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HARISH–CHANDRA HOMOMORPHISMS AND SYMPLECTIC REFLECTION ALGEBRAS FOR WREATH-PRODUCTS by PAVEL ETINGOF, WEE LIANG GAN, VICTOR GINZBURG, and ALEXEI OBLOMKOV To Joseph Bernstein on the occasion of his 60th birthday ABSTRACT The main result of the paper is a natural construction of the spherical subalgebra in a symplectic reflection algebra associated with a wreath-product in terms of quantum hamiltonian reduction of an algebra of differential operators on a representation space of an extended Dynkin quiver. The existence of such a construction has been conjectured in [EG]. We also present a new approach to reflection functors and shift functors for generalized preprojective algebras and symplectic reflection algebras associated with wreath-products. CONTENTS 1. Introduction .... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 91 2. Calogero–Moser quiver ... .... ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 105 3. Radial part map ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 108 4. Dunkl representation .. ... .... ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 117 5. Harish–Chandra homomorphism .. ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 127 6. Reflection isomorphisms .. .... ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 131 7. Appendix A: Extended Dynkin quiver . ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 141 8. Appendix B: Proof of Proposition 3.7.2 . .. ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 144 9. Appendix C: Proof of Theorem 4.3.2 . ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 147 1. Introduction The main result of the paper is the proof of [EG, Conjecture 11.22] that pro- vides a natural construction of the spherical subalgebra in a symplectic reflection al- gebra associated with a wreath-product in terms of quantum hamiltonian reduction of an algebra of differential operators. To state the main result we briefly recall a few basic definitions. 1.1. Quantum Hamiltonian reduction We work with associative unital C-algebras and write Hom = Hom , ⊗= ⊗ , C C etc. Let A be an associative algebra, that may also be viewed as a Lie algebra with respect to the commutator Lie bracket. Given a Lie algebra g and a Lie algebra homomorphism ρ : g → A, one has an adjoint g-action on A given by DOI 10.1007/s10240-007-0005-9 92 PAVEL ETINGOF, WEE LIANG GAN, VICTOR GINZBURG, ALEXEI OBLOMKOV adx : a → ρ(x) · a − a · ρ(x), x ∈ g, a ∈ A.The left ideal A · ρ(g) is stable under the adjoint action. Furthermore, one shows that multiplication in A induces a well defined associative algebra structure on adg A(A,g,ρ) := (A/A · ρ(g)) , the space of adg-invariants in A/A · ρ(g). The resulting algebra A(A,g,ρ) is called the quantum Hamiltonian reduction of A at ρ. Observe that, if a ∈ A is such that the element a mod A · ρ(g) ∈ A/A · ρ(g) is adg-invariant, then the operator of right multiplication by a descends to a well-defined map R : A/A · ρ(g) → A/A · ρ(g). Moreover, the assignment a → R induces an a a adg op algebra isomorphism A(A,g,ρ) = (A/A · ρ(g)) (End (A/A · ρ(g))) . If A,viewedasan adg-module, is semisimple, i.e., splits into a (possibly infinite) direct sum of irreducible finite dimensional g-representations, then the operations of taking g-invariants and taking the quotient commute, and we may write adg adg adg (1.1.1) A(A,g,ρ) = (A/A · ρ(g)) = A /(A · ρ(g)) . adg adg Observe that, in this formula, (A · ρ(g)) is a two-sided ideal of the algebra A . Any A-module M may be viewed also as a g-module, via the homomorphism ρ, and we write M := {m ∈ M | ρ(x)m = 0, ∀x ∈ g} for the corresponding space of g-invariants. Let (A,g)-mod be the full subcategory of the abelian category of left A-modules whose objects are semisimple as g-modules. Let A(A,g,ρ)-mod be the abelian category of left A(A,g,ρ)-modules. One defines an exact functor, called Hamiltonian reduction functor, as follows H : (A,g)-mod → A(A,g,ρ)-mod, (1.1.2) M → H(M) := Hom (A/A · ρ(g), M) = M , where the action of A(A,g,ρ) on H(M) comes from the tautological right action of End (A/A · ρ(g)) on A/A · ρ(g) and the above mentioned isomorphism A(A,g,ρ) = op (End (A/A · ρ(g))) . 1.2. Symplectic reflection algebras for wreath-products Let n be a positive integer. Let S be the permutation group of [1, n]:= {1, ..., n}, and write s ∈ S for the transposition ↔ m.Let L be a 2-dimensional complex m n vector space, and ω asymplectic form on L. Let Γ be a finite subgroup of Sp(L),and let Γ := S Γ be a wreath product n n group acting naturally in L .Given ∈[1, n] and γ ∈ Γ,resp. v ∈ L,wewillwrite γ ∈ Γ for γ placed in the -th factor Γ,resp. v ∈ L for v placed in the -th () n () factor L. HARISH–CHANDRA HOMOMORPHISMS AND SYMPLECTIC REFLECTION ALGEBRAS 93 According to [EG], there is a family of associative algebras, called symplectic re- flection algebras, attached to the pair (L ,Γ ) as above. To define these algebras, write ZΓ for the center of the group algebra C[Γ] and let Z Γ ⊂ ZΓ be a codimension 1 subspace formed by the elements (1.2.1) c = c · γ ∈ ZΓ, ∀c ∈ C. γ γ γ ∈Γ{1} Given t, k ∈ C and c ∈ Z Γ, the corresponding symplectic reflection algebra H (Γ ), o t,k,c n with parameters t, k, c, may be defined, cf. [GG, Lemma 3.1.1], as a quotient of the smash product algebra T(L ) C[Γ ] by the following relations: −1 (1.2.2) [x , y ]= t · 1 + s γ γ + c γ , ∀ ∈[1, n]; () () m () γ () (m) m = γ ∈Γ γ ∈Γ{1} −1 (1.2.3) [u , v ]= − ω(γ u, v)s γ γ , ∀u, v ∈ L,, m ∈[1, n], = m, () (m) m () (m) γ ∈Γ where {x, y} is a fixed basis for L with ω(x, y) = 1. 1.3. Quivers Let Q be an extended Dynkin quiver with vertex set I,and let o ∈ I be an extending vertex of Q . Definition 1.3.1. — The quiver Q obtained from Q by adjoining an additional vertex s CM and an arrow b : s → o is called the Calogero–Moser quiver for Q.Thus, I = I {s} is CM the vertex set for Q , and the vertex s is called the special vertex. CM CM Given α ={α } ∈ Z ,a dimension vector for Q ,write i i∈I CM CM α α i j (1.3.2) Rep (Q ) := Hom(C , C ) α CM {a:i→j | a∈Q } CM = Mat(α × α , C) j i {a:i→j | a∈Q } CM for the space of representations of Q of dimension α.Let D(Q ,α) be the algebra CM CM of polynomial differential operators on the vector space Rep (Q ). CM The group GL(α) := GL(C ) acts naturally on Rep (Q ), by conjuga- CM i∈I CM tion. Hence, each element h of the Lie algebra gl(α) := Lie GL(α) gives rise to a vec- tor field ξ on Rep (Q ). This yields a Lie algebra map ξ : gl(α) → D(Q ,α). CM CM h α The center of the reductive Lie algebra gl(α) =⊕ gl(α ) is clearly isomorphic i∈I i I I to C . Therefore, associated with any χ ={χ } ∈ C , one has a Lie algebra homo- i i∈I 94 PAVEL ETINGOF, WEE LIANG GAN, VICTOR GINZBURG, ALEXEI OBLOMKOV morphism χ : gl(α) → C, x =⊕ x → χ · Tr x . We will use additive notation i∈I i i i i∈I for such homomorphisms and write ξ −χ : gl(α) → D(Q ,α) (rather than ξ ⊗(−χ)) CM for the Lie algebra map h → ξ − χ(h) · 1 .Let Im(ξ − χ) denote the image of the latter map. We may apply Hamiltonian reduction (1.1.1) to the algebra D(Q ,α) and to CM the Lie algebra map ξ − χ . This way, we get the algebra GL(α) (1.3.3) A(D(Q ,α), gl(α), ξ − χ) = D(Q ,α) / J , CM CM χ where GL(α) J := (D(Q ,α) · Im(ξ − χ)) . χ CM Let T Rep (Q ) be the cotangent bundle on Rep (Q ). The total space of CM CM α α the cotangent bundle comes equipped with the canonical symplectic structure and with a moment map ∗ ∗ (1.3.4) µ : T Rep (Q ) → gl(α) gl(α). CM We mayapplythe classical Hamiltonian reduction to C[T Rep (Q )], the Poisson CM algebra of polynomial functions on T Rep (Q ). This way, we get the Poisson alge- α CM −1 GL(α) bra C[µ (0)] of GL(α)-invariant polynomial functions on the zero fiber of the moment map. The algebra in (1.3.3) may be viewed as a quantization of the Poisson −1 GL(α) algebra C[µ (0)] . 