = 1. Now to ~ e VII associate the global function W~(g) = 1-[, W~v(g,). Since for almost all v, ~, is the distinguished unramified vector ~0 in Vn~ and g~ ~ GL,(o,), this product converges absolutely to a continuous function on GL, (A). We first attempt to make an automorphic function from ~ by averaging as much as possible over GL,(k). First note that W~(g) is left invariant under both N,(k) and Z,~(k). To get further invariance, consider the sum U~(g) ----- ~1 W~(yg) = ~] W~ g . Y ~ ~,n(k)\Pn(lc) y' ~ Nn - I(/O\GLn_ l(k) ((0 ~ This sum converges absolutely and uniformly on compact subsets to a continuous function on GL,(A) which is cuspidal along the unipotent radical of any maximal parabolic subgroup of GL. containing B.. As a function on GL,(A), Us(g ) is left invariant with respect to P,(k) and Z,(k) and hence with respect to the full parabolic subgroup P'.(k) associated to the partition (n -- 1, 1) of n. We next construct a second function V~(g) associated to ~ which will be related to U~(g) via the functional equation of the L-function. Put - (i1 11) where (11) is the longest Weyl element of GL,. Then, if we consider W~(~, g), this is left invariant under ,1 /i* ~-1 N.(k) a. = 0 1 o oo ~t which we will denote by N'.(k). Note that N'.(k) C P.(k) where P. is the mirabolic opposite to Pn" 168 J. W. COGDELL AND I. I. PIATETSKI-SHAPIRO Then let us set Vg(g) = Z Wg(0% vg) = Z Wg ~. g . ( (; V ~(k)\Pn(k) T' ~ l(k) To deduce the properties of Vg from those above for Ug, let us recall that if we set "~/~(g) = W~(w. ,g-l), then "W~ is in the Whittaker model for H, the contragredient representation [24, 25, 27]. Then we have V~(g) = X We, g' 7' ~ Nn-t(k) \OI~-l(k) ((o ~ where ,=(o. 11), x(w. 1 l) and oi) . We may conclude that Vg(g) converges absolutely and uniformly on compact subsets to a continuous function an GL.(A). It is left invariant with respect to P'.(k) = ~P'.(k) -1. To any ~ we have attached two functions on GL.(A), one invariant under P'.(k) and the other under P'.(k). Note that together P'.(k) and P'.(k) generate all of GL.(k). Our strategy will be to use the global functional equation to prove that U~(g) = V~(g), which will show that this function is in fact invariant under GL.(k) and hence automorphic. To relate U~ and V~ to the L-function we consider the following integrals. If we restrict U~(g) or V~(g) to GL._a(A)C GL.(A) embedded in the standard way, then u~(h 0 ~)and v~(h 0 01)are rapidly decreasing automorphic forms on GL,_~(A). Let v be an irreducible proper automorphic representation of GLn_x(A ) and let ~ be an automorphic form in the space of v. Set I(~, 9; s) = U~ 1) ~?(h) ] det(h)["-(a/~' dh. ;0 (;0, L n _ l(k) \GLn_ I(A) The integral I(~, q~;s) converges absolutely for Re(s)>> O. If we unfold the series defining u~(h 0 01), we find ~) W~(h) det(h)I '-`1/2' dh i(~,~;s) = f~ w~(h ~ n_ I (A) \GLn_ I (A) = T(W~, W~; s) i"~'n-l(k)\GI.,~,~- CONVERSE THEOREMS FOR GL a 169 where n - 1 (k) \lqn - 1 (A) i.e., W,(h) eSF(% +-~). Similarly, for Vr we may define the integral I(~, ~; s) -- V~ ~) q~(h) I det(h)I '-am dh. L n _ irk) \GL._ I(A) This will converge for Re(s) ~ O. If we unfold it, we find ~r (h 0 O1) ~r l det(h) l'a-"-'a/~' dh = W(Wr W~;1- s), where, as before, we set W~(h) = W~(W._ a 'h--l), ~rr = Wr ,g-a) Both of these families will have an analytic continuation to entire functions of s, bounded in vertical strips. To see this we must relate these global integrals to the global L-function. Up to this point, nothing is used other than general properties of Whittaker func- tions. To prove the continuation of these integrals and relate them, we must use our assumptions on the L-functions. The integrals are related to the global L-functions through their expressions as Whittaker integrals. In fact, we have I(~, ~; s) -- 't'(W~, W~; s) = L(n � ~, s) Eft) ]'(r ~; s) = ~v(~, ~; 1 - s) = L(~ � % 1 -- s) E(s) where E(s) and E(s) are entire functions of s. The analytic continuation of the global L-functions then implies that I(~, ~; s) and I(~, c?; s) both have continuation to entire functions of s which are bounded in vertical strips. The global functional equation for L(H � % s) will allow us to relate I(~, ~; s) and I(~, ~; s) and hence Ur and Vr From the local functional equation we have ,v(W~o, w~o; ,) 'v(~'~o, ,~'~; 1 - s) L(n~ x ~, s) ~(n~ x ~o, s, +~) %(- 1) "-a = ~ L(Ho � %, 1 - s) Using this, the global functional equation will imply that upon taking products we have "F(W~, W~; s) = ~F(W~, W~; 1 -- s) or I(~, $; s) =: I(~, $; s) for all s. 22 j. w. COGDELL AND I. I. PIATETSKI-SHAPIRO Ifwe set Fr = U~- V~ then F~(h 0 ~) is rapidly decreasing on GL._ ~ (k)\GL,_ ~ (A). If we restrict to SLn_l(A) then F~(k 0 ~) will be in L~(SL._I(k)\SL~_I(A)) andif we interpret the above equality in terms of F~, we see that ;SLn_l(k)\ 8Ln_I(A F ~ (h 0 :)~(h)dh=O for all ~ occurring in irreducible automorphic subrepresentations of SL._I(A ). If we then apply the weak form of Langlands' spectral theory we may conclude that Since F~(h 0 ~)--O, we have that Ut o(h O1)=V~(: 01)for all k eSL,~_I(A ) and in particular U~(1) = V~(1). Since this is true for all ~, then U~(g) = Un,g, 4(1) ---- Vn(o)r = V~(g) for all g ~ GL.(A). We now have that Ur is invariant under P.(k), Pn(k), and Zn(k ). Since these generate GL.(k) we see that U~ ~d(GL.(k)\GLn(A)). Thus the map ~-~ U~(g) embeds II into M(GL.(k)\GLn(A)). Hence II is an automorphic subrepre- sentadon. To see that II is cuspidal, since U~(g) is given by the convergent " Fourier expansion " Y ~ Nn_ l(kl\GLn- l(/c) without constant term, we observe that for any parabolic Q., the constant term of U~ along O is 0. Hence U~ ~ ~r176 i.e., U~ is cuspidal and hence II is cuspidal. This is the conclusion of Theorem 1. 4. Preliminary considerations on Whittaker models Before we turn to the rigorous proof of Theorem 1, we would like to gather together some known results which we will need. We begin with the following local and global estimates for Whittaker functions. If v is a place of k then by a gauge on GL.(k.) is meant a function ~, which is left invariant under N.(k~), invariant on the right under Kn,. and which on A.(kv) has the form ~,(a) = l ala2 ... a,-11-' qb(al, a~, ..., an_i) , CONVERSE THEOREMS FOR GL n 171 where al a2 . . . a n ] a~ . .. a n a~ An(k~ a n t is real and non-negative, and q) is a Schwartz-Bruhat function on k~- ~ [22, 27]. A global gauge on GLn(A ) is defined analogously [22]. Then the standard estimates on Whittaker functions are the following. Lemma 4.1. -- a) Let II~ be a generic representation or an induced representation of Whittaker type of GLn(k,) with con, its central character. Let I cony(X)[ = [ x [~. Then for any W, ~ ~//'(II,, +,) there is a gauge 6, such that [ W,(go) l ~< ~,(g,) I det(g,)[a/n. b) Let II be a generic representation or an induced representation of Whittaker type OfGLn(A ) whose central character con is invariant under k x and whose L-function L(II, s) is absolutely convergent in some half-plane. Let [ con(X) [ = [ x [a. Then for any ~ ~ II there exist a global gauge ~ such that I Wg(g) [ ~< ~(g) [ det(g)[a/,. Proof. -- a) When v is a non-archimedean place of k and H~ is generic this is just Proposition 2.3.6 of [22]. As noted in Remark 2.2.5 of [22], the proof is still valid if II~ is induced of Whittaker type. For v an archimedean place this follows from Proposi- tion 2.1 of [27] and the comments following it. b) Let H' = H | co_d/, as in the proof of Lemma 2.3, so that H' has a unitary central character. The product L(H', s) also converges absolutely in some half-plane. Let T be a finite set of places ofk such that II~, and hence II',, is unramified outside T. As in the proof of Lemma 2.3, for v r T we have that II: is the unramified constituent of Ind~ | | ~,, ~). If L(II', s) converges absolutely for Re(s) > e 0 then we have seen that we have the uniform estimate [ ~q,,(~) [ ~ q~. Since II' has a unitary central character, we also have a uniform lower estimate of q~-tn-1)~ ~ [~h,,(t%)[. Hence there exists a uniform do, independent of v, such that q~-ao < ] ~,,,(t%)[ < q~ for all v r T. Now applying Proposition 2.4.1 of [22] we see that we can choose a compatible family of local gauges { ~, } for the II, as in a) such that ~ = 1V[ ~ is a global gauge and gives the estimate in b). [] 5. Prellmln~ry considerations on Langlands' spectral theory For the proof of Theorem 1 we will need the weak form of Langlands' spectral theory for SL n_ x. We recall here what we will need, specialized to SL._ x. For details, see [33], [14] or [36]. j. w. COGDELL AND I. I. PIATETSKI-SHAPIRO Ifwe set Fr = U~- V~ then F~(h 0 ~) is rapidly decreasing on GL._ ~ (k)\GL,_ ~ (A). If we restrict to SLn_l(A) then F~(k 0 ~) will be in L~(SL._I(k)\SL~_I(A)) andif we interpret the above equality in terms of F~, we see that ;SLn_l(k)\ 8Ln_I(A F ~ (h 0 :)~(h)dh=O for all ~ occurring in irreducible automorphic subrepresentations of SL._I(A ). If we then apply the weak form of Langlands' spectral theory we may conclude that Since F~(h 0 ~)--O, we have that Ut o(h O1)=V~(: 01)for all k eSL,~_I(A ) and in particular U~(1) = V~(1). Since this is true for all ~, then U~(g) = Un,g, 4(1) ---- Vn(o)r = V~(g) for all g ~ GL.(A). We now have that Ur is invariant under P.(k), Pn(k), and Zn(k ). Since these generate GL.(k) we see that U~ ~d(GL.(k)\GLn(A)). Thus the map ~-~ U~(g) embeds II into M(GL.(k)\GLn(A)). Hence II is an automorphic subrepre- sentadon. To see that II is cuspidal, since U~(g) is given by the convergent " Fourier expansion " Y ~ Nn_ l(kl\GLn- l(/c) without constant term, we observe that for any parabolic Q., the constant term of U~ along O is 0. Hence U~ ~ ~r176 i.e., U~ is cuspidal and hence II is cuspidal. This is the conclusion of Theorem 1. 4. Preliminary considerations on Whittaker models Before we turn to the rigorous proof of Theorem 1, we would like to gather together some known results which we will need. We begin with the following local and global estimates for Whittaker functions. If v is a place of k then by a gauge on GL.(k.) is meant a function ~, which is left invariant under N.(k~), invariant on the right under Kn,. and which on A.(kv) has the form ~,(a) = l ala2 ... a,-11-' qb(al, a~, ..., an_i) , CONVERSE THEOREMS FOR GL n 173 Now let ~ be an irreducible admissible cuspidal representation of M(A). Then we may form the induced representation Indg~s which we view as the space of functions ~ : SL,_I(A ) ~ C such that for all g ~ SL,_I(A), the function m r-~ ~(mg) is in ~ | 8~/2. Let I(~) denote the subspace of admissible vectors of this induced repre- sentation. So q~ ~ I(~) if it is smooth, K-finite, and satisfies the previous condition. If g e SL,_I(A), then g will have an Iwasawa decomposition g = umk relative to P, where u e U(A) the unipotent radical of P(A), m e M(A), and k ~ K = I-I, K~. Write m = re(g). This is not unique, but its image ,~(m(g)) in M(A)/MI(A) is Uniquely defined. If r ~ I(a) then the function ~z~:g ~ ~(g) t m(g) I ~ is in I(~ | X~). We are now ready to define the Eisenstein series we will use. If M is a Levi subgroup of a parabolic subgroup P of SL,_x, cra unitary cuspidal representation of M, q0 e I(a), ands~C'-I set E~(g; s) -- ~] ~(Yg) I m(yg)18 7 6 P(k)\ SL.- l(k) whenever this converges. The facts we will need about the Eisenstein series are contained in the following theorem. Theorem $1. --The series defining E~(g; s) converges absolutely and uniformly on compact subsets for all s in the positive cone X + ={seX MlRe(s,)-Re(s,+l)> 1} (set s, = 0). In this region, E~(g; s) is a holomorphic function of s and is of moderate growth on SL,_I(k)\SL,_I(A ). Moreover for s ~ X + E~(g; s) e ,~/(SL._I(k)\SL._I(A)). For generic s e X + , I(a| Zs) is irreducible and the map ~ ~-~ E,(g; s) defines an embedding of I(~| as an automorphic subrepresentation of ~r Besides the Eisenstein series we need another family of functions which seem to go by many names (incomplete theta series, pseudo Eisenstein series, etc.). Let us introduce them through the Paley-Wiener functions on X~. If k is a number field, so X~, __ C '-1, then P(XM), the space of Paley-Wiener functions on XM, is the space of holomorphic functions f: X~, -+ C which satisfy an estimate of the following type. For each f e P(XM) there exists a real number r and for each n ~ N there exists a constant C, such that lf(s)l c. + II s I1)-". 174 J. W. COGDELL AND I. I. PIATETSKI-SHAPIRO Ifk is a function field, then X M ~ C'-1/2hi Z,_ 1 and P(XM) is the set of functions /log q which are given by polynomials in q,1, ..-, q,,-x and their inverses. If we define the Fourier transform on functions on X M by 3~(m) = ~.~,o,f(s)[m I' ds, thenaf(m ) is a function on V~ = M(A)/MI(A) and the space of Paley-Wiener functions on X~ has the equivalent characterization by f~P(X~) if and only iff~C~(VM). For ~ ~I(~) andf~P(X~) we define O~./(g) = E r(yg)f(m(Tg)). 7 ~ P(~)\ 8Ln-l(k) Then first basic result on these functions is the following. Theorem $2. -- The sum O~,/(g) converges absolutely and uniformly on compact sets to a rapidly decreasing function on SL._x(k)\SL._I(A ). It has an expansion in terms of Eisenstein series by 0~'t(g) = :m,,, = m,*o, E~(g; s) f(s) Us for any s o ~ X + . To state what we have called the weak spectral theorem, let us recall the convention that for M = SL._ 1 itself, both the Eisenstein series E~(g; s) and the series 0~,t(g ) reduce to just the cusp forms ~ in the cuspidal representation a of SL._ ~(A). Then by weak spectral theory we mean the following result [36, Theorem II. 1.12]. Theorem $3. -- The collection of all functions of the form 0~, f(g) obtained as M runs over all Levi subgroups of SL._I, ~ all unitary cuspidal representations of M (A), ~ E I(~), and f ~ P(XM) are dense in L2(SL._a(k)\SL._I(A)). We will use this in the form of the following standard corollary. We repeat the proof for the convenience of the reader. Corollary. -- Let F(g) be a smooth function of rapid decay on SL._I(k)\SL._I(&). Suppose that :s~_~,~,\S~_x,A, F(g) E~(g; s) dg = 0 for all Einstein series E~(g; s) as M runs over all Levi subgroups ofSL._I, a all unitary cuspidal representations of M(A), ~ e I(a), and all s in a Zariski open subset of X + . Then F(g) - 0. Proof. --Since F(g) is smooth and of rapid decay it lies in L~(SL._ a(k)\SL._ I(A)), and hence by Theorem $3 it suffices to show that I(%f) = :SL._I~k,~L._I,A, F(g) 0~, :(g) dg = 0 CONVERSE THEOREMS FOR GL n 175 for all 0., 1(g) as in the statement of that theorem. If we replace 0., t(g) by its expansion in terms of Eisenstein series from Theorem $2, we have I(%f) = fsL,_x,k,\sI~_~,A)f~*,~,= ~,,o, F(g) E,(g; s)f(s) ds dg. Since F(g) is of rapid decay, ] E~(g; s) [ satisfies a moderate growth estimate depending only on Re(s), and f is Paley-Wiener, we may interchange the order of integration to obtain I(%f) = f~,.,=~,.,, (fsT.._,,k,XST.._,,., F(g) g.(g; s) dg) f(s) ds, By our assumption, fsN_l,k,\sN_x~*, F(g) E~(g; s) dg = 0 except possibly on a set of measure zero in the set Re(s) = Re(s0). Hence I@,f) = .Is,.._l,k,\s~._l,A,. F(g) 0~,1(g ) dg = 0 and we are done. [] 6. Proof of Theorem 1 Let H = | II~ be an irreducible, admissible, not necessarily generic representation of GL.(A) whose central character o H is invariant under k � and whose L-function L(H, s) is absolutely convergent in some half-plane. By the Langlands classification for GL.(k~) at each place v there is an admissible induced representation ~.. of Langlands type such that II~ is the unique irreducible quotient of E, [7]. The representation ~ is induced of Whittaker type and ~., = H, only if H~ is generic [26]. The induced representation E, has a well-defined central character ~os~ and this will be the central character of any constituent of E,. In par- ticular ~, and H, will have the same central character. The point of introducing the ~ is that for non-generic representations like II~ their local L-function is defined through the L-functions of the ~ where an integral representation via Whittaker models can be used [22]. More specifically, from the definition of the local L-functions [24, 27], for every irreducible admissible representation % of GL,(k,) we have L(II~ X %, s) = L(E, x %, s) with a similar equality for the ~-factors. Therefore if we consider the admissible repre- sentation E = | E~, this representation will have the same central character as H and its L-function will be nice for all twists by every cuspidal automorphic representation -r of GL~(A) for all m with 1 ~< m ~< n -- 1. It has the extra advantage that it is induced of Whittaker type. If H was generic to begin with, then H = E. 176 J. W. COGDELL AND I. I. PIATETSKI-SHAPIRO For the convenience of the reader, let us recall how the local L-functions L(~, � %, s) are defined through their Whittaker models. The representations E~ and % are both induced representations of Whittaker type and thus have Whittaker models. For each pair of smooth functions W~(g) e ~f'(~, d?,) and W',(g) e#'(-%, +;-1) there is associated an integral ~(W,,W',;s) =fN W~(0 0 )W',(g) 'det(g)]"-'"-~'/~dg which is absolutely convergent for Re(s) >~ 0 by the estimates in Section 4. Ifk~ is non-archimedean then by Theorem 2.7 of [24] we know the following. The integrals ~F(W~, W'~; s) define rational functions of q~-". As the functions W~ and W'~ run over their respective Whittaker spaces, this family of integrals form a C[q~", q~-"]-fractional ideal in C(q;-'). The local L-factor L(~, � %, s) is the generator of this ideal of the form L(~, � %, s) = P(q~-')-I with P(X) e C[X] a polynomial having P(0) = 1. Moreover, these integrals satisfy a local functional equation of the form 9 (W~, W~;' s) CF(p(w.,,~) W., W.,~'" 1 -- s) L(no x .~, s) r x .., s, +.) o~.o(- 1) ~ = L(H~ x ~%, 1 -- s) In this functional equation, the function UF(Wv, W'~; s) is defined by the integral N t ,r(wo, w,; s) W;(g) I det(g)] "-C~-~'/2 dx dg, m(kv) \GLmlkv) ' m(kv) 0 where k =n--m-- 1. (Note that if re=n-- 1 then CF(W~,W',;s) =tF(W,, W',;s).) The Whittaker functions involved are ~r =W,(w ~g -1) e CC(~,t~-l), and W'(g) = W'~(w,, ,g-l) a~(v~, ~,), where ~%~' is the representation of GL,(k,) on the same space as .~, but with action E',(g) =- E~(tg -1) and similarly for v',. The Weyl elements involved are 1 1), W r the longest Weyl element of GL r and (i ~ 0) As before, p denotes right translation in the Whittaker model. These integrals have the same analytic properties as the u~(W~, W'~; s). The s-factor is of the form CONVERSE THEOREMS FOR GL n 177 r � %, s, +~)= Aq; -z* tbr appropriate constants A and B. The local functional equation is also written as = W~; 1 --s) 'V(W~, W;;s) 7(Zo X "~, s, +o) ~('Wo, ~' where y(E~ X %, s, d?,) ---- t%(-- 1) '~ s(E, X %, s, +~) L(E; x "r~, 1 -- s) L(E~ X %, s) If the local field k~ is archimedean, then the integrals tF(W~, W~; s) extend to meromorphic functions of s. For the L-function L(~ � %,s) and the s-factor s(~, � %, s, +~) we may take the L-function and e-factor of the nm-dimensional repre- sentation of the local WeLl group associated to the pair (E~, %) by the archimedean local Langlands correspondence as in [5, 27, 29]. The ratio W(W~, W~; ' s)/L(~ "~ � ,., s) is again entire and satisfies the same functional equation as in the non-archimedean case. These results are all due to Jacquet and Shalika and the details can be found in [27]. To prove Theorem 1, let us begin with an arbitrary ~ s V=. Since V z is linearly spanned by decomposable vectors we may assume that ~ is decomposable, i,e., ~ -= | ~, with ~ e Vz. Each E~ has a unique Whittaker model W'(E~, d~) and to each ~ is associated a function W~(g~) eqC/'(E,, +~). Now to ~ e V z associate the global function W~(g) = II, W~,(g~). Since ~. is the distinguished unramified vector ~0 in Vz, for almost all v and g, e GL,(o,) for almost all v, this product converges absolutely to a continuous function on GL,(A). The function W~(g) is left invariant under both N.(k) and Z,(k). Consider the sum U~(g) = ]~ W~(vg ) = ~ W~ g . ((; 7 E Nn(k)\Pn(k) V' ~ Nn-l(k)\OLn-t(k) From the global gauge estimate of Lemma 4.1 we may estimate U~(g) and find the following. Lemma 6.1. -- The sum Ur converges absolutely and uniformly on compact subsets to a continuous function on GL,(A). Moreover it is cuspidal along the unipotent radical of any maximal parabolic subgroup of GL, containing B,. If k is a number field, f~ a compact subset of GL,(A) and c > 0 there exists t o suck that if t >t t o then there is a constant c' with property that I W~(ao)] .<. c' II l ada,+t I -"+''"-x-*' I det(a)l a/" i=l for o~ e f~ and a (al a) satisfying [ a,/a,+ 1 I>1 c for 1<<. i <<. n- 2, where d is suck tkat l co=(a) I = l a I a. Proof. -- This is just Propositions 12.2 and 12,3 of [22]. 1:3 23 178 J. w. COGDELL AND I. I. PIATETSKI-SHAPIRO Lemma6.2.--Lel]tEGL._l(A) andcon$idert]lefunctionU~(h 0 0l).TkozU~(h 0 ~) is a ra?idly decreasing automorphic function on GL._z(& ) and furthermore it satisfies the estimate IUr 0 ~) I ~< c, [ det(k) [ -' for suffdently large t > O. m That Ur (h 0 ) is automorphic follows from the formula Proof. "Y' E Nn_ l(kl\GT'n- l(k) First, assume that k is a number field. Then, by reduction theory for GL._I(A), we may write h = yaco where to eft, a compact subset of GL~_I(A), y ~GL._I(k ) and a= with [a~[a~+ zl >c for some c and i= 1,2,...,n--2. a,_ 1 Then from Lemma 6.1 we have (setting a, = 1) the estimate g~ <~ c~ II [ ai/ai+ 1 [-ti+it,-1-il i det(a)f." 4=1 Since the ratios aJa~+ z for 1 <~ i ~< n -- 2 are the simple roots of GL~_a, this shows that Ur (h 0 01) is rapidly decreasing on GI,,-z(A). n--1 On the other hand, since Idet(a) l = II l ada,+l 1' we see that i=1 II [ aJa, +1 1--t, = ] det(a) I-' i=1 [II aJa,+l I '"-1-' c I det(a)I '"-1' 4=1 and therefore ]U~(hO 10)<~c;[det(a)[ -'+''-1'+'~"' ~< c;, I det(a)1-" for t' > 0. Since k = yao~ with [ det(y) [ = 1 and [ det(~) [ bounded for ~ e f~, this gives the estimate when k is a number field. 179 CONVERSE THEOREMS FOR GI n Now assume that k is a function field having a finite field of q elements as its field of constants. It is easy to see from the transformation property defining the Whittaker function that there is a sequence of constants c ----- { r } with c~ = 0 for almost all v such that ff W~(g) 4= 0 with g =: nak, where n e N.(A), k e K = II GL~(oJ, and then [aJ.+l] <..qCvfor l<..i~n--l. Taldnga,~-~l weseethatWg(h 0 ~)vanishes identically for [det(k)[ sufficiently large and so the same will be true for U~(h 0 ~). This establishes the estimate for [ det(h) I large. On a set { h e GL._.I(A) [ I det(h) I = q" } of matrices with fixed determinant, the function U~ (h 0 ~} is compactly supported mod GL._I(k), and hence is rapidly decreasing as an automorphic form on GL._I(& ). To see this, recall that by the reduction theory for GL._I(A ) [16] there exists a set of constants X = { X, } with Xo = 0 for almost all v and a compact subset ~ C N._I(& ) such that, if we set ~(X, a) = { k = nake GL._I(A) [ a = diag(at, ..., a._~) e A._t(A), neff, kel-IGL._l(o.) with for l~<i~<n--2}, then GL._t(/k ) = GL._x(k) ~(~., f~). Hence it suffices to prove that U~ (h 0 ~) has compact support in ~,(~,, f~) = { h e ~(~,, f~) [ I det(h) l = q' }. On such a set if suffices to prove that l al l is bounded if Ur (h 0 :)4:0. IfU~(k 0 01)*0 then there must be a y e GL._I(k ) such that W~ 4= 0. Write y = (y~, 5) as a (,: 01) matrix. First we assume that Y.-1,1 4= 0. Write yh = ynak. It is easy to see that the (n--1, 1) entry of "r~ is y._l, la~. Write yna =bk' with k' eK and b EB._I(& ). Then we have 1Y.-1,1 al I~ ~< [ b._1,._1 [. ~< q~v for all places v. Hence laxl = I Y.- ,laa I = IIl < IIq~ where [ c[ = ]~ c~. The general case proceeds in the same way using the first y~,t such that %,1 4= 0 and u = 0 for all k> ~, giving [ a~ I ~< qlel where co = c~ . 180 J. w. COGDELL AND I. I. PIATETSKI-SHAPIRO Finally, the polynomial estimate for small determinant is now a consequence of the reduction theory and the gauge estimate of Lemma 4.1 as in the number field case. [] Before we proceed, let us note the following. Lemma O. 3. -- The function U~(g) /s not identically O. Proof. -- If we compute the +-Fourier coefficient of U~(g) we find IN U~(ng) +-l(n) dn = I~ ~ W~(yng) +-l(n) dn n(k)\ ~Tn(A) ntk)\ NntA) = 5] W~ ng +-l(n) dn. (k) \ Nn(A ) We now proceed by induction. Let N" C N, be the unipotent radical of P~, i.e., N" = In-1 . 0 ... 0 Then since N" is normal in N. we may integrate over it first, and the formula for an individual term in the b-Fourier coefficient becomes fN"(k)\ N"(A) Now, GL,_I normalizes N" so +((0 But +-l(n") an" = o v' r P._l(k). 1 Y' e P,~_x(k) Hence this term vanishes unless y' c P._l(k). We now proceed by induction in this way and finally conclude f U~(ng) +-l(n) dn = W~(g). Since W~(g) is not identically 0, because ~ ~ W~(g) is injective, this shows that U~(g) cannot be identically 0. [] fNn_l(k)\Nn_l(A) Nn-l(k)\GLn-l(k) Nn(k)kPn(~) CONVERSE THEOREMS FOR. GL n As a function on GL.