1.4. Main result From now on, we fix n ∈ N, a 2-dimensional symplectic vector space L and Γ ⊂ Sp(L), a finite subgroup as in Section 1.2. To (n, L, Γ), we will associate a quiver Q,a dimension vector α, and a character χ as follows. We let Q be an affine Dynkin quiver associated to Γ via the McKay corres- pondence. Thus, the set I of vertices of Q is identified with the set of isomorphism classes of irreducible representations of Γ.Let N be the irreducible representation of Γ corresponding to the vertex i ∈ I,and let δ = dim N . The extending vertex o ∈ I i i corresponds to the trivial representation of Γ,so δ = 1.The vector δ ={δ } ∈ Z o i i∈I is the minimal positive imaginary root of the affine root system associated to Q.Mo- tivated by M. Holland [Ho], we put (1.4.1) ∂ ={∂ } ∈ Z,∂ := n(−δ + δ ), ∀i ∈ I. i i∈I i i h(a) {a∈Q | t(a)=i} CM Given a central element c ∈ ZΓ,write Tr(c; N ) for the trace of c in the simple Γ-module N , i ∈ I.Thus, for any c ∈ Z Γ, see (1.2.1), we have δ · Tr(c; N ) = 0. i o i i i∈I Associated with any data n ∈ N, k ∈ C, and c ∈ Z Γ, weintroducethreevectors CM χ ={χ } ,χ ={χ } ∈ C , i i∈I i∈I CM i CM HARISH–CHANDRA HOMOMORPHISMS AND SYMPLECTIC REFLECTION ALGEBRAS 95 and λ(c) ={λ(c) } ∈ C , such that δ · λ(c) = 1, i i∈I where we have used standard notation δ · λ = δ · λ . These vectors are defined as i i follows (1.4.2) λ(c) := Tr(c; N ) + δ /|Γ|, ∀i ∈ I; i i i χ := n(k ·|Γ|/2 − 1) + 1, χ := λ(c) − ∂ − k ·|Γ|/2, o o o χ := λ(c) − ∂ , ∀i ∈ I{o}; i i i χ := χ − 1 = n(k ·|Γ|/2 − 1), χ = χ , ∀i ∈ I. We are going to consider representations of the quiver Q with dimention vec- CM tor CM (1.4.3) α ={α } ∈ Z , where α := 1, and α := n · δ , ∀i ∈ I. i i∈I s i i CM ≥0 I GL(α) CM Let χ ∈ C be as in (1.4.2), and let J = (D(Q ,α) · Im(ξ − χ )) be χ CM the corresponding two-sided ideal in D(Q ,α), cf. (1.3.3). Write e := g CM g∈Γ |Γ | n for the ‘symmetrizer’ idempotent viewed as an element of the symplectic reflection algebra H (Γ ). t,k,c n We are now in a position to state our main result about deformed Harish– Chandra homomorphisms for symplectic reflection algebras associated with a wreath- product. According to [EG], the importance of the deformed Harish–Chandra homo- morphism is due to the fact that this homomorphism provides a description of the spherical subalgebra eH (Γ )e ⊂ H (Γ ) in terms of quantum Hamiltonian reduction t,k,c n t,k,c n of the ring of polynomial differential operators on the vector space Rep (Q ).In the CM special case of a cyclic group Γ ⊂ SL (C), that is, for quivers Q of type A (equipped 2 m with the cyclic orientation), the deformed Harish–Chandra homomorphism has been already constructed in [Ob], see also [Go]. In all other cases, a construction of the deformed Harish–Chandra homomorphism Φ will be given in the present paper. k,c Our main result reads Theorem 1.4.4. — Assume that Γ ⊂ SL (C) is not a cyclic group of odd order (i.e. Q is not of type A ), and put t := 1/|Γ|. Then, for any n ∈ N, k ∈ C, c ∈ Z Γ, there is an 2m o algebra isomorphism Φ : A(D(Q ,α),gl(α), ξ − χ ) k,c CM GL(α) ∼ = D(Q ,α) / J → eH (Γ )e. CM χ t,k,c n 96 PAVEL ETINGOF, WEE LIANG GAN, VICTOR GINZBURG, ALEXEI OBLOMKOV Furthermore, the map Φ is compatible with natural increasing filtrations on the algebras k,c involved and the corresponding associated graded map gives rise to a graded Poisson algebra iso- morphism, cf. (1.3.4): −1 GL(α) gr Φ : C[µ (0)] gr(eH (Γ )e). k,c t,k,c n This theorem is a slightly modified and corrected version of [EG, Conjecture 11.22] (in [EG], as well as in the main body of the present paper, everything is stated in terms of the quiver Q rather than in terms of the Calogero–Moser quiver Q , CM see Definition 5.2.1 and Theorem 5.2.4 in Section 5.2 below; however, the two ap- proaches are easily seen to be equivalent). Theorem 1.4.4 is a common generalization of two earlier results. The first one is [GG2, Theorem 6.2.3], cf. also [EG, Corollary 7.4]; it corresponds to the (somewhat degenerate) case of Γ ={1}. The second result, due to M. Holland [Ho], is a special case of Theorem 1.4.4 for n = 1,where the symplectic reflection algebra is Morita equivalent to a deformed preprojective algebra of [CBH]. Also, in the special case of a cyclic group Γ = Z/mZ the isomorphism of Theorem 1.4.4 has been recently constructed in [Go] using the results from [Ob]. A ‘classical’ counterpart of Theorem 1.4.4 involving classical Hamiltonian re- duction (at generic values of the moment map (1.3.4)) has been proved in [EG, The- orem 11.16]. Combining Theorem 1.4.4 with (1.1.2), and using the same argument as in the proof of [GG2, Proposition 6.8.1], we deduce Corollary 1.4.5. — There exists an exact functor of Hamiltonian reduction H : (D(Q ,α),gl(α))-mod → eH (Γ )e-mod. CM t,k,c n This functor induces an equivalence (D(Q ,α),gl(α))-mod/ Ker H → eH (Γ )e-mod. CM t,k,c n We expect that the Hamiltonian reduction functor induces an equivalence be- tween the subcategory of (D(Q ,α),gl(α))-mod formed by D -modules whose char- CM acteristic variety is contained in the Nilpotent Lagrangian,see [Lu1, §12], and the cate- gory of finite dimensional eH (Γ )e-modules. t,k,c n 1.5. Four homomorphisms Our construction of the isomorphism Φ in Theorem 1.4.4 is rather indirect. k,c It involves four additional algebras and four homomorphisms between those algebras, which are important in their own right. HARISH–CHANDRA HOMOMORPHISMS AND SYMPLECTIC REFLECTION ALGEBRAS 97 The first algebra, to be denoted Π (Q ), is a slightly renormalized version of CM the deformed preprojective algebra, with appropriate parameters, cf. [CBH], associ- ated to the Calogero–Moser quiver Q . The second algebra, to be denoted B,con- CM tains the spherical algebra eH (Γ )e as a subalgebra. The algebra B is a ‘Calogero– t,k,c n Moser cousin’ of generalized preprojective algebras introduced by two of us in [GG, (1.2.3)], see also Definition 6.1.3 below. The third algebra, T , is a ‘matrix-valued’ counterpart of the algebra introduced in (1.3.3). To define this algebra, we introduce the following vector spaces ∗ ∗ n (1.5.1) N =⊕ N , where N := N = C, and N := N ⊗ C , ∀i ∈ I. i∈I i s i CM o i ∼ i Thus, we have N = C , so the group GL(α) acts on N in an obvious way, and this gives the tautological representation τ : gl(α) → End N. Following M. Holland [Ho], we apply the quantum Hamiltonian reduction to the algebra D(Q ,α) ⊗ End N and CM to the Lie algebra homomorphism ξ − (χ − τ) : gl(α) → D(Q ,α) ⊗ End N, CM h → ξ ⊗ Id − 1 ⊗ (χ(h)Id − τ(h)), h N D N where χ : gl(α) → C is as in (1.4.2). This way, we get an algebra GL(α) (D(Q ,α) ⊗ End N) CM (1.5.2) T := . GL(α) ((D(Q ,α) ⊗ End N) · Im(ξ − (χ − τ))) CM 1 × Now, let P = (L{0})/C be the projective line. We will consider an appropri- 1 n ate Γ -equivariant vector bundle of rank dim N on X,where X ⊂ (P ) is a Γ -stable n n Zariski open dense subset in the cartesian product of n copies of P .Further, wewill define a certain algebra D(X, p,) of twisted differential operators acting in that vector bundle, see Section 3.1 for the notation and also (3.6.1). One has the following diagram of four algebra homomorphisms, all denoted by various Θ’s, involving the four algebras introduced above Π (Q ) CM lQ lQ lQ HollandlQ l Quiver Θl Θ lQ lQ l vv (1.5.3) B Rm Rm Rm Rm Dunkl Radial Rm Θ Θ Rm Rm vvm D (X, p,) Holland In this diagram, the map Θ is (a slightly renormalized version of ) an Dunkl algebra homomorphism introduced by M. Holland in [Ho]. The map Θ is 98 PAVEL ETINGOF, WEE LIANG GAN, VICTOR GINZBURG, ALEXEI OBLOMKOV a Γ-analog of the Dunkl representation for rational Cherednik algebras, cf. [EG]. The Radial map Θ is obtained by a ‘radial part’ type construction with respect to an appro- priate transverse slice to generic GL(α)-orbits in Rep (Q ). We produce such a slice CM ⊕n using a map L → Rep (Q ), which is generically injective and is such that its CM image is generically transverse to GL(α)-orbits in Rep (Q ). Our radial part con- CM struction associates to a polynomial GL(α)-invariant differential operator GL(α) Radial u ∈ (D(Q ,α) ⊗ End N) a Γ -invariant twisted differential operator Θ (u) ∈ CM n D(X, p,) . Quiver The fourth map, Θ , is new. The main idea behind the construction of this map, as well as the definition of the algebra B, will be outlined in Section 1.7 below and a more rigorous treatment will be given later, in Section 2.2. Remark 1.5.4. — In the special case of a cyclic group Γ = Z/mZ, the Dunkl op- erators that we consider are not the same as those introduced earlier by Dunkl-Opdam in [DO]. 1.6. Strategy of the proof of Theorem 1.4.4 The proof of the main theorem is based on the following key result Theorem 1.6.1. — Diagram (1.5.3) commutes, i.e., we have: Radial Holland Dunkl Quiver Θ ◦ Θ = Θ ◦ Θ . The proof of this theorem is long and messy; it occupies about one half of the paper. In the proof, we explicitly compute both sides of the equation Radial Holland Dunkl Quiver Θ ◦ Θ (x) = Θ ◦ Θ (x), for an appropriate set {x, x ∈ Π (Q )} CM of generators of the algebra Π (Q ). CM To deduce Theorem 1.4.4 from Theorem 1.6.1, one has to be able to replace in diagram 1.5.3 the algebra T , of ‘matrix valued’ twisted differential operators, by a ‘smaller’ algebra of scalar-valued twisted differential operators of the form A(D(Q ,α), gl(α), ξ − χ), that appears in Theorem 1.4.4. CM To this end, let p ∈ End N denote the idempotent corresponding to the projec- tion N = N N .For χ, χ as in (1.4.2), one proves j s j∈I CM GL(α) (1.6.2) pT p D(Q ,α) / J = A(D(Q ,α), gl(α), ξ − χ ) =: A . s χ s CM χ CM χ Write e for the idempotent in the algebra Π (Q ) corresponding to the triv- i CM Quiver ial path at i. It is easy to see that the map Θ sends the subalgebra e Π (Q )e s CM s ⊂ Π (Q ), spanned by paths beginning and ending at the special vertex s,into CM eH (Γ )e, a subalgebra in B. Furthermore, restricting diagram (1.5.3) to the sub- t,k,c n HARISH–CHANDRA HOMOMORPHISMS AND SYMPLECTIC REFLECTION ALGEBRAS 99 algebra e Π (Q )e , one obtains four algebra homomorphisms along the perimeter s CM s of the following diagram e Π (Q )e s CM s hS hS HollandS h Quiver hS Θ Θ hS hS hS hS hS ttS h tt k,c GL(α) eH e D(Q ,α) / J (1.6.3) t,k,c CM χ U h Uk Uk H Uk Uk Uk Uk Uk Uk Uk Dunkl RadialUk Uk Θ Θk uuk D (X, p, ) Here, D(X, p, ) stands for an appropriate ring of scalar-valued Γ -invariant twisted s n differential operators on X. The perimeter of diagram (1.6.3) commutes by Theorem 1.6.1. In addition, one proves Holland Dunkl Lemma 1.6.4. — In diagram (1.6.3),the map Θ is surjective and the map Θ is injective. It is clear that the lemma yields Holland Radial Holland Dunkl Quiver Ker Θ ⊂ Ker(Θ ◦ Θ ) = Ker(Θ ◦ Θ ) Quiver = Ker Θ . Holland Quiver The resulting inclusion Ker Θ ⊂ Ker Θ implies that we may (and will) define the dashed arrow Φ in diagram (1.6.3) to be the composite k,c GL(α) Holland −1 D(Q ,α) (Θ ) e Π (Q )e proj e Π (Q )e CM s CM s s CM s Holland Quiver Ker Θ Ker Θ Quiver eH e. t,k,c To complete the proof of Theorem 1.4.4, one observes that all the objects ap- pearing in diagram (1.6.3) come equipped with natural filtrations, and all the maps in the diagram are filtration preserving. Therefore, to prove that the map Φ is bi- k,c jective, it suffices to show a similar statement for gr Φ , the associated graded map. k,c The latter statement follows readily from the results of [CB] and [GG2] concerning the geometry of moment maps arising from representations of affine Dynkin quivers. Quiver 1.7. The algebra B and the map Θ To define the algebra B that appears in diagram (1.5.2), we will first introduce in (2.2.1) certain idempotents e ∈ C[Γ ], i ∈ I. Then, we let i,n−1 n (1.7.1) M := H (Γ )e (⊕ H (Γ )e ). t,k,c n i∈I t,k,c n i,n−1 100 PAVEL ETINGOF, WEE LIANG GAN, VICTOR GINZBURG, ALEXEI OBLOMKOV op Thus, M is a left H (Γ )-module, and we put B := (End M) . This endomor- t,k,c n H (Γ ) t,k,c n phism algebra is built out of Hom-spaces between various H (Γ )-modules which t,k,c n appear as direct summands in (1.7.1). The Hom-spaces are easily computed, and we find (1.7.2) B = B , where B = eHe, and i, j s,s i, j∈I CM B = eHe , B = e He, B = e He , ∀i, j ∈ I. s, j j,n−1 i,s i,n−1 i, j i,n−1 j,n−1 Each direct summand B here is a subspace of the algebra H (Γ ),and mul- i, j t,k,c n tiplication in the algebra B is given by ‘matrix multiplication’ B × B → B where, i, j j,k i,k for each i, j, k ∈ I , the corresponding pairing is induced by the multiplication in CM H (Γ ). t,k,c n Quiver Our construction of the map Θ is based on an exact functor (1.7.3) H (Γ )-mod → Π(Q )-mod, M → M. t,k,c n CM To define this functor, let L ,resp. Γ , be a copy (inside the algebra H (Γ )) (1) (1) t,k,c n of our 2-dimensional vector space L,resp. copy of thegroup Γ, corresponding to the ⊕n first direct summand in L .Further,let S be the subgroup of S which permutes n−1 n n−1 [2, n],and let Γ = S Γ ⊂ Γ be the wreath-product subgroup corresponding n−1 n−1 n n ⊕n to the last n − 1 factors in Γ . It is clear from the commutation relations in T(L ) C[Γ ] that any element of the subalgebra H ⊂ H (Γ ), generated by L and Γ , n (1) t,k,c n (1) (1) commutes with Γ . n−1 Now, let M be an arbitrary left H (Γ )-module. We deduce that the space t,k,c n n−1 M ⊂ M,of Γ -invariants, is stable under the action of the subalgebra H .Thus, n−1 (1) n−1 to each vertex i ∈ Q we may attach the vector space M := Hom (N , M ),the i Γ i (1) corresponding Γ -isotypic component. Further, following the strategy of [CBH] and (1) Γ Γ n−1 n−1 using the McKay correspondence, we see that the action map L ⊗ M → M (1) induces linear maps between various isotypic components M .Thisway,the collection {M } acquires the structure of a representation of the quiver Q . In addition, the sub- i i∈I Γ Γ Γ n n−1 n−1 space M := M ⊂ M is clearly contained in M = Hom (N , M ) = M as s o Γ o (1) a canonical direct summand. Therefore the imbedding b : M → M and the projection s o b : M → M provide additional maps, making the collection {M } arepresen- o s i i∈I CM tation of the quiver Q . One can check that this representation descends to a rep- CM resentation of the algebra Π(Q ), which is a quotient of the path algebra of Q . CM CM Thus, to any H (Γ )-module M we have assigned a Π(Q )-module M =⊕ M . t,k,c n CM i∈I i CM This gives the desired functor (1.7.3), cf. Section 1.8 below for a generalization. Finally, we apply the functor M → M to M := M,the H (Γ )-module in (1.7.1). t,k,c n It is immediate from (1.7.2) that one has a natural bijection B M. The bijec- tion gives B the structure of a left Π(Q )-module, moreover, the action of Π(Q ) CM CM HARISH–CHANDRA HOMOMORPHISMS AND SYMPLECTIC REFLECTION ALGEBRAS 101 on B commutes with right multiplication (with respect to the algebra structure) by the elements of B. It follows that the Π(Q )-module structure on B comes, via left CM multiplication, from an algebra homomorphism Π(Q ) → B. The latter homo- CM morphism clearly restricts to a homomorphism e Π(Q )e → B = eH (Γ )e, s CM s s,s t,k,c n Radial