(A), U~(g) is left invariant with respect to P.(k) and Z.(k) and hence with respect to the full parabolic subgroup P'.(k) associated to the partition (n-- 1, 1) ofn. Consider now a second function V~(g) associated to ~. Let us set ((; :)) Y E.-" N~(k)\ Pn(k) y' ~ lff n_ l(k)\ OL n _ x(k) where the notation is as in Section 3. To deduce the analytic properties of V~ from those above for Ur let us recall that if we set W~(g) = W~(w. tg-1) then ,~r is in the Whittaker model for ~'. Then we have where 1(w._11) ~'=~ ~. -101) and We may conclude that V~(g) converges absolutely and uniformly on compact subsets to a continuous function an GL,(A). It is left invariant with respect to P~(k) = tP',(k)-1. The function V~(g) does not vanish identically. Furthermore, if we consider the function v~(h 0 ~) on GL.- l(&) it is a rapidly decreasing automorphic functi~ ~ GL,- l(&)- The only difference is that our determinant estimate becomes for t>O. These facts follow from Lemma 6.2. From ~ we have produced two functions on GL.(A), one invariant under P'~(k) and the other under P',(k). Note that together P'~(k) and P~(k) generate all of GL,(k). To relate U~ and V~ to the L-function consider the following integrals. Let x be an irreducible proper automorphic subrepresentation of GL,_I(A ) and let 9 ~ Vs. Set I~n - 1 (k)\ GL n _ 1(A) 182 J.W. COGDELL AND I. I. PIATETSKI-SHAPIRO As a function on GL~_I(&), ~0(h) is of moderate growth and transforms via a central character o~. On the other hand, U~ (h 0 01) is rapidly decreasing on GL,_ I(k)\GL~_ ~(A) and, in terms of the determinant, satisfies lUg(: ~)l,<c, ldet(h)l-'forevery t>O. Hence I(~, ~; s) converges absolutely for Re(s) >> O. On the other hand, if we unfold the series defining U~ (h 0 ~) we find I(~, q~; s)= Io Ut (: 01)~o(h):1 det(h)I'-a/2'dh IJn_ l(k) \O]',n_ l(A) 1,A detlh ,s ,1,2 = ~(W~, W~; s), where WAh) = / ~(nh) +(n) an n- l{k)\ Nn- 1{ A) i.e., W~(h) eYCz(% qb-1). Hence we have: Lemma 6.4. -- For any % irreducible automorphic subrepresentation of GL~_I(A), and o eV, the integral I(~,~0;s) converges for Re(s)>>0. Moreover, in this range, I(~, ~; s) = ~F(W~, W~; s). Hence 1(4, ~; s) = 0 if\ is not generic. Similarly, for V~ we may define the integral ]~(~. q~; s) = fo 'v~(h 001)~(k)[det(h)'8-a'2'dh Ln_l(k}\GLn_ 1( } which will converge for Re(s) ~ 0. If we unfold this, then we find T(~, ~; s)= f~._.,A,\o._~,. ~r~ (; Ol)'W~(h)'det(h)[n-"-n'~'dh CONVERSE THEOREMS FOR GL n where, as before, we set W~(g) = W~(w, ,g-X), ~,(h) = W,(w._, 'h-'). Hence we have proven: Lemma 6.5. -- For v any irreducible automorpkic representation of GL,_~(A) the integral I(~, ~; s) converges for Re(s) < 0. Moreover, in this range, I(~, ~?; s) = W(~V~, ~; 1 -- s). Hence I(~, ~; s) = 0 if .~ is not generic. Both of these families will have an analytic continuation to entire functions of s, bounded in vertical strips. To see this we must relate these global integrals to the global L-function. We will work with I(~, ~; s) in detail, then I(~, q~; s) proceeds in the same way. Proposition 6.1. -- The integral I(~, ~ ; s) has an analytic continuation to an entire function ofs. Proof. -- We will consider two cases, although this is not really necessary. First, assume that 9 is cuspidal. We take I(~, q~; s) = ~F(W~, W~; s). Assume that and q~ are decomposable. (This is possible since the decomposable vectors span V= and V,.) Then we have I(L ~; s) = II ~'(w~, w~o; s). Now, from the local theory of L-functions ~F(W%, W~o; s) = E~(s) L(E~ x ~, s) is an entire function of s. If v is non-archimedean, E,(s) eC[q~, q~-'] and if both ~ and % are the distinguished unramified vectors, which is true for almost all v, E.(s) -- 1 [24]. If v is archimedean, then E~(s) is an entire function of s [27]. Hence, setting E(s) = l-i v E,(s), we find I(L ~; s) = 11 L(~ � ~o, s) E,(s) = L(Z � ~, s) E(s). So if v is cuspidal, then by our hypothesis on L( ~ � v, s) we have that L(E � v, s) is an entire function. The same holds true of E(s) and hence for 1(4, ~; s). Now suppose v is not necessarily cuspidal. Since I(~, q~; s) - 0 unless 9 is generic, we may assume that v is an irreducible generic automorphic subrepresentadon of GL,_ I(A). Then by the work of Langlands [34] there exists a partition (q, ..., r~) of n -- 1 and irreducible cuspidal representations % of GL,i(A) such that x is a subrepresentation of T = I,,~OL,-lCAIt_ C~ | %), where Q is the standard parabolic associated to this partition. The theorem as stated in [34] only gives ~ as a subquotient of T. But if one begins with v an automorphic subrepresentation, the proof presents v as a subrepresen- 184 J. W. COGDELL AND I. I. PIATETSKI-SHAPIRO tation of the induced Y. Let us sketch Langlands' proof, referring to [34] for more details. Langlands begins with the realization of the automorphic representation ~ on V/U where V is a space of automorphic forms generated by a single form q~ and U is a subspace of V. Since we are assuming that T is an irreducible subrepresentation of the space of automorphic forms, we may take V irreducible and U ---- { 0 }. Langlands then realizes in the space of constant terms q0p ~ Vp of the forms q0 ~ V along a suitably chosen para- bolic P with Levi M. Since our V is irreducible, we get a realization of'r on a subspace Vp of these constant terms and Up : { 0 }. Again, using the fact that Up : { 0 ) for this realization of r the argument of Lemma 6 in [34] produces a generator ~r of Vp such that r ---- x(a) 9v(g) with ~( a character of the center of M(A) for all g e G(A) and a in the center of M(A). Langlands then projects each ~0 v e Vp to a function ~v in the space of constant terms transforming by an irreducible cuspidal representation ~ of M(A) having the central character ~(. Since Vp is still irreducible as a G(A) representation, this mapping is an injection and realizes Vp as a subspace V~ of these functions, i.e., a subspace of the induced representation from this cuspidal representation ~ of M(A) to G(A). Taking G = GL,_ 1 and P ---- Q we obtain the conclusion stated above. Now, locally for each place v of k, % will be a generic irreducible subrepresentation of Y, ~--- .XtUQ(kv )]'-'~G:Sn-x(~v)t-kul, ~ ~ ... | Since each ~r~ is cuspidal and hence generic, the local components ~, ~ must also be generic. Then the results of Rodier [40] and Jacquet [19] imply that each T~ is of Whittaker type, that is, has a one-dimensional space of Whittaker functionals. Hence it has at least one generic constituent. If k, is non-archimedean, the results of Bernstein and Zelevinsky [4] imply that there is a unique generic constituent and so T, must be it. If k. is archimedean, then using the Casselman subrepresentation theorem for each %, and the transitivity of induction we can embed Y, into a representation Y'~ which is induced off the Borel subgroup. Now the results of Kostant [31] imply that T'~ has a unique generic constituent. Since Y, is a subrepre- sentation of Y~ we have that Y~ can have at most one generic constituent. Since we already have seen that it has one generic constituent, namely "r,,, we have that % must be the unique generic constituent of Y~ in the archimedean case as well. % cannot lie in the kernel of the map from Y~ to its Whittaker model, since if it did this would imply that Y~ would have at least two generic constituents. Hence, the Whittaker model of % will be a subspace of the Whittaker model of Y,. In particular, the family of integrals defining L(E~ � %, s) will be a subspace of those defining L(E~ � T~, s). At those places where % is unramified, these families agree. Hence from the computation of the local L-functions in [24] we see that at all non-archimedean places we have and if E, and % are unramified then CONVERSE THEOREMS FOR GL n 185 Hence for v non-archimedean we have L(~o � 7,, ~) = ~I L(ao � ~,,., s) E:(~), where E'~(s) is endre, bounded in strips, and identically one for almost all v. If o is archi- medean then from [27] we have L(z~ � 7. ~) = fl L(=. � ~,,,~, s). Hence, globally we have ,m L(~ x 7, ~) = E'(~) 11 L(~ x ~,, ~). By our hypothesis, each L(E � ,~, s) is entire and bounded in strips since each ,~ is cuspidal. The same is true of E'(s) = I-i v E~(s). Hence it is true of L(E X 7, s). (We will use the boundedness in strips in the proof of Proposition 6.3.) If we now write I(~, ~; s) = L(~. � ~, s) F.(s) as above we see that I(~, q~;s) is entire as desired. [] Proposition 6.2. -- The integral I(~, ~ ; s) has an analytic continuation to an entire function ofs. Proof. ~ In this case we write Y(~, ~; s) = "I'(~r ~; 1 s). By the local theory of L-functions, we may relate this integral to the global L-function L(Et x 7~; 1 -- s) and proceed as before. [] We next relate these integrals, again using the properties of the global L-function --this time the functional equation. th/s Proposition 6.3. -- As entire functions of s, I(~, ,; s) = I(~, ~; s). Moreover, function is bounded in vertical strips. Proof.-- If we write these integrals as Euler products I(~, ~; s) := 11 ,v(w~o, w~0; s) I'(~, ~?; s) ---- ~ ~F(l~rcv , l~r; 1 -- s) J. W. COGDELL AND I. I. PIATETSKI-SHAPIRO then the local factors are related by the local functional equation [24, 27] v(W~o, w~v; s) V(~'~o, X~',po; 1 -- s) r x %, s, +,3 co.~,,(-- 1)"-: = L(~ x .~,l-s) L(E, X %, s) or 9 (W~v, W,v; s) y(g, x %, s, ~,) = ~(X~r~, X~r,,; 1 -- s). Therefore, taking the product over v, we have 9 I'(W~, w~; s) v(~ x % s) = 'I'(~, "~; 1 - s). Since L(E x % s) is assumed to satisfy the global functional equation, T(g X % s) --- 1 if-~ is cuspidal and hence I(~, $; s) = I(~, 9; s) in the cuspidal case. If'~ is not cuspidal, then we may assume it is generic (since otherwise both sides are identically zero) and hence, as in the proof of Proposition 6.1, is a subrepresentation of an induced repre- sentation where each o, is cuspidal and generic on GL,I(A ). Now, from the local theory, i=1 and hence globally ~(~ x % ~) = ~I ~(-: x ~,, ~). i=1 Now, by assumption, since each ~, is cuspidal the global y(g x ~, s) = 1. Hence in the case of non-cuspidal -~ we still have I(~, ,; s) = I'(~, 9; s). We need to show that this function is bounded in vertical strips. Note that from the integral representations, I(~, 9; s) is bounded in vertical strips in its half-plane of absolute convergence Re(s) >> 0 and T(~, 9; s) is bounded in strips in its half-plane of absolute convergence Re(s) < 0. To verify that it is bounded in any vertical strip we just need to see that it grows sufficiently slowly that the Phr~igmen-Lindel6ff principle applies. From the proof of Proposition 6.1, we have I(~., ,p;s) = L(~ x %s) IIS~(s). t~ The factor L(E � % s) is bounded in any vertical strip as in the proof of Proposition 6.1. The factor E~(s) is identically 1 for almost all places. At the remaining non-archimedean places E~(s) belongs to C[q*, q-'] and is thus bounded in any vertical strip. If v is archi- medean, then E,(s) = 'I'(W~o, W~o; s) L(~ x "~, s) CONVERSE THEOREMS FOR GL n From the local archimedean theory [27] the numerator decreases like 1 over a polynomial in s at infinity in vertical strips while the denominator is a linear exponential factor times a product of P-functions. Then Stirling's formula applied to this product of U-functions gives a bound on I L(E, � v,, s)[-1 of the form Ce a~ at infinity in any vertical strip, where we have written s = a 4- it as usual. Hence Phr~tgmen-Lindel/Sff applies to I(~, ~; s) and we may conclude that it is indeed bounded in any vertical strip. [] This concludes our use of the L-function. We now maneuver ourselves into a position where we can apply the weak form of Langlands' spectral theory for auto- morphic representations. For each idele a let us set and similarly for I1(~ , ~; a). The integrals I1(~ , @; a) and I1(~ , @; a) are continuous functions on kX\A x. Note that if we replace x by x | co for a (unitary) character co then I1(~, ~.co; a) = co(a) I1(~, ~; a) and similarly for Ix(~, ~; a). Hence we may write I(~,~.co;s) =f Ii(~,~;a) co(a)la[~-(1/2'd� for Re(s) >>0 Jk X\A� I(~,q~.co;s) =9 Ii(~,@;a ) co(a)la['-(lmd � for Re(s) <0. J~ x\A� We may now apply the following elementary lemma of Jacquet-Langlands [21, Lemma 11.3.1]. Lemma. -- Let fl and f~ be two continuous functions on kX\A x . Assume there is a constant c so that for all (unitary) characters co of k� X the integral fk fx(a) co (a) I a l" dX a x\ A� is absolutely convergent for Re(s) > c and the integral co(a) I a I' dX a is absolutely convergent for Re(s) < -- c. Assume that the functions represented by these integrals can be analytically continued to the same entire function and that this entire function is bounded in strips. Then fl and f2 are equal. 