denoted Θ . There is a modification of the above construction, to be explained in Section 2.2, in which the algebra Π(Q ) is replaced by the renormalized algebra Π (Q ).This CM CM way, one obtains similar algebra homomorphisms Quiver (1.7.4) Θ : Π (Q ) → B, and CM Quiver Θ : e Π (Q )e → B = eH (Γ )e. s CM s s,s t,k,c n 1.8. Applications to reflection functors and shift functors In Section 6, we study reflection functors and shift functors for generalized pre- projective algebras and symplectic reflection algebras associated with wreath-products, cf. [GG]. More generally, let Q be an arbitrary (not necessarily extended Dynkin) quiver, with vertex set I.Write C = (C ) for the generalized Cartan matrix of Q and W ij for the Weyl group W, defined as the group generated by the simple reflections r for i ∈ I.The group W acts on C as r : λ = λ e → λ − C λ e . i j j ij i j j∈I j∈I For any λ ∈ C , one has an algebra Π (Q ), a renormalized version of the cor- responding deformed preprojective algebra studied in [CBH]. Further, for any integer n ≥ 1, and complex parameters ν ∈ C and λ ∈ C , we have associated in [GG, (1.2.3)], see also Definition 6.1.3 below, a generalized preprojective algebra A (Q ). n,λ,ν For each i ∈ I, there are reflection functors F for the corresponding deformed preprojective algebras Π (Q ), introduced in [CBH], and also their analogues for gen- eralized preprojective algebras, introduced in [Ga]: (1.8.1) F : A (Q )-mod → A (Q )-mod. i n,λ,ν n,r (λ),ν We will show in Section 6.6 that these functors satisfy standard Coxeter rela- tions: Proposition 1.8.2. — For the reflection functors F for generalized preprojective algebras, one has: (i) If λ ± pν = 0 for p = 0, 1, ..., n − 1,then F = Id. (ii) Suppose C = 0.If λ ± pν = 0 and λ ± pν = 0 for p = 0, 1,..., n − 1,then ij i j F F = F F . i j j i (iii) Suppose C =−1.If λ ± pν = 0, λ ± pν = 0 and λ + λ ± pν = 0 for ij i j i j p = 0, 1, ..., n − 1,then F F F = F F F . i j i j i j 102 PAVEL ETINGOF, WEE LIANG GAN, VICTOR GINZBURG, ALEXEI OBLOMKOV Part (i) of the proposition has been already proved in [Ga, Theorem 5.1]; Parts (ii) and (iii) are new. In the special case n = 1, the proposition is due to [CBH], [Na], [Lu2], and [Maf]. However, we believe that, even in that special case, our proof appears to be simpler. Next, given c as in (1.2.1), we put −1 σ c := (2t − c ) · γ and e := (−1) σ(e ⊗ ··· ⊗ e ). γ − o o n! γ ∈Γ{1} σ ∈S Using our main Theorem 1.4.4 and reflection functors, we will deduce Corollary 1.8.3. — For t = 1/|Γ| and any c as in (1.2.1), there are algebra isomorphisms eH e e H e e H e . t,k,c − t,k−2t,c − − t,k−2t,c − We will prove the first isomorphism above in Section 5.3 and the second in Sec- tion 6.7. Using the composite isomorphism in Corollary 1.8.3, we define the shift functor to be the functor (1.8.4) S : H -mod → H -mod, V → H e ⊗ eV. t,k,c t,k−2t,c t,k−2t,c − eH e t,k,c Finally, we can extend the construction exploited in the definition of the map Quiver Θ to an appropriate, more general, context as follows. Let T be any nonempty subset of I. Generalizing the definition of Calogero– Moser quiver, let Q be a quiver obtained from Q by adjoining a vertex s, called the special vertex, and arrows b : s → i,one foreach i ∈ T.Recall that e denotes the i i idempotent in the path algebra corresponding to a vertex i.Thus,given λ ∈ C ,we write λ = λ e , and we also put e := e . i i T i i∈T In Section 6.2, for any n ≥ 1,λ ∈ C ,ν ∈ C, we introduce an exact functor (1.8.5) G : A (Q )-mod → Π (Q )-mod. n,λ,ν T λ−νe +nνe T s The construction of reflection functors for generalized preprojective algebras, see (1.8.1), implies readily that, for any i ∈ I, one has the following commutative dia- gram A (Q )-mod A (Q )-mod n,λ,ν n,r (λ),ν (1.8.6) G G Π (Q )-mod Π (Q )-mod. T T λ−νe +nνe r (λ)−νe +nνe T s i T s The functor (1.8.5) is a generalization of the functor M → M considered in Section 1.7 in the following sense. Let Q be the extended Dynkin quiver associated HARISH–CHANDRA HOMOMORPHISMS AND SYMPLECTIC REFLECTION ALGEBRAS 103 to a finite subgroup Γ ⊂ SL (C).Given adata (n, k, c), as in (1.4.1), put t = 1/|Γ| and ν = k ·|Γ|/2. The generalized preprojective algebra A (Q ) is Morita equivalent, n,λ,ν according to [GG], to the symplectic reflection algebra H (Γ ), so one has a cat- t,k,c n egory equivalence H (Γ )-mod A (Q )-mod. Therefore, composing this equiva- t,k,c n n,λ,ν lence with (1.8.5), yields a functor H (Γ )-mod → Π (Q )-mod. t,k,c n T λ−νe +nνe T s The latter functor reduces, in the special case of the one point set T ={o}, to the functor M → M considered in Section 1.7. 1.9. Quantization of the Hilbert scheme of points on the resolution of Kleinian singularity The shift functor (1.8.4) is the Γ-analogue of the shift functor introduced in [BEG] in the case of the trivial group Γ. The latter functor has been used by Gordon- Stafford [GS] to construct quantization of the Hilbert scheme of n points of the plane C . Now, let X → L/Γ be the minimal resolution of the Kleinian singularity L/Γ and let Hilb X be the Hilbert scheme of n points in X. It should be possible to use the shift functor (1.8.4) and Theorem 1.4.4 to construct quantizations of Hilb X.This would provide a common generalization to the case of wreath-products Γ = S Γ n n of the results of Gordon-Stafford [GS] in the special case Γ = 1 and n ≥ 1,and also of the results of Boyarchenko [Bo] in the special case of arbitrary Γ ⊂ SL (C) and n = 1, cf. also [Mu] for the case of cyclic group Γ (and n = 1). In a different direction, the construction of the algebra eH (Γ )e in terms of t,k,c n Hamiltonian reduction provided by Theorem 1.4.4 gives way to applying the machin- ery of [BFG] to symplectic reflection algebras over k, an algebraic closure of the finite field F . In more detail, fix a finite group Γ ⊂ SL (C) and a positive integer n. Then, a routine argument shows that, for all large enough primes p>n, each of the schemes n −1 X, Hilb X,and µ (0), cf. (1.3.4), has a well defined reduction to a reduced scheme −1 over k.Further,let M be the irreducible component of µ (0), cf. (1.3.4), as de- fined in [GG2, Theorem 3.3.3(ii)]. Then, the action of the group GL(α)/G on M m n is generically free. Moreover, according to H. Nakajima, there exists a GL(α)-stable Zariski open dense subset M ⊂ M of stable points, such that one has a smooth uni- versal geometric quotient morphism M → Hilb X.Furthermore,in thiscase all the Basic assumptions of [BFG, 4.1.1] hold. Next, let Q [Γ] be the group algebra of Γ with rational coefficients. Write Z(Γ, Q ) for the center of Q [Γ],and Z (Γ, Q ) for the corresponding codimension 1 subspace, cf. (1.2.1). Fix k ∈ Q and c ∈ Z (Γ, Q ) and let eH (Γ , Q )e be the o t,k,c n Q -rational version of the C-algebra eH (Γ )e. Then, there exists a large enough t,k,c n 104 PAVEL ETINGOF, WEE LIANG GAN, VICTOR GINZBURG, ALEXEI OBLOMKOV constant N(k, c) > max(n, |Γ|) such that for all primes p> N(k, c) the Q -algebra eH (Γ , Q )e has a well defined reduction to a k-algebra eH (Γ ,k)e. t,k,c n t,k,c n On the other hand, one can apply a characteristic p version of quantum Hamil- tonian reduction, as explained in [BFG, §3], in our present situation. This way, for all large enough primes p, Theorem 4.1.4 from [BFG] provides a construction of a sheaf n (1) n of Azumaya algebras A on (Hilb X) , the Frobenius twist of the scheme Hilb X. k,c Mimicing the proof of [BFG, Theorem 7.2.4(i)–(ii)], and using our The- orem 1.4.4, one obtains the following result Theorem 1.9.1. — Fix k ∈ Q and c ∈ Z (Γ, Q ). Then, there exists a constant d(k, c) > max(n, |Γ|), such that for all