188 J. w. COGDELL AND I. I. PIATETSKI-SHAPIRO Therefore we may conclude that I1(~, ~; a) = I'x(~, ~; a) for all a e A � and all ~. In particular, for a = 1, we have U~ (h 0 ~)v(h)dh= Vr (h 0 01)~(h)dh. Note that since U~ and V~ are rapidly decreasing on GL._~(k)\GL._~(A) they are also on SL._I(k)\SL._I(A ) and hence Ur162 ~L*(SL._I(k)\SL._I(A)). Let F~(g) = U~(g)-- V~(g). Then F~(h 0 ~)israpidlydecreasing and (h 0 ~) ~(h) dh ~- 0 for all ~ occurring in irreducible automorphlc subrepresentations of GL~_I(A). Proposition6.4.--Wehave F~(h 0 01) =0. Proof. -- We wish to apply the weak form of Langlands spectral theory for SL,_ 1 as formulated in Section 5. Before doing so we must relate automorphic representations of SL,_I(A ) to the restrictions of automorphic representations of GL,_I(A ). By a result of Labesse-Schwermer [32] (see also Lemme 5.6 of Clozel [11]) given any irreducible cuspidal representation -r 1 of SL,_I(A ) there exists an irreducible unitary cuspidal representation -r of GL,_a(A) whose restriction to SL,_I(A) contains -q. The argument in [32] extends to yield that any irreducible cuspidal representation -r 1 of M 1 = S(GL,I(A ) � ... � GL,,(A)) extends in this way to an irreducible unitary cuspidal representation v of M = GL,I(A) � ... x GL.,(A). If we apply this fact in the construction of Eisenstein series, we find that for any partition (nl,..., n,) of n- 1 and any irreducible cuspidal representation a� of the Levi subgroup M1 = S(GL,I(A) � 9 .. � GL.~(A)) the space of Eisenstein series E,(h; s) for ~ c I(~i) and s ~ X+ is obtained by the restriction of Eisenstein series on GL._I(A ) formed with the extention a of ai to M = GL.I(A ) � ... � GL,,(A). In the realm of absolute convergence for these GL._ I(A) Eisenstein series, the induced representations are irreducible for all parameters in a Zariski open subset, and hence for these values of the parameter the Eisenstein series generate irreducible automorphic subrepresen- tations. Hence for all s in a Zariski dense subset of X+ the Eisenstein series E,(h; s) are obtained from restriction of irreducible automorphic subrepresentations of GL,_I(i ). As a consequence, we see that fSLn_x(k)\SLn_liA)F~ fs~_~(~)\sr~._~(A~ Is~_~(~\s~._lCA) CONVERSE THEOREMS FOR GL n for all Eisenstein series E,(h; s) as M 1 runs over all Levi subgroups of SL,_ 1, al all unitary cuspidal representations of MI(A), ~ e I(al) , and all s in a Zariski open subset ofX+~. Hence by the Corollary to Theorem S3, F~(h0 ~)-0. D Since F~(h 0 ~)--0, we have that u~(h 0 ~)= v~(h 0 ~)for all heSL,_~(A) and in particular U~(1) = "V~(1). Since this is true for all ~, U~(g) = Umo,~(1 ) ----- Vr~,g,~(1) = V~(g) for all g e GL,(A). We now have that U~(g) is invariant under P,(k), P,(k), and Z,(k). Since these generate GL,(k) we see that U~ e ~r Thus the map ~-*U~(g) embeds "z into d(GL,(k)\GL,(A)). Hence E is an automorphic sub- representation. In fact the map ~ ~-~ U~(g) embeds E in the space of cusp forms. To see this, we must show that for any parabolic Q, with unlpotent radical NQ, the constant term of U~ along Q is 0, i.e., f~ U~(ng) dn = O. q(k)\ Nq(A) Since U~(g) is left invariant under GL~(k) and all k-rational Borel subgroups of GL. are conjugate under GL.(k) it suffices to compute the constant term along the unipotent radicals of standard parabolic subgroups O.~D B., so that N-Q C N.. If Q' is a maximal parabolic subgroup such that Q' D Q.D B., then N a, is a normal subgroup of NQ and in computing the constant term along Na we can integrate along NQ, first. Hence to show that U~(g) is cuspidal it suffices to show that it is cuspidal along the unipotent radical of any standard maximal parabolic subgroup. But this is guaranteed by Lemma 6.1. Hence U~ e ~r176 i.e., U~ is cuspidal for every ~ and hence ~ is cuspidal. As a constituent of ~., H will then be cuspidal automorphic as well. However, we can say a little more. Since H is cuspidal, it is generic. Thus each local component II, is generic. But as we have pointed out, when II, is generic, II, = ~, ~. Hence 1I ~ [] 7. A second converse theorem Theorem 1 is a generalization of results of Jacquet and Langlands for GL(2). It gives the most information about H, namely that it is not only automorphic but also cuspidal. However Theorem 1 requires information about L(II � v, s) which is usually not available. More precisely, in Theorem 1 we assume that L(H � % s) is entire for twists by all euspidal automorphic forms on all GL~(A) with m < n. It is very difficult to obtain such information. Andr6 Weft, even before Jacquet-Langlands, suggested a 190 J. w. GOGDELL AND I. I. PIATETSKI-SHAPIRO different method of proving this type of theorem, which will allow us to obtain a result suitable for applications. In the method of Weil the first step is the construction of some periodic holomorphic function which is supposed to be an automorphic form. From given information about the functional equations satisfied by the associated Dirichlet series and their twists, Well derived the conclusion that this function was an automorphic form with respect to some congruence subgroup. In the following Theorems 2 and 3 we will follow the method of Well disguised in the language ofautomorphic representations. For each finite set of places S of k containing all archimedean places and for each integer m, let ~s(m) = { n : zc is an irreducible generic automorphic representation of GL,,(A), unramified at all v r S }. Similarly, let ~~ be the set of cuspidal elements of f~s(m). Theorem 2. -- Let n >1 3. Let II be an irreducible admissible generic representation of GL,(A) whose central character o~ n is invariant under k x and whose L-function L(I-I, s) is absolutely convergent in some half-plane. Fix a non-empty finite set of places S of k containing all archimedean places such that the ring o s of S-integers of k has class number one. Suppose that for every m with 1 <<, m <~ n -- 1 and every ,r E f~~ ) the L-function L(II � v, s) is nice. Then there exists an irreducible automorphic representation II' of GL,(A) such that II, ~_ II'~ for all v ~ S and for all non-archimedean v such that I-I, is unramified. In order to prove Theorem 2, we will first use the framework of Theorem 1 to construct an embedding ofII s in the space of smooth functions on rs\G s for a congruence subgroup r s of G s with respect to an appropriate Hecke algebra. Let us recall that according to the general Duality Theorem [ 13], it is known that " classical " automorphic forms with respect to a group r are in duality with embeddings of given irreducible representations of GLz(R) into the space L2(I'\GL2(R)). In the case n I> 3 there is a simplification compared with Well's theory, which in fact says that the set of assumptions (i.e. necessary twists) does not depend on the conductor of the representation II. The reason for this simplification is that the congruence subgroup theorem is true for SL, for n/> 3. There are two extensions of this which follow after some extra arguments. Currently they are separate statements, but we hope that they will eventually coalesce. Corollary 1. -- With the hypotheses of Theorem 2, there exists a proper automorphic repre- sentation 17" with II'~' ~_ l-I, for all non-archimedean v for which II, is unramified. The next Corollary is the one which is most useful for the application to Langlands' lifting. Corollary 2. -- With the hypotheses of Theorem 2, there is a unique irreducible generic auto- morphic representation II" such that II'~' ~ II,for all v ~ S and all non-archimedean v for which I-I, is unramified. GONVERSE THEOREMS FOR GL n 191 8. The conductor of a representation Let II = @ H, be an irreducible admissible generic representation of GL,(A). Let S be a finite set of places ofk containing all archimedean places. For almost all places ~ S, the representation II~ is unrarnified, that is, II~ contains a vector which is fixed by the maximal compact subgroup K~ = GL,(o~). This vector is unique up to scalar multiples. Let T denote the smallest finite set of places containing S such that II~ is unramitied for v r T and let T' = T\S. So T' is the set of places not in S for which II, is ramified. For those places v ~ T', it is known from [23] that there is a unique integer m, > 0 such that if we set Kl,,(p~) := g e GL.(o,) : g =- (mod p~) 0 ... 0 then the dimension of the space of Kl,~(p~o)-fixed vectors in YI~ is one. Set m~=0 for veT. We will call the compact subring n= II p~'~CA s the S-conductor of II. If S is precisely the set of archimedean places, hence is empty in the function field case, we will call rt the conductor of II. It determines (and is determined by) an ideal of o s by n s = k c3 k s rt C o s . To simplify notation we will denote rt s simply by u, since they can be distinguished by context. Note that Ds[u -~ 1-I~s odP~ ~. If we set Kx(u) = g e,~s GL.(o~) : g = (mod u) 9 0 ... 0 = II Kl,,(pr 0) c G S, then the dimension of the space of K1(u)-fixed vectors in II s is exactly one. We may similarly define (mod p~) K0,,(p~) = g ~ GL.(o~) : g - 0 ... 0 1 E 192 j. W. COGDELL AND I. I. PIATETSKI-SHAPIRO and (mod n) Ko(n ) = ge II GLn(o.):g--- ,$8 0 ... 0 = II Ko, o(p~ ) C G s. The group K~, o(p~ .) will then be a normal subgroup of K0, o(p~ ~) with abelian quotient given by Ko, o(p~)/Ki, o(p~ .) _ (oo/p~) � and Ki(rt ) is a normal subgroup of Ko(rt ) with quotient I-[**s(oJp~o) � ~ (osflt)� Then the action of Ko(rt ) will preserve the one-dimensional space of Kl(rt)-fixed vectors and act on it by a character of K0(rt ) trivial on Ki(~t). It is easy to compute the action of K0(n ) on the space of Ki(rt)-fixed vectors. Let C ~ be a non-trivial Kl, o(p~,)-fixed vector in H o for v ~ S. Then the tensor product C ~ = (~ C ~ is a non-trivial Ki(n)-vector in II s. If v r T then Kx, o(p~ ) = K0,,(p~o) = GLn(o,) and so for go e Ko, o(p~ o) we have Ho(go) ~o ___ ~o. If v e T' and g. = (g~, s) e Ko,.(p~) then from the congruence condition we have [ gn. j [~ < 1 for 1 ~< j < n. Since go e GL.(oo) we must have max{[g.,~[o}= 1 and hence [g.,nlo= 1 and so gn, ne0~ x. Then we may write go = (g,, n In) g~, o with g~, ~ e Kx, o(p~). Then iio(g~, ) ~o = iio(g,, ' n In) ~o = O~,,(gn. n) C ~ where % is the central character of H,. So we may define a character Z = @ X, of K0(rt ) by xo(g~) = 1 if v ~ T and xo(go) = c~ n) if v e T'. This is guaranteed to be a character by construction. If we wish to emphasize the dependence on the central character co of II we will write X = Xo,. We have IIS(g) ~0 = X,o(g) ~o for g e K0(rt ). There is another useful construction of X,~. Consider the central character co of H. If v r T then for any local unit u, ~ o~ x we have u, In ~ GL~(oo) = Kl.o(P~V) and so c%(uo) C ~ = IIo(u o I,) C ~ = ~o so that %(u,) = 1. Similarly, ff v e T' and u o is a local unit of the form 1 + p~v then o~,(uo) = 1. So co o is unramified at v r T and has conductor at least p~v at the places v e T'. Since (Os/rt) � _ IIo(oo/p~ v) x, the character o~ defines a character Xo of (Os/n) � via this isomorphism by X~ ----Iioes %. Then, through the isomorphism K0(rt)/Ki(n ) -(os/rt) � this character X,o defines a character of K0(rt) trivial on Kx(rt ) which is easily seen to be the same character as defined above. Hence we could write zo(g) ---- xo(g,, ,) for g = (&, j) ~ K0(n). CONVERSE THEOREMS FOR GL n 193 9. Generation of congruence subgroups Let n >/ 3. Let S denote a non-empty finite set of places of k containing all archi- medean places, Let os denote the S-integers of k. Since OL.(os) = GL.(k) n Gs HsGL.