primes p>d(k, c) and t = 1/|Γ|∈ k, we have 0 (1) H (Hilb X) , A eH (Γ ,k)e; k,c t,k,c n moreover, i n (1) H (Hilb X) , A = 0, ∀i>0. k,c 1.10. Directions of further research Quiver The map Θ introduced in this paper turns out to be useful in the theory of deformed double current algebras developed by N. Guay [Gu1, Gu2, Gu3]. Namely, it is possible to view the integer n in the definition of the algebra eH e as a parameter t,k,c Quiver and to make an “analytic continuation” of the construction of the map Θ with respect to that parameter. This way, one obtains a new construction of Γ-deformed double current algebras (for gl(1)) as appropriate quotients of the algebras e Π (Q )e . s CM s This will be discussed in a forthcoming paper [EGR]. Dunkl We expect that the map Θ will be helpful in developing a Borel–Weil–Bott style theory for representations of symplectic reflection algebras for wreath products. Such a theory would provide a geometric realization of finite dimensional represen- tations of these algebras (including those studied in [Mo, Ga]) in the spaces of global 1 n sections of appropriate coherent sheaves on (P ) satisfying appropriate vanishing con- ditions. First steps in this direction are taken in [E], and forthcoming work of S. Mon- tarani. 1.11. Acknowledgments We are grateful to Iain Gordon for a careful reading of a preliminary draft of the paper. The work of P. E., W. L. G., and V. G. was partially supported by the NSF grants DMS-0504847, DMS-0401509, and DMS-0303465, respectively. The work of P. E., V. G., and A. O. was partially supported by the CRDF grant RM1- 2545-MO-03. HARISH–CHANDRA HOMOMORPHISMS AND SYMPLECTIC REFLECTION ALGEBRAS 105 2. Calogero–Moser quiver 2.1. Intertwiners Let Q be the double quiver of Q,obtainedfrom Q by adding a reverse edge a : j → i for each edge a : i → j in Q.For anyedge a : i → j in Q,we write its tail t(a) := i and its head h(a) := j . We have an identification L → L : u → ω(u, ·).Let a ∈ Q be an edge, then ∗ ∗ for each intertwiner φ : L ⊗ N → N , we have a corresponding intertwiner t(a) h(a) ∗ ∗ φ : N → L ⊗ N . a t(a) h(a) Suppose Q is not of type A . Following [CBH] (cf. also [Me]), we normalize the intertwiners such that for each edge a ∈ Q,wehave φ ∗φ = δ Id ,and so a h(a) N t(a) ∗ ∗ ∗ ∗ φ φ =−δ Id .Thus, φ φ is δ times the projection of L ⊗ N to N ,and a t(a) N a h(a) a a h(a) t(a) h(a) ∗ ∗ φ φ is −δ times the projection of L ⊗ N to N . Hence, for any vertex i, a t(a) a t(a) h(a) ∗ ∗ (2.1.1) φ φ − φ ∗φ = δ Id . a a i L⊗N a a a∈Q ;h(a)=i a∈Q ;t(a)=i Suppose now that Q is of type A .Then Γ is the group with 2 elements 1,ζ . Moreover, ζ x =−x and ζ y =−y. Write the vertices of Q as o and i,where N is the trivial representation of Γ and N is the sign representation of Γ.We have y y x x a decomposition L = N ⊕ N where N is spanned by x and N is spanned by y. i i i i x x Let pr : L ⊗ N → N be the projection map to N ⊗ N = N ,and pr : L ⊗ N → N o i o i o i i i be the projection map to N ⊗ N = N .Let pr : L ⊗ N → N be the projection map o i i o i o y y to N ⊗ N = N ,and pr : L ⊗ N → N be the projection map to N ⊗ N = N . i o i o i o i o i Denote the edges of Q by a and a .If a : o → i,then let φ = pr and φ = pr . 1 2 1 a a 1 i o x x If a : i → o,thenlet φ = pr and φ =−pr .If a : o → i,thenlet φ = pr and 1 a a 2 a 1 o i 2 i y y ∗ ∗ φ =−pr .If a : i → o,then let φ = pr and φ = pr . Itiseasy tosee that with a 2 a a o 2 o i 2 2 these choices, we again have (2.1.1). 2.2. Quiver map For convenience, we shall fix an isomorphism N = C ,where δ = dim N .We i i i have CΓ = End N = Mat (C).Let e (1 ≤ p, q ≤ δ ) be the element of CΓ i δ i i∈I i∈I i p,q with 1 in the ( p, q)-entry of thematrixfor the i-th summand and zero elsewhere. Let e be the idempotent e .In particular, e = γ/|Γ|,where o is the extending vertex 1,1 γ ∈Γ of the affine Dynkin quiver Q.Notethat N = C[Γ]e and φ ∈ e (L ⊗ C[Γ]e ). i i a h(a) t(a) Here, the left action of Γ on L ⊗ C[Γ] is the diagonal one. When Q is not of type A , φ spans e (L ⊗ C[Γ]e ).When Q is of type A with vertices o and i, e (L ⊗ C[Γ]e ) a h(a) t(a) 1 o i and e (L ⊗ C[Γ]e ) are both 2 dimensional and spanned by the intertwiners φ which i o a they contain. 106 PAVEL ETINGOF, WEE LIANG GAN, VICTOR GINZBURG, ALEXEI OBLOMKOV We put e := σ ∈ C[S ].For any i ∈ I,let n n σ ∈S n! n (2.2.1) e := e (e ⊗ e ⊗ ··· ⊗ e ) ∈ C[Γ ], and i,n−1 n−1 i o o n e := e (e ⊗ ··· ⊗ e ) ∈ C[Γ ]. n o o n The idempotent e is same as the one that appears in Theorem 1.4.4 of the Introduc- tion. For each vertex i of the Calogero–Moser quiver Q , cf. Definition 1.3.1, the CM idempotent e is the trivial path at the vertex i.Let λ be the trace of t · 1 + c on N , i i i let λ = λ e ,and let i i i∈I ν := k|Γ|/2. Let Π = Π (Q ), the deformed preprojective algebra of Q with parameter λ−νe +nνe CM CM o s λ − νe + nνe as defined in [CBH]. So Π is the quotient of the path algebra CQ o s CM of Q by the following relations: CM ∗ ∗ ∗ [a, a ]+ bb = λ − νe , b b =−nνe . o s a∈Q We shall define a functor from H-modules to Π-modules. Let M be a H-module. We want to define a Π-module M.For each i ∈ I,let M := e M. Also, let M := i i,n−1 s eM.If a is an edge in Q , then define a : M → M to be the map given by the t(a) h(a) element φ ⊗ e ⊗ ··· ⊗ e ∈ H.Define b : M → M to be the inclusion map, and a o o s o define b : M → M to be −ν · (1 + s + ··· + s ). o s 12 1n Lemma 2.2.2. — With the above action, M is a Π-module. Proof. —Itisclear that (1 + s +···+ s )e = ne .On M,wehave b b =−nν, 12 1n n−1 n and bb =−nνe =−ν · (1 + s + ··· + s ). n 12 1n ⊗n ⊗n By [GG, (3.5.2)], we have an isomorphism f H f = A (Q ) where n,λ,ν ⊗n f = e , cf. Definition 6.1.3 below. Now f M is a A (Q )-module, and i n,λ,ν i∈I ⊗n ⊗n e M = e f M, eM = e f M. The action of the edge a : M → M is i,n−1 i,n−1 t(a) h(a) ⊗(n−1) the action given by the element a ⊗ e ∈ A (Q ). n,λ,ν Now on M,at a vertex i = o, s,wehave ∗ ∗ aa − a a = λ a∈Q ;h(a)=i a∈Q ;t(a)=i by the relation (i) in Definition 6.1.3. At the vertex o,we have ∗ ∗ ∗ aa − a a = λ + ν · (s + ··· + s ) = λ − ν − bb , o 12 1n o a∈Q ;h(a)=o a∈Q ;t(a)=o using again the relation (i) in Definition 6.1.3. HARISH–CHANDRA HOMOMORPHISMS AND SYMPLECTIC REFLECTION ALGEBRAS 107 It is clear that the assignment G : M → M is functorial. We will give a more general construction in Section 6.2. Applying the functor M → M to the H-module M introduced in (1.7.1) we Quiver construct, as has been explained in Section 1.7, the algebra homomorphism θ : Π → B. Quiver ∗ Observe that θ (b ) is 0 when ν = 0. For this reason, we shall need a slight modification of the above constructions. Define Π to be the quotient of the path algebra CQ by the following rela- CM tions: ∗ ∗ ∗ [a, a ]+ νbb = λ − νe , b b =−ne . o s a∈Q We have a morphism of algebras Π → Π defined on the edges by ∗ ∗ ∗ a → a for a = b , b → νb . This is an isomorphism only when ν = 0. Given a H-module M,wedefine a Π -module structure on M as above, except that now, we let b : M → M be −(1 + s + ··· + s ). Hence,as above,we obtain o s 12 1n Quiver amorphism Θ : Π → B, cf. (1.7.4). 