(o,) , we may view GL,(os) as a subgroup of GL,(k) embedded in G s. Then GL,(os) is a discrete subgroup of G S. For the proof of Theorem 2 we will need a preliminary result on the generation of certain congruence subgroups of GL~(os). The heart of this proof is Lemma 9.1 which is extracted from the proof of Theorem 4.2 of Bass [1]. This result from the stable algebra of GL, plays a role in the solution of the congruence subgroup problem for SL, [2]. This is the place where the restriction n >/ 3 comes from, as in the congruence subgroup theorem. Let T' be a finite set of places disjoint from S and let T = S w T'. For each v e T' let m~ be a positive integer and for v ~ T set m~ = 0. Let n = I-I~ $ s P~~ C k s. As in Section 8, n defines an ideal, again denoted 1t, in a S. The congruence subgroups of GL,(os) we are interested in are (mod 1I) Pl(rO = 7eGL.(os):7- 0 I and P0(n ) ---- y~GL.(os):7 = 9 (mod 11) I" /o 0 If we define Ki(n ) C G s for i = 0, 1 as in Section 8 then we can also characterize F,(rt) by P~(n) = GL~(k) c3 Gs.K~.(n ). Consider the following subgroups of F,(rt). Set P;(os) = P'.(k)n P'.(k)n G.K S, where as usual we have set K S = 1I,$ S K,. This is the set of all matrices in GL~(os) whose last row is of the form (0, ..., 0, ,) ifi= 0 or (0, ..., 0, 1) ifi = 1. It is inde- pendent of 11. Set P~(rt) = P'~(k) c3 Gs.K,(rt ). This is the subgroup of F,(n) consisting of those matrices whose last column is t(0, ..., 0, ,). There is a congruence condition on the last row of these matrices. 25 194 J.W. COGDELL AND I. I. PIATETSKI-SHAPIRO Proposition 9.1. -- The groups P'(os) and P'(rt) together generate the congruence sub- group P,(rt) fir i = 0, 1. --t For now, let A,(n) denote the subgroup of GLs(os) generated by P~(os) and P~ (n). Note that Al(n ) C A0(rt ). Lemma 9.1. -- Let (a 1, ...,as)~o~ be a unimodular sequence suck that (al, ..., a,)- (0,..., 0, d)modrt. Then there exists an element 7 e Aa(tt) such that (hi, ..., as) "~ = (0, ..., 0, d). Proof. -- The sequence (al, ..., a,) is unimodular in the sense that there exist cl, 9 9 c. in os such that 1 = ~c~ a,. Therefore al = ]~a 1 c, a t = a 1 cl a~ q- 1~=.~ a~ ci a~. If we substitute this expression for a 1 into 1 = Y~c, a~, and let q = a 1 c~, we find 1 =c lqa l-k- ~ c~(q+ 1) a~. S=2 Since a 1 e 1t, we have al ca = q ~ 1t, and we see that the sequence (qal, as, ..., as) is again unimodular. Since 0s is a Dedeldnd domain, n = 2 defines a stable range for D s in the sense of [1]. (Note that there is a shift of one in the definition of stable range between [1] and [2].) This implies that there exist a'~ = a~ § b~ qal with b~ s Ds such t ! that the sequence (a2, ..., a,) is unimodular. Let t ! Then (a,, ..., as) v a = (al, as, ..., as). Note that we still have t t (al, a=, ..., a,) - (0, ..., 0, d) (mod n). r t p t Since (a~,...,a,) is unimodular, we may write 1 =Z"~=2qa~. Write a' s=d+q. t t t with qs e 1t. Then we have q, -- ale 1t and by the unimodularity of (a~, ..., a's) we may write this element as q', -- al = Y~,"=2 di a" with d i e ft. Now let T~ = 9 kd, 0 ... 1 ' ~,) ,~ (q',, a~,.., Then v 2 e P[( n) since d s elt. So (al, a~, .. ., = ' . a's) . Now set (11) CONVERSE THEOREMS FOR GLn 195 t i e i l so that (q,, a~, ..., a,) = =-- (q',, a=, ..., a,_l, d). Note that we still have t t , a t q,,a~, . .~ ,-1 err. So if we set 7 8 ~--- 9 . 9 -- ~._~ - - r then % ~P;(u) and (q',, ~, ..., a',_l, 1) % = (0, ..., O, d). Therefore (al, ..., a,) 71 7 2 ~7 s = (0, ..., 0, d). Since 71, ~ ~ Pl(os) and 72, % e P~(u) we see that 71 7~ aTs E AI(U ). [] Proof of the proposition.- Since P;(os), P;(u) C F~(u), it is clear that A,(u) C P,(u). Now let y e F~(n), so y - (mod u) 0 ... 0 with d = 1 if i = 1. Let u = det(y). This is a unit in Os and the diagonal matrix diag(u, 1, ..., 1) is in P~(Os). Then diag(u -1, 1, ..., 1) Y has determinant 1. Hence its last row is unimodular in the sense of the lemma and we still have y - (mod u). 1 0 ... 0 Now, by our lemma, there exists Y1 ~ AI(u) such that YYI = = P" 1 0 ... 0 But then p e P~(Os). Hence Y = diag(u, 1, ..., 1) py~ -1 e P~(Os) Al(u) C A,(u). [] 196 J. W. COGDELL AND I. I. PIATETSKI-SHAPIRO 10. Proofs of Theorem 9, and its corollaries Let u denote the S-conductor of H and co = to n its central character. Let t ~ = @~s t ~ be a non-zero Kl(rt)-fixed vector in II s as in Section 8. So t ~ transforms by the character Z~ of Ko(rt ) as in Section 8. Proof of Theorem 2. -- For each ~s ~IIs consider the functions U~| ) and V~| to(g) associated to the vector ~ = ~s | ~0 ~ II. The function Ut~| is left invariant under P,(k) and V~| to is left invariant under P.(k). Both are invariant under Z,(k). In addition, U~| and V~| are right invariant under Kl(n ) for all ~s ~ IIs. Now, if we restrict these functions to GL,_I(A ) we find: Lemma 10.1. -- In addition to the properties from Lemma 6.1 and Lemma 6.2, the t x # (: )are a, ,e .re #naion k- invariant under KS ,-1 = II,~s GL,-I(~ Proof. -- For v such that II, is unramified this is clear since t ~ is fixed by GL,(o,) D GL,_x(o,). At the remaining places t ~ is fixed by K~,~(p~,)DGL,_~(o,). [] Now consider the integrals I(~ s | t ~ q~; s) and ]'(~s | t ~ q~; s) for q~ lying in a proper automorphic representation v of GL,_I(A ) as defined in Section 6. Since U~| to (h 0 ~) is unramified, we find that I(~s | ~~ ~; s) = f~ Ut~| to (h 0 ~)*?(h)ldet(h)] *-nmdh L._ l(k)\OL. _ llA) = fl%_ gn)\Gi~_ltn)Wts @ t~ (h 0 ~)W~ (h)[det(h)''-(lmdh K,_ a. The same is true for Hence I(~ s | ~o, q~; s) - 0 unless v contains a vector fixed by s ~(~s | ~ Hence, if vr 1), then I(~ s| ~ =0=I(~s |176 On the other hand if-c e~~ 1), then by our assumption on the L-function we have as in the proof of Theorem 1 that I(~ s | t ~ c?; s) -- I(~s | t ~ c?; s). If v ~ fls(n -- 1) but is non-cuspidal, then v must still be generic for the integrals to be non-zero and as before we have that v is a subrepresentation of a representation induced from cuspidals. Since 9 is unramified for v 6 S and generic, these cuspidal representations must also be unramified for v $ S and generic. Then arguing as before, we find that, in this case as well, I(~ s | t ~ q~; s) ----- "I(~s | t ~ c?; s). Hence we have the following result. CONVERSE THEOREMS FOR GL n 197 Proposition 10.1. -- For all proper automorphic representations -r of GL,_I(A ) we have I(~s | ~o, r s) = I(~s | ~o, ~; s) for all ~s e 1-I s . From here, applying the weak form of Langlands' spectral theory as in the proof of Theorem 1, we find Proposition 10.9.. -- The equality U~| ) = V~s| holds for all ~s ells. Since ~s was arbitrary in I/s and ~ transforms by the character X~ of K0(u ) as in Section 8, we find that in fact U~| to(g) = Vt~ | to(g) for all g e G s . K0(u ) C GL,(A) and all ~s e H s . However, since we have fixed the vector ~0 at places v r S, we cannot conclude equality for all g e GL,(A). For this reason we are unable to embed II as a subrepresentation of d(GL,(k)\GL,(A)). We will only be able to embed 1-I s as a subrepresentation of a space of classical modular forms on G s trans- forming by the Nebentypus character X~ ~ of F0(rt ). To simplify notation, let us introduce the functions (I)~(gs) = Ut~| IS)) = V~| lS)), where 1 s = YI~ s 1~ e G s and (gs, lS) e G s G s : GL,(A). This associates to each ~s e H s a function on G s. Let Po(os) and Po(rt) be the discrete subgroups of G s defined in Section 9. These are both subgroups of Fo(rt ). Since Fl(rt ) is a normal subgroup of F0(rt ) with abelian quotient (Os/rt) � the central character ~ of II induces a character Z~ of F0(n ) through the character Z,~ of (os/rt) � defined in Section 8. Lemma 10.2. -- The function dp~ is left invariant under Pl(os) and PI('It) and transforms by the character X~ 1 under Po(os) and Po(rt). Proof. --This is the standard argument. Write an element g ~ GL,(A) as g = (gs, gS) with gs ~ Gs, gS ~ G s. Then for y e Po(as) we have @~(Ygs) = U~|176 IS)) 9 Since U~s| is left invariant under P',(k) this is dP~s(ygs) = U~| 7-1)) = U~| to((gs, IS)) 9 But now y -1 e K0(rt ). Since U~s| transforms by Zo under K0(rt ) we have r = X~I(Y)@~(gs). The argument for P'o(rt) is the same, but using V~| Since Xo is trivial on the subgroups P'~(Os) and P~(n) we obtain the invariance of ~ under these groups. [] 198 J.W. COGDELL AND I. I. PIATETSKI-SHAPIRO By Proposition 9.1, the groups P~ (Os) and P'(rt) generate the congruence subgroup F~(rt) fi GL,(os) for i = 0, 1. Hence we may conclude that for every ~s e 1-I s the function r is left invariant under Fl(rt ) and transforms by the character X= 1 under P0(rt). Let d(r0(n)\Gs; cos, Z= -~) be the set of automorphic forms 9 on G s in the sense of [6] which also satisfy (1) r = X='(V) O(gs) for T e Fo(rt) (2) r gs) = r r for z s e Z.(ks) N k~, where r s is the central character of 1-I s. The character X= x is referred to as the Neben- typus character [18]. We then have the following. Proposition 10.3. -- The map ~s ~ (I)r embeds H s as an irreducible subrepresentation of d(ro(n)\Os; o,s, From Section 1 of the appendix, we know that d(Fo(n)\Gs; cos, X= 1) is naturally isomorphic to the space d(GL,(k)\GL,(/k); co)x,~m of Kl(rt)-invariant functions in the space of automorphic forms transforming by the character co under the center. To relate irreducible subrepresentations of ~r cos, Z~ x) to automorphic repre- sentations of GL.(A) occurring in ~r co) we need to know that the representation consists of Hecke eigenforms for an appropriate Hecke algebra. In the appendix we explain this relationship and the Hecke algebras involved when the S-class number is equal to one. We refer the reader to the appendix for the notation to be used. Let T be the smallest finite set of places containing S and such that II~ is unramified at all v r T. Let T' ---- T~S. So T' consists of those places dividing the S-conductor ft. Then rIT is an irreducible unrami_fied representation of G T and hence corresponds to a character A of the Hecke algebra ~(G T, K T) of compactly supported KT-bi-invariant functions on G T. Since ~0 is the unique KI(n ) D KZ-fixed vector in [I s we see that for an g g(G K 1-i~(~) ~o __ A(O) ~o. There is a natural Hecke algebra, which we will denote by ~',(rt), acting on the space d(I~o(rt)\Gs; cos, X~I) 9 To describe ~c(rt), let M = GL,(k) c~ (II~e ~, K0,.(p~)) G T. This M consists of those rational matrices y ~ GL~(k) such that for all v e T' the v-compo- nent y~ lies in K0,~(p~ ). Then Fx(rt) C M. Let ~,~,(rt) depote the C-span of the double cosets Fdn)\M/Pl(n ). The algebra ~,(n) is related to the following adelic Hecke algebra. Let GS(rt) ----- (II~ T, K0,~(p~)) G T. Then GS(rt) D Kx(rt ) and we may from the associated Hecke algebra ~(GS(rt), Kx(rt)) of compactly supported Kl(n)-bi-invarlant func- tions on GS(rt). From the appendix we know that this algebra is isomorphic to GONVERSE THEOREMS FOR GL n 199 C[(os/rt) � | ~, K T) and so contains o~~ ~, K T) as a subalgebra. Then there is a natural isomorphism ~ :o*'c(tt ) -+og~ which takes the double coset Fx(n)tI'l(n ) to the normalized characteristic function Ot of the double coset Kl(n ) tKa(rt ). The algebra structure on o~~ is the pull back of that of ~(tt) via ~. In particular, 3$~ has a subalgebra o~ corresponding to o~ff(G T, K ~) via ~. If Fx(rt ) tI'l(rt ) e d~'o(n) then the associated Hecke operator $'~ acting on d(I'0(rt)\Gs; cos, X= a) is defined as follows. For f~ ~r ~0s, X~ ~) and Pl(rt ) tPl(rt ) = II aj Pl(tt ) the action is (g',f) (gs) = ~. f(a; 1 gs)" The algebra ~(rt) acts on ~r o) x~la) by convolution (0 * 9) (g) = forum ~"b(h) eo(gh) ah for 9 eg~~ and ~? e d(GL,(k)\GL,(&) ; c0) xa~m. These facts can be found in Section 3 of the appendix. Proposition 10.4. -- For each ~s ~ IIs the function cb~ is a Hecke eigenform for #f'~ with eigencharacter A, i.e., 8" t q)~ = A(O,) O~ for each I'l(rt) tI'l(rt) e ~o~. Proof. ~ Let Fl(rt) tI'l(tt ) be a double coset in ~,o~ and $'~ the associated Hecke operator. Write Pl(u ) tPl(rt) = H a~ El(u) with a~ = p~ g~ e P~(k) P0(rt ). This choice of coset representatives is possible by Lemma A.2 of the appendix. Then since Or transforms by the Nebentypus character Z= ~ we have = Z Xo(Y~) U~|162 1)). Since U~| is left invariant under P'.(k) and t ~ transforms by Xo under Ko(rt ) we have X,~(V~) U~| ~o((p[ ~ gs, 1)) = Zo(Vj) U~. ~o((gs, P~)) = U~| ~0((gs, lS)). Thus (~', qSCs) (gs) = U~s| r lS)). As noted above, ~o is an eigenfunction for oY'(G T, K T) with eigencharacter A. Thus (g', Or.) (gs) --= A(O,) U~| IS)) t~a = A(O,) *v.(gs). [] J. W. COGDELL AND I. I. PIATETSKI-SHAPIRO We now have that l-I s is an irreducible subrepresentafion of d(F 0 (u)\Gs; ~s, ~(~ 1) which consists of Hecke eigenvectors for the Hecke algebra 3r with eigencharacter A. We may then apply Theorem A of the appendix to conclude that there exists an irre- ducible automorphic representation 1-I' of GL,(A) such that ri~ _~ rI s (since I-I s is irreducible) and such that II '~ is the unique irreducible representation of G ~ with eigencharacter A for ~#(G T, K~). Thus II '~ ___ II ~ and II' satisfies the conclusions of Theorem 2. [] Proof of Corollary 1. -- We begin with the representation II' from Theorem 2. II' is an automorphic representation with the desired properties, but it may be only a subquotient of the space of automorphic forms. The fact that our original representation II is generic will allow us to pass from II' to a proper automorphie representation, that is, a subrepresentation of the space of automorphic forms. Since our original representa- tion II was generic, then II' is quasi-generic in the sense that II' = ~) II~ is irreducible and for almost all v, II~ is generic. Hence to complete the corollary as stated, it is enough to prove the following result. Proposition 10.5. -- Let II' be an irreducible automorphic quasi-generic representation of GL,(A). Then there exists an irreducible proper automorphic representation II" which is quasi- isomorphic to II'. Moreover, II', N II',' for all non-archimedean places v where II'~ is both generic and unramified. To prove this we will use the following well-known fact. Lemma 10.3. -- Let II, be an irreducible admissible unramified generic representation 0fGL,(kv) over a non-archimedean local field k~. Then there exist unramified characters Xl, ~, 9 9 X,. of GLI(k~) = k~ such that T-,~aL,(ko)t . @ 9 | X,,~). I~v ~ ~lXUBn(k v) kLl, v " 9 Proof. -- By the theory of spherical functions [8] we know that there are unramified characters 11,,, 9 9 Z,,, of GLI(k,) such that II, is the unique unramified constituent of Indg~'(Zl,~| ... | Without loss of generality we may write each Z,,v(x) = Ix ]~" with the u i E C and assume Re(u1)/>... /> Re(u,). Following Jac- quet [20], if we group the characters into families with Re(u~) equal and induce these up to the appropriate GL 8 we get a sequence of quasi-tempered representations vl, ~, ..., vr,,. Since these induced representations v~,~ are irreducible [19] we may use induction in stages to get i.,Aa~,(ko)t~, | | Z,,,) ---- Ind~"ck~)(va, | | v,,,) XXUBn(kv) kill, v " " " , " " " for an appropriate parabolic Q_~. Then this induced representation is actually an induced representation of Langlands type. As Jaequet observed in [20] II~ is in fact the Langlands CONVERSE THEOREMS FOR GL n 201 quotient of this representation. Since H, is generic we know by Jacquet and Shalika [26] that this induced representation of Langlands type must actually be irreducible and hence H~ -= Ind~Ln(ko)('rl ~ | | % ~) = T,,,tGI~%)f,, | | X,, ~)" [] , " " " , ~'~'-tBn(k v) kA,1, v " " " Proof of the Proposition. -- Since II' is automorphic, then by Langlands [34] there exists a partition (rl, ..., r,,) of n and irreducible cuspidal representations a, of GLri(A ) - ~ -~Gr,.(A)~ | | tr,~), where Q, is the standard such that II' is a subquotient of .~ = lnuQ(A) tal ... parabolic subgroup associated to the partition. Let v be a non-archimedean place where II', is both generic and unramified. By Lemma 10.3, there exist unramified characters ZI,,, ..., Z,,, of GL~(k,) such that Ilvt = ~"~Bn(k v | " 9 @Z,,,)" On the other hand, H', is a generic unramified ~ ~ox~(k~/_ | | ~,~ ~). By [3, Lemma 2.24] and Rodier [40], constituent of ~ = lnua,~ )tox, ~ . .. , each %~ must be generic and unramified. By Lemma 10.3, each ~,,, must be fully induced from unramified characters of Bri(k,) and, by transitivity of unitary induction, there are unramified characters ~q,,, ..., ~t,,, of GL,(k,) such that E~ = Ind~)(~x,~ | | ~,, ~). Since II; and E~ are both fully induced off the Borel and have a common constituent, namely II~, by [4] they have the same Jordan-HSlder constituents. But II', is irreducible. Hence, so is ~.~ and E. =-- II',. Since ~ is now irre- ducible at almost all places and has a finite composition series at the remaining finite number of places, we see that the global representation ~ will have a finite composition series and each composition factor will be admissible. Using the theory of Einstein series, at least one constituent of ~ embeds into the space ,~(GL,(k)\GL,(A)) as a proper automorphic representation. In fact, Lemma 7 of Langiands [34] gives a non-zero intertwining of a subrepresentation of E to the space of automorphic forms. Taking any irreducible submodule of the image gives a consti- tuent of E embedded as a proper automorphic representation. Let II" be this component. At all v where E~ is irreducible, we must have II~' ----- E~. In particular II~' = H~ at all non-archimedean v where II" is generic and unramified. [] This completes the proof of Corollary 1. [] Proof of Corollary 2. --Take II' from the conclusion of Theorem 2. As in the proof of Proposition 10.5 we have that there exists a partition (rl, ..., r,,) of n and irreducible cuspidal representations ~i of GL,i(A) such that II' is a subquotient of -- T~s Q Q t~m). By [25] the components II', __ 1-I~ for non-archimedean v where II, is unramified completely determine the partition and the a~, so this data is completely determined by II. Furthermore, as in the proof of Proposition 10.5, at the places where II~ is unra- miffed E, is irreducible and II, __ II" _~ E,. Set II~' = E, at these places. Now consider any other non-archimedean place v. Since the a~ are cuspidal, v)TnrlGLn(kv)[~t\/,.1, 202 j. w. COGDELL AND I. I. PIATETSKI-SHAPIRO they are genetic and the same is true of their local components. Hence at any finite place, .~ has a unique generic constituent. Let II" be this constituent. At those places v e S, let II~'= II~ "~ I/~. This is a genetic constituent of ~,. Let 1-I" = | II~'. Then 17" is the unique generic constituent of ~ subject to lq~' = 1-I, for v e S. By Langlands' result [34] II" is automorphic. This is the desired representation. [3 11. A third converse theorem In the next version of the converse theorem we relax the condition that/I be genetic. The cost is that we can no longer guarantee that the automorphic representation II' we produce agrees with II at the places v ~ S. We now repeat the statement, already given in Section 2, of the precise result: Theorem 8. -- Let n >>. 3. Let 17 be an irreducible admissible representation of GL,(A) whose central character co n is invariant under k � and whose L-function L(II, s) is absolutely convergent in some half-plane. Let S be a non-empty finite set of places of k, containing all archi- me&an places, such that the S-class number of k is one. Suppose that for every m with 1 <~ m <~ n -- 1 and every v ~ ~~ ) the L-function L(I-I � % s) is nice. Then there exists an irreducible auto- morphic representation II' of GL,(A) such that II', ~_ II, for all non-arehimedean places v where II~ is unramified. This is only a mild modification of Theorem 2. Proof. -- For each v let E~ be the representation of Langlands type having II, as its unique irreducible quotient. Each ~ is of the form E,, = Ind,~'k"'(p,,,, I I',~ | | Pmo,, I[ "m''') where Q,, is a standard parabolic subgroup associated to a partition (rl,,, ..., r~.,) of n, p,,, is an irreducible tempered representation of GLuey(k,) and the u~., are real numbers satisfying ul,, > ... > u~,,. As we noted in the proof of Theorem 1, each E, has the same central character as II, and each E, is an induced of Whittaker type and hence injects into its Whittaker model. By the local theory of L-functions for non-genetic representations [24, 27] we have by definition L(n. x .~, s) = L(S. x .., s) ~(n,, x ,~,, s, +,,) = ~(~,, x ~,~, s, +,,) for all irreducible admissible % of GL,~(k~) with 1 ~< m ~< n -- 1. Now if we form the representation E -----| ~, then ~ is a global induced repre- sentation of GL~(A) of Whittaker type having an automorphic central character and such that the L-function L( .~. � "5 s) is nice for every -~ ~ ~(m) with 1 ~< m ~< n -- 1. CONVERSE THEOREMS FOR GL n 203 To proceed as in Theorem 2 and embed =s into a space of classical automorphic forms we need to choose a standard vector in each F,, for all v r S. For each v r S for which II, is unramified, E, must also be and it must have a unique K, = GL,(o,)-fixed vector t ~ which projects to the distinguished K~-fixed vector of II,. Since l-I, is the unique irreducible quotient, Eo must be cyclic and generated by ~o. Since t ~ is the unique K,-fixed vector in E~, it must transform by a character A, under the local Hecke algebra o~~ K,) of compactly supported K,-bi-invariant functions on GL,(k,). Since the quotient map E, ~ II, is intertwining, the image of t ~ in II, will also transform by this character and II, is the unique irreducible unramified representation of GL,(k~) associated to this character. For the places v not in S where II, is not unramified, if we let ~g'(E,, +) be the Whittaker model of E, then by Jacquet and Shalika [26] the restriction of the functions in ~r d?,) to the mirabolic P,,, contains all smooth functions on P,,, which are left quasi-invariant under N,, ~, i.e., the space of Ind~,,~(~b,). Choose a function W', which is fixed by K, n P,,,. The corresponding function W [ in~/.~(E,, +~) will have a stabilizer containing Kx(p~ ~ for some m, >/ 0. We take the corresponding vector ~0 as our standard vector at this place. If we let ~o = @~s ~, then t ~ e =s and t~ fixed by KI(U ) where u = II,e ~, p~~ The argument of Section 8 still gives that t ~ transforms by the character X~, under Ko(rt ) even though =. is not irreducible since E has central character ~ ---- ~n. We now proceed as in Theorem 2. For each ts ~ '='s we form the functions U~| ~o(g) and Vv~| ~o(g). From the methods of Theorem 1 and 2, U~s| lS)) = V~s| lS)) = q)~(gs) for gs ~ Gs and the map ts ~"* ~s(gs) embeds E s into d(l"0(rt)\Gs; o~s, )(gl). Since E s has II s as its unique irreducible quotient, if we take a vector ~s ~ Es which has a non- zero projection to II s then ts must be a cyclic generator for E s. Hence the image of Es in d(P0(rt)\Gs; COs, )~1) is cyclic with a generator f0. As noted before, for all places v r T (as before T is the smallest set of places contai- ning S outside of which l-I,, is unramified) t ~ is a Heeke eigenvector. Hence our standard vector ~0 is an eigenvector for o~g'(G T, K ~) with eigencharacter A = @,~TA,. Then Proposition 10.4 shows that for every ~s ~ =s the function ~ is a Hecke eigenfunction for o~ with eigencharacter A. We now have that "='s is a cyclic subrepresentation of d(P0(n)\Gs; COs, Zg 1) which consists of Hecke eigenvectors for the Hecke algebra oct~ with eigencharacter A. Applying Theorem A of the appendix, we conclude that there exists an irreducible automorphic representation II' of GL,(A) such that II~ is a constituent of E s and II '~ is the unique representation of G ~ with eigencharacter A. But as we have seen above, IF is also the unique representation of G T with eigencharacter A. Therefore II" _ II, for all non-archimedean v where II, is unramified. [] J. W. COGDELL AND I. I. PIATETSKI-SHAPIRO 12. Our final converse theorem Theorems 2 and 3 have the drawback that the automorphic representation II' associated to 11 need not be cuspidal nor unique. However it is possible to associate to II a unique collection of cuspidal representations ~, on general linear groups GL,I(A ) with (rl, ..., r~) a partition of n. Theorem 4. -- Let II be an irreducible admissible representation of GL,(A) satisfying the hypotheses of Theorem 3. Then there exists a partition (q, ..., r,~) of n and irreducible cuspidal representations a~ 0fGL,i(A) such that for all non-archimedean places where H, is unramified we have that a~,, is unramified and L(II,, s) = Il~ L(a~,,, s). Moreover, the sequence (~, ..., a,,) is unique up to permutation. Proof. -- By either Theorem 2 or Theorem 3 we have associated to 11 an automorphic representation 11' such that II,_ 11', for all non-archimedean places v where II, is unramified. By Langlands [34] there is a partition (rz, ..., rm) of n and irreducible cuspidal representations a~ of GL,~(A) such that 11' is a constituent of ~- = ,naQ(A ~- | ... | am) where O is the standard parabolic associated to the parti- tion. By Jacquet and Shalika [25] this sequence of cuspidal representations is unique up to ordering. Moreover, the sequence is uniquely determined by the unramiffed constituent of ~, at those places where ~, is unramiffed. But this unramified constituent is exactly II,. Hence the sequence (az, ..., am) is uniquely determined by 11. In computing the local L-factors, at the places where a representation is unramified, the local L-function can be computed from the unramified vector using Hecke theory. At those places where .