2.3. Holland’s map In this subsection, we recall a construction of Holland from [Ho]. Let ∈ Z denote the coordinate vector corresponding to the vertex i ∈ I.Let δ = δ , the minimal positive imaginary root of Q .Let α := nδ + ,adimension i i s i∈I vector for Q .Thus, α = nδ for i ∈ I,and α = 1. We shall assume that λ · δ = 1, that CM i i s is, t = 1/|Γ|. a a Let e and t (a ∈ Q , 1 ≤ p ≤ α , 1 ≤ q ≤ α ) be, respectively, the CM h(a) t(a) p,q p,q coordinate vectors and the coordinate functions on Rep (Q ).We write e for the CM q,p transpose of e . Now define a representation of Q on O(Rep (Q ))⊗N,the space CM CM p,q α of N-valued regular functions on Rep (Q ), as follows. For any a ∈ Q ,wedefine α CM CM the following End N-valued differential operators of order 0 and 1, respectively a a a a := − e ⊗ t , resp., a := e ⊗ . ˆ ˆ p,q p,q q,p ∂t p,q p,q p,q The assignment a → a, a → a extends by multiplicativity to an algebra homo- ˆ ˆ morphism Holland GL(α) (2.3.1) θ : CQ → (D(Q ,α) ⊗ End N) , CM CM 108 PAVEL ETINGOF, WEE LIANG GAN, VICTOR GINZBURG, ALEXEI OBLOMKOV where CQ denotes the path algebra of the double quiver Q .By[Ho,The- CM CM Holland orem 3.14] and [Ho, Lemma 3.16], θ descends to homomorphisms, cf. notation in (1.4.2): Holland Holland θ : Π → T and θ : e Πe → A . χ s s χ We remind that A is the algebra in (1.6.2). Holland Later, we will define a homomorphism Θ : e Π e → A . s s χ 3. Radial part map From now on, we assume that Q is not of type A where n is even. Equivalently, that means that Γ has the (necessarily unique) central element of order 2, to be denoted ζ ∈ Γ. 3.1. Twisted differential operators × m Let T = (C ) be a torus with Lie algebra t := Lie T, and p : X → X aprin- cipal T-bundle. For any h ∈ Lie T, the infinitesimal h-action yields a vector field ξ on X.Let D be the sheaf of algebraic differential operators of X. The action of T on X makes D a T-equivariant sheaf of algebras, and we write Γ(X, D ) for the al- X X gebra of T-invariant global differential operators on X. The assignment h → ξ gives a Lie algebra homomorphism t → Γ(X, D ) . Let ρ : t → End F be a finite dimensional representation. We form D := X,F D ⊗ End F, a sheaf of associative algebras on X.Let ξ − ρ : t → D = X X,F D ⊗ End F, h → ξ ⊗ Id − ρ(h) be the diagonal Lie algebra homomorphism. X F adt We write Im(ξ − ρ) for the image of this homomorphism, and ( p D ) for the ∗ X,F subsheaf of those sections of the push-forward sheaf p D ,on X,which commute ∗ X,F adt with Im(ξ − ρ).Thus, Im(ξ − ρ) is a central subspace of ( p D ) , a sheaf of asso- ∗ X,F adt ciative algebras on X.We write ( p D ) · Im(ξ −ρ) for the (automatically two-sided) ∗ X,F adt ideal in ( p D ) generated by the image of ξ − ρ. Thus, the quotient ∗ X,F adt adt ( p D ) /( p D ) · Im(ξ −ρ) is a well-defined sheaf of associative algebras on X. ∗ X,F ∗ X,F Let adt adt (3.1.1) D(X, p,ρ) := Γ X,( p D ) /( p D ) · Im(ξ − ρ) ∗ X,F ∗ X,F be the algebra of its global sections, to be referred to as the algebra of twisted differential operators on X associated with the principal T-bundle p : X → X and representation ρ. For any open set U (in the ordinary, Hausdorff topology), we write H ol(U, F) for the vector space of all holomorphic maps U → F. Given such an open subset HARISH–CHANDRA HOMOMORPHISMS AND SYMPLECTIC REFLECTION ALGEBRAS 109 U ⊂ X, put (3.1.2) H ol (U) := { f ∈ H ol(U, F) | ξ f = ρ(h) f , ∀h ∈ t}. There is a natural action of the algebra D(X, p,ρ) on the vector space H ol (U) given, in local coordinates, by differential operators with End F-valued coefficients. Given a decomposition F = F ⊕···⊕F into a direct sum of t-subrepresentations, 1 r we have End F = Hom(F , F ). This gives the induced direct sum decompo- m l 1≤m,l≤n sion D(X, p,ρ) = D(X, p,ρ, F → F ). m l 1≤m,l≤n Thus, for each (m, l ), the direct summand D(X, p,ρ, F → F ) has a natural structure m l of left D(X, p,ρ| )-module and of right D(X, p,ρ| )-module. F F l m 3.2. The radial part construction Let G be a linear algebraic group and Y a smooth G-variety. Assume in addition that we have a smooth subvariety X ⊂ Y which is stable under the action of a torus T ⊂ G, and we also have a smooth morphism p : X → X, which is a principal T-bundle with respect to the induced T-action on X. Thus, we have the following diagram g×x →g(x) x →1×x oo oo (3.2.1) G × X X X Y. p act Let g := Lie G and let ρ : g → End F be a finite dimensional representation. For any open subset U ⊂ Y, wemay consider thevectorspace H ol (U ) defined Y ρ Y similarly to (3.1.2) but with respect to the g-representation ρ.Write ρ = ρ| for the t t restriction of ρ to the Lie algebra t = Lie T. Restriction of functions gives the map Res : H ol (U ) → H ol (X ∩ U ), f → Res f := j f . ρ Y ρ Y Let D(Y, F) = Γ(Y, D ) = Γ(Y, D ) ⊗ End F be the algebra of End F-valued Y,F Y differential operators on Y. As above, wehave the Liealgebra map ξ − ρ : g → adg D(Y, F) and the subalgebra D(Y, F) , of the operators commuting with the image of that map. Let K ⊂ G be a finite subgroup that preserves X and normalizes the torus T. The action of K on X,resp. on D , induces a natural K-action on X,resp. on the algebra D(X, p,ρ ), of twisted differential operators on X.Wewrite D(X, p,ρ ) for t t the subalgebra of K-invariants. 110 PAVEL ETINGOF, WEE LIANG GAN, VICTOR GINZBURG, ALEXEI OBLOMKOV One has the following standard result. Proposition 3.2.2 (Radial part map). — Assume that G is connected and the differ- ential of the map act,in (3.2.1), is an isomorphism at every point j( x), x ∈ X.Then, Radial adg K (i) There is a natural radial part homomorphism θ : D(Y, F) → D(X, p,ρ ) such that, for any open (in the Hausdorff topology) subset U ⊂ Y, we have Radial adg θ (D) · (Res f ) = Res(Df ), ∀D ∈ D(Y, F) , f ∈ H ol (U ). ρ Y adg (ii) The two-sided ideal (D(Y, F) Im(ξ − ρ)) is contained in the kernel of the radial Radial part map θ . (iii) Assume, in addition, that X is affine and the restriction of ρ to t is diagonalizable. Then, there are canonical algebra isomorphisms, cf. (1.1.1), A(D(X, F),t, ξ − ρ) adt adt = D(X, F) /(D(X, F) Im(ξ − ρ)) D(X, p,ρ ). 3.3. A slice in Rep (Q ) α CM We choose the following orientation on Q:the vertex o is a sink, and any other vertex is a source or a sink. Thus, the second order element ζ acts by 1 at sinks and by −1 at sources. Note also that, see (1.4.2) ∂ =−n Tr | (ζ), i ∈ I. i N The collection of intertwiners φ = (φ ) introduced in Section 2.1 gives rise a a∈Q to a linear map ∗ ∗ φ : L → Rep (Q ), where φ : L → Hom(N , N ), u → φ (u). a a t(a) h(a) We also define a linear map j : L → Rep (Q ) by CM j( u ,..., u ) = (1, 1, ..., 1), and 1 n b j( u ,..., u ) = (φ (u ), ..., φ (u )), ∀a ∈ Q . 1 n a a 1 a n Lemma 3.3.1. — Let u, w ∈ L. Suppose there are β ∈ End(N ) for i ∈ I such that φ (u)β = β φ (w) for all edges a ∈ Q.Ifthe β are not all zero, then u is proportional to a t(a) h(a) a i γ w for some γ ∈ Γ. The lemma will be proved later, at the end of Section 8.2. From this lemma, we deduce HARISH–CHANDRA HOMOMORPHISMS AND SYMPLECTIC REFLECTION ALGEBRAS 111 n n Corollary 3.3.2. — There exists a Γ -stable Zariski-open dense subset L ⊂ L such that reg n n j( L ) is contained in the set of generic representations of Q and, moreover, j( L ) meets generic reg reg GL(α)-orbits in Rep (Q ) transversely. CM Proof. — First, we claim that the affine space j( L ) meets generic GL(α)-orbits in Rep (Q ). CM Recall that generic representations of Q with dimension vector nδ are direct sums of n representations with dimension vector δ, it suffices to show that the subspace consisting of the representations {φ (u)} for all u ∈ L intersects generic GL(δ)-orbits a a∈Q in Rep (Q ). By the preceding lemma, the (rational) map Rep (Q ) → P defined δ δ in [Ri, Theorem 6.2] (which parametrizes generic orbits) is nonconstant on L. The corollary now follows from the standard Bertini–Sard theorem, cf. e.g. [Ve]. 