~., is unramified, each a,., must also be and ~, has a unique unra- miffed vector which in turn projects to the unramified vector in its unramified quotient, namely II" Thus when 11" is unramiffed L(H~, s) -- L(~,, s) -= II L(a,,,, s). But when 11, is unramified, 11, ~_ II~. Thus : L(II~, s) = IlL(a,,,, s) as desired. ! t t t Next, suppose that (rl, ...,r,) is another partition of n and ai, ..., a, are cuspidal automorphic forms on the GL,~(A) such that for all v where 1I, is unramified we have L(FI,, s) = IIL(,~,,, s) = IlL(a,,,, s). Then let -- TnAOL"r ~ | a~) ~.~t = .... Q'(A) \~ ~ " " " ~OT,,(AI,~al CONVERSE THEOREMS FOR GL n 205 where Q' is the standard parabolic subgroup associated to the partition (r~, ..., rs At the places v where II, is unramified we then have L(.'~.,, s) = L(~.;, s). For GL,, the L-function of an unramified representation % completely determines the Satake parameter t., e GL,(C) of % since L(II~, s) = det(I, -- t~ q;-,)-l. Thus we see that for these places E~ and E~ must have the same unramified constituent. Hence again Jacquet and Shalika [25] let us conclude that n = m, r~ = r~, and (h -~ (h after reordering. [] APPENDIX We retain the notation and conventions of Sections 7-10. In particular, k is a global field, S is a non-empty finite set of places of k containing all archimedean places, and Os is the ring of S-integers of k. Let o~ be a character of A � which is trivial on k � Let d(GL.(k)\GL.(A); co) denote the space of automorphic forms on GL.(A) which transform under the center Z.(A) by the character o~, i.e., f(zg) = co(z)f(g) for g c GL.(A) and z c Z~(A). The purpose of this appendix is to explain the connection between a space of classical auto- morphic forms with Nebentypus d(r0(rt)\Gs; o~s, X~ 1) and the subspace of the adelic automorphic forms d(GL,,(k)\GL.(A); o~) which are fixed by Kl(rt ). The automorphic forms in d(Fo(n)\Gs; COs, =1) are analogous to the functions in ~qC(F0(N)\SL,(R); Z) which are obtained by lifting classical modular forms on the upper half-plane ~ with respect to Po(N ) and Nebentypus character Z to functions on the group SL2(R). For this reason, we will refer to the functions in ~qC(F0(rt)kGs; COs, ~(~x) as " classical " automorphic forms. The functions in ~r ; co) we will refer to as " adelic" automorphic forms. In the case of class number one fields, and forms without Neben- typus, this is explained in [6]. For the convenience of the reader, we recall the extensions to the S-arithmetic case, still assuming the S-class number is one. 1. Relation between automorphic forms Assume that the S-class number of k is one. One consequence of this is that A � = k � k~ u s. As a consequence of strong approximation for SL. [30] and the fact that det(Kl(rt)) = u s we have that GL.(A) may be decomposed as (A.1) GL.(A) = GL.(h) G s K~(rt) as in [6]. Since Kl(rt ) C K0(n ) we also have (A.2) GL.(A) ----- GL.(k) G s Ko(rt ). From the decomposition (A. 1) we have d(GL.(k)\GL.(A))K~("' ___ d(r,(n)\G,) (A.3) where the isomorphism associates to each Kl(n)-invariant automorphic formfon GL.(A) the classical form fc given by f0(gs) =f((gs, lS)) 9 CONVERSE THEOREMS FOR GL n 207 For our purposes we need to keep track of the central character. Let us suppose that ~(GL,(k)\GL,(A) ; co) xl~"~ is non-empty. WHte n = H,~ s p~ with m~/> 0 and m~ =0 for almost all v. Let T'={vlm, 4=0}and T=S~3 T'. Sore,=0 for vr Let f(g) be a non-zero function in this space. If v r T then for any local unit u~ e o~ we have u~ I N e GL,(o,)----Kl.,(p~) which is naturally embedded in Kx(rt ) and so %(u,)f(g) -----f(u, I,g) =f(g) so that %(u,) = 1. Similarly, if v ~T' and u, is a local unit of the form 1 + p~", then %(u,) = 1. So % is unramified at v ~ T and has conductor at least p~"o at the places v e T'. Since (Os/rt) � _ II,es(OJp~)� co defines a character X~ of (Os/n) � via this isomorphism by X~----II,~s c%. The central character co allows us to define a character Z~ ---- IIX, of K0(rt ) as in Section 8. The construction there was not dependent on the space of Kl(rt)-fixed vectors being one, just on the existence of K~(rt)-fixed vectors and the central character. Since the second construction of this character in Section 8 is through the character Z,o of (Os/rt) � we see that Z,~ also defines a character of Fo(rt ) through the quotient map ro(n)/r~(n) ~ (os/n) � as in Section 10. Now let ~r c%, ~1) be the space of classical automorphic forms f~ on G s satisfying (I) f~(Ygs) = X~'(Y)f~(gs) for y e ro(rt ) C O s (2) f~(Zs gs) = r for z s e Z,(ks) ___ k~. Then from the decomposition (A. 2) we have (A. 4) ~r ; o~) ~''' _~ d(ro(n)\G s; '~s, z~') where the isomorphism associates to every Kl(rt ) invariant automorphic form f on GL,(A) the classical form f, on G s given by f,(gs)=f((gs, IS)) and to a classical form f, on G s with Nebentypus character Z~ 1 the adelic form given by f(Ygsko) =f~(gs)X,o(k0) where y e GL,,(A) and k 0 e K0(n ) as in the decomposition in (A.2). 2. Comparison of Heeke algebras Both the spaces in (A.3) and (A.4) have natural Hecke algebras which act on them. We will describe these algebras and compare their actions. If G is any locally compact totally disconnected topological group and K is an open compact subgroup of G we will let .~(G, K) denote the space of K-bi-invariant compactly supported functions on G. This space is an algebra under convolution: the Hecke algebra of G with respect to K. The space d(GL,(k)\GL,(A)) xll"~ is most naturally a module for the Hecke algebra 9ff(G s, Kl(rt)) = (~,$s ~~ KI,,(P~V)) acting by fight convolution. Since Kl(rt ) is not the maximal compact subgroup of G s, the algebra 9ff(G s, Kl(n)) is not 208 J. w. COGDELL AND I. I. PIATETSKI-SHAPIRO necessarily commutative. However ~(G T, K T) = @~ ~ 3r K~) is commutative since K T = 1-I~r K~ and K~ is the maximal compact subgroup at all places v r T. The algebra Yg(G T, K ~) is naturally a subalgebra of 3/QG s, Kl(n)) by the embedding (P ~-~ (Px~, | where ~K~, is the normalized characteristic function of K~, = II,e r, Kl(p~ "~) C Gr,, i.e., the Characteristic function of K T, divided by the volume of KT,. Then 3r T, K T) is the subalgebra of functions whose support lies in K. r, G r. For any g e G s let (P~ denote the characteristic function of the double coset K,(n) gK~(n) divided by the volume of K~(U). The commutative algebra s/g(G T, K ~) acts naturally on ~'(GL~(k)\GL.(A)) Klm~ by right convolution (r 9 ~) (g) = fG~r ~(gh) dh for ap eaiP(G T, K T) and ~ e auf(GL,(k)\GL,,(A)) K*''~. When keeping track of the actions of K0(n ) and P0(rt) it is most convenient to use an intermediate algebra between ~(G s, KI(n)) and 3/~ T, KT). Let us set GS(n) = ( [I K0,,(p~")) G ~. rUT' Then G s DGS(rt)DG T and GS(n)DKo(n)DK,(rt). Since K0(n) normalizes Kl(n ) and has as quotient Ko(n)/Kl(n ) ,~ II,e T, Ko(pD)/Kl(p~ "~ ~ [I,em,(odpD) � - (os/rt) � we see that the double coset algebra aC~(GS(n), Ka(n)) is naturally isomorphic to C[(os/n) � |162 T, K r) and is therefore again commutative and contains ~(G r, KT). Let us set #g(n) = ~(GS(n), K~(n)). As before, the commutative algebra ~(rt) acts on aC(GL,(k)kGL,(A); (o) xllm by right convolution dh (r 9 (g) = r ~(gh) for @ eaCg(n) and q~ ~ aC(GL.(k)\GL.(A); co) xl"'. There is a corresponding classical Hecke algebra, which we will denote by Wc(rt), which acts on d(l~0(n)\Gs; COs, Z~I). To describe afro(n), let M = GL.(k) n G s GS(n). The group M consists of those rational matrices "r e GL.(k) such that for all v e T' the v-component y. lies in K0,.(p~ ). We may view M as a subgroup of GL.(k) and hence of both O s and G s. Then r,(n) = GL.(k) G s Ks(n)C M. Lemma A.I. -- The map ~: rl(n)\M/Vl(n)-+Ki(n)kGS/K~(n) which is given by ra(rt)tPx(n)~-*Kx(n)tKl(rt) is injective with image Kl(n)kGS(n)/Kx(n). Further- more, if we have the decomposition into right cosets Fl(n)tFa(rt)= lla~ Pl(rt) then also K,(n) tKl(rt) = H aj K,(n). Proof. ~ This argument is modeled after that of Shimura [41], who proved a similar result for GL2. CONVERSE THEOREMS FOR GL n 209 Let V be the n-dimensional vector space on which GL,(k) acts having basis { el, 9 9 e, } with respect to which K and Kx(n) are defined. IfA is the space of os-lattices in V then there is a natural action of both GL,(k) and G s on A [37]. Let L o = o s e 1 + ... + o s e, be the free os-lattice such that GL,(o s) is the stabilizer of L o in G s and GL,(os) is the stabilizer of Lo in GL.(k). Set L1 = Os e~ + ... + Os e~_ ~ + rte., so that Ko(n ) is the set ofg e G s such that gL 0 = Lo and gL1 = L1, and K~(n) is the subgroup of elements g which act trivially on L0/L1. We define F~(rt) in GL~(k) by the same conditions. Let u e Kx(rt) and t e M. We first claim that there exists 7 e F~(n) and u~ e K~(rt) such that ut = 7tu~. To see this, consider the lattices L~ = tL o and L3 = utLo. After scaling by an element of Z,(k) if necessary, which will not effect our conclusion, we may assume L,, L~C L 0. By the theory of invariant factors, Theorem 81.11 of [37], there exists a basis x~, ..., x, of V and os-ideals at, 9 .., a. and fractional ideals b~, 9 .., b~ such that Lo=aiXl+ ... +a.x. L~=alblxl+... +%b.x. and L3 = uL,. Consider a place v ~T'. Then u, ~Kl, v(~0~v ) and t, ~Ko,~(p~v ). Therefore 1 lt~, Lo,, = L2,, -- L3, ~. Hence u~ 1 L2 ' , = Ls ' , with u~ = Consider a place v C T. Then Ls, ~ = u~L.,,~ = u~(]~(a~), (b~)~xi). Now write u~ = u~ d, where det(u~) = 1 and d~ is the diagonal matrix diag(det(u~), 1, ..., 1) with respect to the basis {xl,...,x,}. Then d~L0, ~=Lo, ~ so that d veK~ and hence 1 La, ~ Hence 1 Le,, Ls,,. u~ 1 e K~. Also u~ L2, ~ = u~ 14(Z(a,L x,) = . uo = Let u I = II u,. t Then u 1 e Kl(rt ) SL,(A s) is such that u IL~ = L3. Let c c k � be such that L1, Lg., L3 D cL o. Then by strong approximation for SL, there exists 7 eSL,(k) such that 7 -- ul (modcos). Then yL2 = L3, 7Lo = Lo, and since y,- 1 (modco,) for v ~T', y, LI,, = Lx., and Y acts trivially on L0/L1. Hence 7 ~ SL,(k) c~ G s Kx(u ) C Ial(n), We now have 7tLo = 7L, = L3 = utLo. Hence there exists u 1 in the stabilizer of L 0 in G s, namely K s, such that ut = 7tul. Since t, u, and 7 are all in Ko(u), we must have u 1 e Ko(u ) as well. However, since u 1 = t -1 7 -1 ut we see that u 1 acts trivially on Lo/L x and so u 1 s Kx(rt ). Thus ut = 7rut with Y s Px(u) and ut ~ Ka(rt). We are now ready to prove injecfivity. Let tl, t, e M be such that Kt(n) tl Kl(n) = Kt(rt) t~ Kt(tt). Then there exist ul, u s c K10t ) such that u 1 t 1 = t 2 u,. Write u 1 t 1 = Y1 tl ua with Y1 e FI(lt ) and u s e Kt(rt ). Then Y1 tl us = ts u,. Hence t~ -1 "#1 c GL.(k) c~ G s KI(n ) ----- Ft(n ). Thus Yt tt ----- t, y, and rl(rt) tt Pt(n) = Pt(n) ta Pt(rt). Thus the map ~ is injecfive. The fact that the image is Kt(rt)\GS(rt)/Kl(rt) is clear. 27 210 J. W. COGDELL AND I. I. PIATETSKI-SHAPIRO Now suppose that for t e M we have Px(U) tPx(u) = Hs~l a s Pi(u). Multiplying by Ki(u) on the right we have Pl(u) tKi(u) = Ua~ Ki(u) and it is easy to see that a~ Ki(u ) = a s Kl(u) implies that a~ Pi(rt) = a s Pl(u) so that the right hand side is a disjoint union. But as we have seen above, any ut with u e Kx(rt) and t e M can be written as ut = ytu i with y e Pi(u) and ui e Kl(u ). Hence Kl(u ) tK~(u) = P~(u) tKi(u). This finishes the proof. [] We will also need the following result on the choice of coset representatives. Lemma A.2. -- For t ~ M there exists a decomposition Pi(n ) tPl(u ) ----Ha s Pi(rt) with a s e P'.(k) Po(n). Proof. -- This lemma is a consequence of the class number one assumption. To better illustrate this, let us first consider the case where there is no level, so u = o s and P = Pi(n ) = Po(U) = GL,(os), and remove the class number assumption for the moment. Then M = GL,(k). We claim that ]P',(k)\GL,(k)/P] = h s. To prove this, let us first recall some facts about the classification of lattices over the Dedekind domain os [28]. If L is a os-lattice of rank n then L has the form L=aiXl+ ... +a.x. with a~ fractional Os-ideals. The group GL,(k) acts on these lattices and this action has a complete invariant, namely the Steinitz invariant St(L) = cl(al ... a.) where cl(b) represents the ideal class of the fractional ideal b. So, given two rank n lattices L1 and L, there exists an element y e GL,(k) such that yLi = L, if and only if St(L1)= St(Lz) [28, Theorem 10.14]. Since P = GL,(os) is the stabilizer in GL,(k) of the standard lattice Lo---:Osei+ -.. +Ose, then the set GL,(k)/P is in one-to-one correspondence with the set A 0 of all rank n os-lattices with trivial Steinitz invariant. Now consider the action of P',(k) on the space A 0. Geometrically P',(k) is the subgroup of GL,(k) which preserves the subspace ( el, ..., e,_ i ) spanned by the first n- 1 of the standard basis vectors. It has the structure of a semi-direct product of GL~_I(k ) � GLI(k) acting on k "-1. If L 1 eA 0 then we may associate to L i the rank n- 1 sublattice L~ = L 1 c~ (el,..., e,_ 1 ). We claim that the Steinitz invariant of L~, i.e., St(L 1 rn ( el, ..., e,_l )) is a complete invarlant of the action of P',(k) on A o. Suppose that L1, L~ e A o and L1----pL~ with p e P',(k). Let L~ = L~ n <el, ..., e,_l >. CONVERSE THEOREMS FOR GL. 211 Then by Theorem 81.3 of O'Meara [37] there exists y, ---- ~a,, ~ e~ with a,,. 4= 0 and fractional ideals o~ such that (A.5) L, = L~ + o~y,. Let the action of p on (el,..., e._l ) be given by the element A ~ GL._I(k) then from Lx = pLz we find L~ + alYl = AL~ + a2PY2. Since neither Yx nor PY2 lie in (ex, ..., e~_t) we find pL, n (el, ..., e,_l) = Lt n (e~, ..., e,_x) = L[ = AL~ and hence St(L, n ( el, ..., e,_x )) = St(l_~) = St(AL~) = St(pL n ( el, ..., e,_ 1 )). Hence St(L n ( el, ..., e,_l )) is a P',(k) orbit invariant. Now suppose that L1, Lz ~ A 0 are such that St(L1 n (el, ..., e,_l )) : St(L2 n (el, ..., e,_l)). Let L~ ----- L, n ( el, ..., e,_ ~ >. Then there exists A e GL,_i(k) such that L~ = AI~. Write each I~ as L, : L~ + 0~y, as in (A. 5). Since St(L1) ---- St(Lz) and St(L~) = St(I~) we see that ai and r are in the same ideal class. So, modifying yz by a non-zero scalar if" necessary, we may assume L, = L~' + ay,. Since each y, = ]~a,, ~ea with ai,, 4= 0 we may solve the equation (with A e GL._I(k ) as above) for the (n -- 1) � 1 vector b and the non-zero scalar d. Thenp = ( A ~)EP',(k)andpL,=Ll, so that Ll and L~lie in the same P',(k)-orbit. Hence St(L n (el, ..., e,_ 1 )) is a complete invariant for the action of P,(k) on A 0 _ GL,(k)/P. Since this invariant can take on any ideal class as a value, we see that [ P',(k)\GL,(k)/P [ = h s as desired. In the case n = 2 this is the usual proof that the number of cusps for the full Hilbert modular group is equal to the class number of the underlying field. Now let us return to the class number one case, i.e., we again assume h s = 1. Then the above argument gives that GL,(k) = P',(k) GL,(os) which implies the lemma when there is no level 1t. In the case of level, we claim that M = (P'~(k) n M) ro(11 ) from which the lemma follows. Of course, we have M D (P~(k) c~ M) I'0(1t ) so we need only prove the opposite inclusion. Let m e M. Since h s = I we may write m = py with p e P'~(k) and y e GL,(as). View GL~(os) as GL,(k) n G s K s. Then for v e T' we have m~ =p, y, or p~ = m~ 1 y,. Since m, e K0,,(p~ ) and T~ e GL,(o,) we see that p~ ~ GL,(o~) at these places. But P',(k~) n GL,(o~) C K0. ~(p~'o). Hence p, e Ko,,(p~) for v e T' and p e P'.(k) n M. Now consider T" Since y e GL.(os) = GL.(k) t~ G s K s we know that for all v r S we have T. e K.. Now if v ~ T' we have Y. = P~- 1 m. e K0, .(p.~.). J. w. COGDELL AND I. I. PIATETSKI-SHAPIRO Hence y ~ GL,(k) n G s K0(rt ) ---= F0(n ). Thus M =- (P',(k) c~ M) I'o(n ). This then proves the lemma for h s =- 1. [] Let o~(rt) denote the C-span of the double cosets Fi(rt)kM/Fi(tt ). Then the map ~ induces a C-linear bijection ~ :#f0(rt)-+Yg(n) which takes the double coset Fx(rt) tPi(n ) to the normalized characteristic function qS,. The algebra structure on 3r is the pull back of that of 3f'(n)via ~. If I'i(rt ) tFi(rt ) e ~r then the classical Hecke operator ~ acting on ~/(Fo(n)kGs; o~,, Z~ ~) is defined as follows. For fe J~'(Fo(rt)\Gs; r Z~ ~) and Fi(n ) tFi(n ) = IJaj Fi(rt ) the action is (~f) (gs) = ~. f(a-[ ~ gs). If we recall that o~(n) acts on d(GL,(k)\GL,(A); o~) Kl(n) by convolution (0 * (g) = fG,,., .(h) (gh) dh for q) eW(rt) and q~ e d(GL,(k)\GL,(A); ~)~:1~"), then we have the following result. Proposition A. 1. -- The bijection d(GL,(k)\GL.(A) ; co) xl'"' --% ~(F0(n)\G s; co s, Z~ 1) given in (A. 4) is an isomorphism of I-Iecke modules under the identification of algebras given by 3. Comparison of automorphic representations We would now like to compare certain automorphic subrepresentations of d(Fo(rt)\Gs; COs, ?(~1) which consist of Hecke eigenfunctions for the subalgebra oct~ of ~(n) which corresponds via ~ with the subalgebra 3r T, K T) of ~(rt), with the representations they generate in d(GL,(k)\GL,(A); o~). For the sake of envisioned applications we work in the context of cyclic representations rather than irreducible ones. Theorem A. -- Let 1-1 s be a cyclic automorphic subrepresentation of d(D0(n)\Gs; co, Z~ 1) which cons#ts of Itecke eigenvectors for 3~~ ~_ ~,vt~ K T) with eigencharacter A. Then there exists an irreducible automorphic representation II' of GL,(A) such that II' s is a constituent of l-I s and H '~ is the unique irreducible representation of G T with eigencharacter A. Proof. -- Using the isomorphism of (A.4) we may embed l'I s as a Gs-invariant subspace of d(GL,(k)\GL,(A); co) consisting of Ki(n)-fixed vectors. Let f0 be a cyclic generator of II s in d(GL,(k)\GL,(A); ~). Let (Hi, V1) be the GL,(A) subrepresen- tation generated by 1-I s. Then H 1 will also be cyclic, generated by f0- Let II ~ be the unique irreducible admissible G%module associated to the cha- racter A of W(G T, K T) [3]. CONVERSE THEOREMS FOR GL n 213 Since f0 is a Hecke eigenfunction for o~ with eigenfunctional A, then as an element of d(GL,(k)\GL,(A) ; ~) it is an eigenfunction for W(G T, K ~) as well. Let U be a maximal GL,(A)-invariant subspace of V1 not containing f0 (such a U exists by Zorn's lemma). Then V~/U is a non-zero irreducible subquotient of the space of automorphic forms and hence admissible by [6], paragraphs 4.5 and 4.6. Call the representation of GL,(A) on this quotient II'. Then II' is an irreducible automorphic representation and II' = | II~. Since II' is irreducible and contains a K ~ fixed vector with eigenfunctional A, namely the image off0, we see that II '~ ~ IF. Now consider II~. Since the map V~ -+ II' is intertwining, we see that II~ is an irreducible quotient of V 1. Since II 1 was generated by the G s module IIs, II~ must be isomorphic to an irreducible constituent of II s. [] REFERENCES [1] H. BAss, K-theory and stable algebra, Publ. Math. IHES, 29. (1964), 5-60. [2] H. BAss, J. MILOR, and J.-P. S~.RRE, Solution of the congruence subgroup problem for SLn(n ~> 3) and SP2n(n >1 2), Publ. Math. IHES, 33 (1967), 59-137. [3] J. N. BEm~STEm and A. V. ZELEVINSKY, Representations of the group GL(n, F) where F is a non-archimedean local field, Russian Math. Surveys, 31"3 (1976), 1-68. [4] J. N. BERNSTEIN and A. V. ZELEVINSKY, Induced representations of reductive p-adic groups, I, Ann. scient. Ec. Norm. Sup., 4 e s~rie, 10 (1977), 441-472. [5] A. BO~RL, Automorphic L-functlons, Proc. Symp. Pure Math., 33, part 2 (1979), 27-61. [6] A. BOR~L and H. JACQUET, Automorphic forms and representations, Proc. Symp. Pure Math., 83, part 1 (1979), 189-202. [7] A. BOREL and N. WALLACH, Continuous Cohomology, Discrete Subgroups, and Representations of Reductive groups, Annals of Mathematics Study, No. 94, Princeton, Princeton University Press, 1980. [8] P. CARTIER, Representations of p-adic groups: a survey, Proc. Syrup. Pure math., 88~ part 1 (1979), 111-155. [9] W. CASS~.LMA~r, Introduction to the theory of admissible representations of p-adic reductive groups, manuscript, 1975, [10] W. CASSELMAN, Canonical extensions of Harish Chandra modules, Can. J. Math., XLI (1989), 385-438. [11] L. CLOZEL, Representations galoisiennes associ6es aux repr6sentations automorphes autoduales de GL(n), Publ. Math. IHES, 73 (1991), 97-145. [12] D. FLATH, Decomposition of representations into tensor products, Proc. Syrup. Pure Math., 88~ part 1 (1979)o 179-183. [13] I. M. GELFAN9, M. I. Gga~.V and I. I. PIATETSla-SrIAPmO, Representation Theory and Autornorphic Functions, Boston, Academic Press, 1990. [14] R. GODEraEWr, Introduction ~t la th6orie de Langlands, S~minaire Bourbaki, 1966-1967, No. 321. [15] H. HAMBUROm% l~lber die Riemannsche Funktionalgleichung der ~-Ftmktion, Math. Zeit., 10 (1921), 240-254; 11 (1922), 224-245; 13 (1922), 283-311. [16] G. HARDER, Minkowskische Reductionstheorie tiber Funktionenk6pern, Inv. Math., 7 (1969), 33-54. [17] E. HEctor, l~Iber die Bestimung Dirichletscher Reihen durch ihre Funktionalgleichung, Math. Ann., 119. (1936), 664-699. [18] E. HECKS, Mathematische Werke, G/Sttingen, Vandenhoeck und Ruprecht, 1959. [19] H. JACQUET, Generic representations, in Non-commutative Harmonic Analysis, Lecture Notes in Mathematics No. 587, Berlin-Heidelberg-New York, Springer Verlag, 1977, p. 91-101. [20] H. JACQIJET, Principal L-functions of the linear group, Proc. Syrup. Pure Math., 88, part 1 (1979), 111-155. [21] H.JAcQm~T and R. P. LAnOLANDS, Automorphic Forms on GL(2), Lecture Notes in Mathematics No. 114, Berlin- Heidelberg-New York, Springer Verlag, 1970. [22] H. JACQUET, I. I. PIATSTSI~I-SHAI'mO and J. SrlALIKA, Automorphic forms on GL(3), I & II, Ann. of Math., 109 (1979), 169-258. J. W. COGDELL AND I. I, PIATETSKI-SHAPIRO [23] H. JACUET, I. I. and J. SHALIKA, Conducteur des repr6sentations du groupe lin6aire, Math. Ann., 256 (1981), 199-214. [24] H. JACQUET, I. I. PL~TETSKI-SHAPmO and J. SHALIKA, Rankin-Selberg Convolutions, Am. J. Math., 105 (1983), 367-464. [25] H. JACkeT and J. SHALXX~, On Euler products and the classification of automorphic representations, I, Am. J. Math., 108 (1981), 449-558; II, 103 (1981), 777-815. [26] H. JAC~UET and J. SHALIKA, The Whittaker models of induced representations, Pacific J. Math., 109 (1983), 107-120. [27] H. JACOtmT and J. SHALtr,.A, Rankin-Selberg Convolutions: Archimedean Theory, Festschrift in Honor of L I. Piatetski-Shapiro, part I, Rehovot, Weizmann Science Press, 1990, p. 125-207. [28] N. JAcoBso~, Basic Algebra II, San Francisco, W. H. Freeman and Co., 1980. [29] A. KNAPP, Local Langlands correspondence: Archimedean case, Proc. Syrup. Pure Math., 55, part 2 (1994), 393-410. [30] M. KNESER, Strong approximation, Proc. Syrup. Pure Math., 9 (1966), 187-196. [31] B. KosT^~vr, On Whittaker vectors and representation theory, Inv. Math., 48 (1978), 101-184. [32] J.-P. L~RssE andJ. SCn'~WR~Ea, On liftings and cusp cohomology of arithmetic groups, Inv. Math., 83 (1986), 383-401. [33] R. P. LANOLANDS, On the Functional Equation Satisfied by Eisenstein Series, Lecture Notes in Mathematics No. 544, New York, Springer Verlag, 1976. [34] R. P. L~OLANDS, On the notion of an automorphic representation, Proc. Syrup. Pure Math., 33, part 1 (1979), 203-207. [35] H. MAASS, l~ber eine neue Art yon nichtanalytischen automorphen Funktionen und die Bestimmung Dirich- letscher Reihen durch Funktionalgleichungen, Math. Ann., 121 (1944), 141-183. [36] C. MOEGLIN and J.-L. WALDSPU~:OER, Ddcomposition spectrale et Sdries d'Einstein. Une paraphrase de l'dcrlture, Boston, Birkh~iuser, 1994. [37] O. T. O'ME~.A, Introduction to Quadratic Forms, Berlin-G6ttingen-Heidelberg, Springer Verlag, 1963. [38] I. I. PmTETSFa-St-mPmo, Zeta-functions of GL(n), Preprint, University of Maryland, 1976. [39] I. I. PIATETSKX-S~emO, The converse theorem for GL(n), Festschrift in Honor ofl. L Piatetski-Shaplro, part II, Rehovot, Weizmann Science Press, 1990, p. 185-195. [40] F. RODmR, Whittaker models for admissible representations of reduetive p-adic split groups, Proc. Symp. Pure Math., 26 (1973), 425-430. [41] G. SmMURA, On Dirichlet series and Abelian varieties attached to automorphic forms, Ann. Math., 76 (1962), 237-294. [42] A. WEIL, Ober die Bestimung Dirichletscher Reihen durch Funktionalgleichungen, Math. Ann., 168 (1967), 149-156. J. w. G. Department of Mathematics, Oklahoma State University, Stillwater, OK 74078 I. I. P.-S. Department of Mathematics, Yale University, New Haven, GT 06520 Manuscrit re(u le 11 mai 1992. Rgvisg le 21 octobre 1993. PIATETSKI-SHAPIRO,

Journal

Publications mathématiques de l'IHÉSSpringer Journals

Published: Aug 30, 2007

References