3.4. Discriminant function × × Put L := L{0}. The multiplicative group C is imbedded in GL(L) as scalar × × × × 1 matrices, and we have the standard C -bundle L → P := L /C = P ( projective line). The group S := C ∩ Γ consists of two elements, S ={1,ζ },where ζ ∈ Γ is our second order element. Given ∈ P ,write Γ ⊂ Γ for the isotropy group of the line ⊂ L. Clearly, one has S ⊂ Γ . Therefore, |Γ |/2 =|Γ /S| is a positive integer, and we put κ := |Γ /S|− 1. Thus, we have κ >0 if and only if the group Γ ⊂ Γ,is strictly larger sing sing than S. The lines with this property form a finite set P ⊂ P .For each ∈ P , L L we choose and fix a vector representative v ∈ {0}⊂ L. We introduce the following function ∆ := ω(v , −) ∈ C[L], sing ∈P which is uniquely defined up to a nonzero constant factor depending on the choice sing of representatives v , ∈ P . Further, we introduce a discriminant function on L , L reg defined by 1 1 (3.4.1) ∆ (u , ..., u ) := , for u ,..., u ∈ L. n 1 n 1 n ∆ (u ) ω(u ,γ u ) k m l γ ∈Γ k=1 m =l Let L be a Zariski open set as in Corollary 3.3.2. Shrinking this set if neces- reg sary, from now on we assume in addition that L is an affine T-stable subset con- reg × n tained in (L ) and, moreover, that the denominator of the function ∆ does not van- n n n ish on L .Thus, theset L is Γ T-stable, and we have ∆ ∈ C[L ]. n n reg reg reg 112 PAVEL ETINGOF, WEE LIANG GAN, VICTOR GINZBURG, ALEXEI OBLOMKOV The natural action of the torus T on L induces an action of the Lie algebra reg n n t = Lie T on the coordinate ring C[L ].Given h = (h , ..., h ) ∈ t = C , we write 1 n reg n n the action map as h : C[L ]→ C[L ], f → ξ f , and also put Tr(h) = h + ... + h . 1 n reg reg Lemma 3.4.2. — The discriminant ∆ ∈ C[L ] is a weight vector for the t-action, reg specifically, we have ξ ∆ = (n|Γ|− 2) Tr(h) · ∆ , ∀h ∈ t. n n sing Proof. —Note that κ =|Γ|−2.Hence, ∆ is a homogeneous polynomial ∈P of degree |Γ|− 2. We see that any vector u appears on the RHS of (3.4.1) with −1 degree −(|Γ|− 2), in the factor ∆ (u ) ,and with degree −(n − 1)|Γ|,in the factor 1/ ω(u ,γ u ). m l 3.5. Compatibility with group actions × n Let T := (C ) be the torus, and form the wreath product Γ T = × n S (Γ × C ) . We are going to define a group imbedding (3.5.1) j : Γ T → GL(α). Lie n To this end, we recall the direct sum decomposition (1.5.1), so dim N = α and i i one may identify the group GL(α ) with GL(N ),for any i ∈ I . Now, by the structure i i CM of group algebras, we have the canonical algebra isomorphism C[Γ] ⊕ End(N ). i∈I Thus, we have a group imbedding Γ → GL(δ) and, therefore, an imbedding Γ → GL(δ) × ··· × GL(δ) → GL(α ). Further, we define a homomorphism S → i n i∈I ∗ n n GL(N ) by σ → Id ⊗ σ , where σ ∈ GL(C ) stands for the permutation matrix i N C C corresponding to σ ∈ S . This way, combining together the above defined imbeddings of Γ and S , we obtain a group imbedding j : Γ → GL(α),suchthat its compon- n Lie n ent Γ → GL(α ), at the special vertex s, sends every element of Γ to 1. n s n It remains to define the torus imbedding j : T → GL(α), t → g(t) ={ g (t) ∈ Lie i GL(α )} . The latter is given as follows. We put g (t) = 1, ∀t ∈ T, and, for any i i∈I s CM i ∈ I,let −1 g (t) := t ⊗ Id if i is a source in Q , i N g (t) := Id if i is a sink in Q , i N −1 n where, for t = (t , ..., t ) we let t ∈ GL(C ) denote the diagonal matrix with diag- 1 n −1 −1 onal entries t , ..., t .Wenotethatthe image of T under the above imbedding is not contained in the center of the group GL(α). × n n The torus T := (C ) acts naturally on L ; the element t = (t , ..., t ) ∈ T 1 n sends (u , ..., u ) ∈ L to (t u , ..., t u ).Thisactionof T commutes with that of the 1 n 1 1 n n HARISH–CHANDRA HOMOMORPHISMS AND SYMPLECTIC REFLECTION ALGEBRAS 113 n n group Γ . Thus, we get an action of the group Γ T on L .It is easyto show that the group imbedding j : Γ T → GL(α) agrees with the slice imbedding Lie n j : L → Rep (Q ).Specifically,one has CM (3.5.2) j( g(u)) = j ( g)(j( u)), ∀u ∈ L , g ∈ Γ T. Lie n Radial Radial 3.6. The homomorphisms θ and Θ CM The element χ ∈ C , in (1.4.2), gives a Lie algebra homomorphism χ : gl(α) → C. We also have the tautological representation τ : gl(α) → End N, see (1.5.1), and we let χ − τ : gl(α) → End N, h → χ(h) · Id − τ(h), ∀h ∈ gl(α). The group imbedding (3.5.1) induces the corresponding Lie algebra imbedding j : t = Lie T → gl(α).Welet ρ := (χ − τ) ◦ j be the pull-back of the represen- Lie Lie tation χ − τ to the Lie algebra t via the imbedding t → gl(α). We are now in a position to apply the general radial part construction of Sec- tion 3.2 in our special case. Specifically, the n-th cartesian power of the C -bundle × × n n n × n L → P gives a principal T-bundle (L ) → (P ) .We set X := L ⊂ (L ) , L L reg and let X ⊂ (P ) be the image of X.Write p : X → X for the restriction of the bundle projection to X.Thus, X is a Γ -stable Zariski open dense subset of (P ) , n L and p : X → X is a principal T-bundle. We apply Proposition 3.2.2 to × n G = GL(α), T = (C ) , K = Γ , Y = Rep (Q ), n α CM n n p : X = L → X = L /T. reg reg Thus, we obtain an algebra homomorphism, cf. (1.5.2): GL(α) (D(Q ,α) ⊗ End N) CM Radial Γ (3.6.1) θ : T = → D(X, p,ρ) . GL(α) ((D(Q ,α) ⊗ End N) Im(ξ − (χ − τ))) CM Further, we introduce another representation : t → End N, h → (h) by the formula (h) := ρ(h) + (n|Γ|− 2) Tr(h)Id . It is easy to see that each of the direct summands in the decomposition N = ⊕ N , cf. (1.5.1), is stable under the t-action via either representation ρ or .Thus i∈Q i CM we can write ρ =⊕ ρ ,and =⊕ ρ . To describe these representations more i∈I i i∈I i CM CM explicitly, let c be the coefficient in (1.2.1) corresponding to our second order element ζ ∈ Γ, and put c ·|Γ| (3.6.2) µ := − + 1 , and ψ := δ · χ ∈ C. i i {i∈I | i is a source in Q } 114 PAVEL ETINGOF, WEE LIANG GAN, VICTOR GINZBURG, ALEXEI OBLOMKOV One finds that the representations ρ and are given by the following explicit i i formulas: ρ (h) = ψ · Tr(h) · Id , (h) = µ + · Tr(h) · Id i N i N i i if i = s or i is a sink in Q;and ρ (h) = ψ · Tr(h) · Id − h ⊗ Id , i N N (h) = µ + · Tr(h) · Id − h ⊗ Id i N N if i is a source in Q,where,for any h = (h ,..., h ) ∈ t = C , we write Tr(h) := 1 n n n h + ... + h , and where the tensor factor h,in h ⊗ Id , stands for the map C → C 1 n N given by the diagonal matrix with diagonal entries h , ..., h . 1 n Next, according to Lemma 3.7.6 below, we have c ·|Γ| (3.6.3) 2(µ − ψ) =−2 + 1 + 1 − 2ψ = (|Γ|− 2) + (n − 1)|Γ|= n|Γ|− 2. Hence, Lemma 3.4.2 shows that − ρ = (n|Γ|− 2) · Tr(−) is nothing but the weight of ∆ , the square root of the discriminant function. Thus, given D ∈ D(X, p,ρ), we may conjugate D by the operator of multiplication by the function ∆ to obtain a twisted differential operator ◦ D ◦ ∆ ∈ D(X, p,), such that for any open set U ⊂ L the induced action on functions is given by the map reg Radial Γ(U,) → Γ(U,), f → (1/ ∆ ) · θ (D)( ∆ · f ). n n We note that although the square root of the discriminant function ∆ is ill de- fined as a function, conjugation by the operator of multiplication by such a function is an unambiguously defined operation on twisted differential operators. Furthermore, the result of conjugation by ∆ is clearly independent of the choice of nonzero con- stant factor involved in the definition of ∆ , cf. Section 3.4. Thus, we have a canoni- cally defined algebra homomorphism Radial Γ Θ : T → D(X, p,) , Radial Radial u → Θ (u) := √ ◦ θ (u) ◦ ∆ . Radial Holland 3.7. Formulas for the map θ ◦ θ We introduce some notation. Given a map f : L → U and any 1 ≤ l ≤ n,we write f for the map L → U,(u ,..., u ) → f (u ).Thus, given γ ∈ Γ, we have the l 1 n l composite L → L → U, and the corresponding map ( f ◦ γ) : L → U. l HARISH–CHANDRA HOMOMORPHISMS AND SYMPLECTIC REFLECTION ALGEBRAS 115 Let ω denote the symplectic form on L.For anyvector v ∈ L,wehavethe linear function v : u → ω(v, u).Thus, for any v ∈ L and γ ∈ Γ,wemay consider the following functions ∨ ∨ −1 n (3.7.1) (γ v) := (v ◦ γ ) : L → C, −1 (u , ..., u ) → ω(γ v, u ) = ω v,γ u and 1 n l l ω(γ ; m, l ) : L → C,(u ,..., u ) → ω(u ,γ u ), ∀1 ≤ m = l ≤ n. 1 n m l n n The definition of the open subset L ⊂ L insures, see Section 3.4, that none of the reg n n functions ω(γ ; m, l ) vanishes on L .Hence,we have 1/ω(γ ; m, l ) ∈ C[L ]. reg reg In Section 2.1, for each edge a ∈ Q,wehaveintroduced intertwiners ∗ ∗ ∗ ∗ φ : L ⊗ N → N and φ : N → L ⊗ N . Below, we shall view φ as a a a t(a) h(a) a t(a) h(a) ∗ ∗ Hom(N , N )-valued linear function on L, written as u → φ (u).The n-th carte- t(a) h(a) sian power of this function gives a Γ -equivariant linear map n n ∗ ∗ n φ : L → Hom(N , N ) = Hom(N , N ) ⊗ End C , t(a) h(a) a t(a) h(a) (u , ..., u ) → φ (u ) ⊗ E , 1 n a l ll 1≤l≤n where E stands for the n × n-matrix unit with the only nonzero entry at the place ll (l, l ). ∗ ∗ Similarly, we shall view φ as a Hom(N , N )-valued constant vector field on a t(a) h(a) L whose value at each point u ∈ L is equal to φ .Thus, given m ∈[1, n], we shall ∂ ∗ ∗ n write for the Hom(N , N )-valued first order differential operator on L cor- ∗ h(a) t(a) (∂φ ) a m ∗ ∗ responding to the constant vector field φ ∗ ∈ Hom(N , N ) ⊗ L along the m-th a h(a) t(a) direct factor L in L . Holland Radial Next, recall the map θ , introduced in Section 2.3. The composite θ ◦ Holland θ associates to every edge a ∈ Q a twisted differential operator from the algebra n adt D(X, p,ρ). By definition, such an operator is a coset modulo the ideal D(L , N) · reg Im(ξ − ρ), see Proposition 3.2.2(iii), of an element Radial Holland n n θ ◦ θ (a) ∈ D L ⊗ Hom(N , N ) ⊂ D L , N . t(a) h(a) reg reg ∗ ∗ We will write such an element D as an n × n-matrix with entries in Hom(N , N ), t(a) h(a) and write D for (m, l )-th entry of that matrix. ml Proposition 3.7.2. — Let a ∈ Q.Then Radial Holland n (i) θ ◦ θ (a) is a zero-order differential operator on L given by multiplication reg by the function φ . a 116 PAVEL ETINGOF, WEE LIANG GAN, VICTOR GINZBURG, ALEXEI OBLOMKOV Radial Holland ∗ (ii) For l = m,the (m, l )-entry of θ ◦θ (a ) is is a zero-order differential operator on L given by multiplication by the function reg (φ ∗ ◦ γ) a l (3.7.3) −k/2 γ, ω(γ ; m, l ) Radial Holland ∗ and the (m, m)-entry of θ ◦ θ (a ) is a first order differential operator −1 1 ∂ (φ ◦ (γ + Id)) a mm −1 (3.7.4) + − 1 +|Γ|c γ |Γ| ∂(φ ∗ ) ω(γ ; m, m) a m γ =1,ζ 1 (φ ◦ γ) + . |Γ| ω(γ ; m,) =m (iii) For the edge b : s → o we have Radial Holland b (3.7.5) θ ◦ θ (b) =− e , p,1 Radial Holland ∗ b b θ ◦ θ (b ) = (1 − δ χ ) e = ν e . j j 1,p 1,p j∈I p p The proof of Proposition 3.7.2 will be given later, in Section 8. We will use Lemma 3.7.6. — We have c + n = (1 − 2ψ)/|Γ|, and k = 2(1 − δ χ )/|Γ|. ζ j j Furthermore, c = 1 − χ (δ − Tr | (γ)) /|Γ| for γ = ζ. γ j j N Proof. —Since λ = Tr | (t · 1 + c), we obtain by orthogonality relations that i N c = 1/|Γ| λ Tr | (γ). Hence, for ∂ as in (1.4.2), we compute γ j N j∈I χ (δ − Tr | (γ)) j j N ∗ ∗ = λ · δ − ν − ∂ · δ − λ Tr | (γ) + ν − n Tr | (ζ) Tr | (γ) j N N N j j j j = 1 −|Γ|c − n Tr | (ζ) Tr | (γ). γ N N If γ = ζ , then this is equal to 1 −|Γ|c .If γ = ζ, then itisequal to 1 −|Γ|c − n|Γ|. γ ζ Moreover, χ (δ − Tr | (ζ)) = 2 χ δ = 2ψ. j j N j j j {j∈Q | j is a source} We also have χ δ = 1 − ν = 1 − k|Γ|/2. j j j HARISH–CHANDRA HOMOMORPHISMS AND SYMPLECTIC REFLECTION ALGEBRAS 117 4. Dunkl representation 4.1. Dunkl operators n n Recall the principal T-bundle p : X = L → X = L /T, and other notation reg reg introduced in Section 3.6. We are going to define a certain representation η of the Lie algebra t = Lie T which is normalized by the natural Γ -action on t.Thus, thereis an action of Γ on D(X, p,η), the corresponding algebra of twisted differential oper- ators. Our goal is to define certain elements in D(X, p,η)Γ which may be thought of as Γ-analogues of Dunkl operators. The construction of these operators will be given in five steps. sing Step 1. —Write L for the preimage of P P under the projection reg L L P .Thus L ⊂ L is an open dense subset formed by the points v ∈ L such L reg that, for any γ ∈ Γ{1,ζ }, we have γ(v) ∈ Cv. For any γ ∈ Γ, we have a quadratic function ω : L → C, u → ω(u,γ u).This function does not vanish on L , thus, we have 1/ω ∈ C[L ].Given v ∈ L, we also reg reg have the linear function v : u → ω(v, u), on L. Recall the coefficients c ∈ C given by (1.2.1). To each v ∈ L we assign the following element ∂ (γ v + v) v −1 (4.1.1) D := 2|Γ| + c γ ∈ D(L ) Γ. γ reg ∂v ω γ =1,ζ Step 2. —Let F = C be a 2-dimensional vector space with fixed basis + − ( f , f ),and identify End F with the algebra of 2 × 2-matrices. We set D(L , F) = reg D(L ) ⊗ End F, and form the smash product D(L , F) Γ = (D(L ) ⊗ End F) Γ, reg reg reg where Γ acts trivially on F and on End F. For each v ∈ L, we introduce the following element written as a matrix with entries in D(L ) Γ: reg 0 −v 00 01 v v ∨ (4.1.2) D := = D · − v · ∈ D(L , F) Γ. reg D 0 10 00 Step 3. —For any affine variety Y and n ≥ 1, one has the standard alge- n ⊗n ⊗n ⊗n ∼ ∼ bra isomorphism D(Y ) = D(Y ) .Since End(F ) = (End F) , we deduce an n ⊗n ⊗n algebra isomorphism D(Y , F ) = D(Y , F) . The symmetric group S acts nat- n ⊗n n ⊗n urally on Y and on (End F) , hence, also on the tensor product D(Y , F ) ⊗n D(Y , F) . We take Y = L and put X := L , cf. Section 3.4. Thus, X is a Γ -stable reg n reg affine open dense subset of (L ) , and we have a chain of natural algebra imbed- reg 118 PAVEL ETINGOF, WEE LIANG GAN, VICTOR GINZBURG, ALEXEI OBLOMKOV dings ⊗n n ⊗n n ⊗n n (D(L , F) Γ) = D (L ) , F Γ → D(X, F ) Γ reg reg ⊗n → D(X, F )Γ , where the middle imbedding is given by restriction from (L ) to X. reg v,n n ⊗n For any v ∈ L and l = 1, ..., n, let D ∈ D(L , F )Γ denote the image of F,l reg ⊗(l−1) v ⊗(n−l ) the element 1 ⊗ D ⊗ 1 , cf. (4.1.2), under this imbedding. Step 4. —For any l = 1, ..., n, and γ ∈ Γ,let γ ∈ Γ denote a copy of the element γ placed in the l-th factor of Γ . In particular, given any 1 ≤ m = l ≤ n,and γ ∈ Γ, we have the corresponding transposition s ={m ↔ l}∈ S and the element ml n −1 s γ γ ∈ Γ .Given γ ∈ Γ, v ∈ L and any 1 ≤ m = l ≤ n,we alsohaveregular ml m n ∨ n functions v and 1/ω(γ ; m, l ) on L , see (3.7.1). l reg With this notation, for any v ∈ L and 1 ≤ m = l ≤ n,wewillnow define an element v ⊗n ⊗n ⊗n (4.1.3) R ∈ (C[X]⊗ End F )Γ = Hom(F ,(C[X]⊗ F )Γ ). n n ml To this end, write (4.1.4) f = f ⊗ ... ⊗ f | f = f , i = 1, ..., n 1 n i ⊗n for the standard basis of the vector space F . Given such a basis element f = f ⊗ − − ... ⊗ f and 1 ≤ m ≤ n,let f := f ⊗ ... ⊗ f ⊗ f ⊗ f ⊗ ... ⊗ f .Now, vieweach n 1 m−1 m+1 n v ⊗n ⊗n R in (4.1.3) as a linear map F → (C[X]⊗ F )Γ , which we define as follows ml 1 (γ v ) v − −1 + + (4.1.5) R ( f ) = ⊗ f · s γ γ , if f = f , f = f , ml m m l ml m l 2 ω(γ ; m, l ) γ ∈Γ 1 v v − −1 + − R ( f ) = ⊗ f · s γ γ , if f = f , f = f , ml m m l ml m l 2 ω(γ ; m, l ) γ ∈Γ v − R ( f ) = 0, if f = f . ml ⊗n ⊗n We identify the algebra C[X]⊗ End F with the subalgebra of D(X, F ) formed by zero order differential operators. Therefore, we may view the elements R , ml ⊗n in (4.1.5), as being elements of D(X, F )Γ , which have zero order as differential operators. Given k ∈ C and v ∈ L, we define first order differential operators v v,n v ⊗n (4.1.6) Dunkl := D + k R ∈ D(X, F )Γ , ∀1 ≤ l ≤ n. l F,l lm l =m Step 5. —Let µ ∈ C be the constant defined in (3.6.2). We introduce a rep- resentation of the 1-dimensional Lie algebra C on the vector space F. Specifically, we HARISH–CHANDRA HOMOMORPHISMS AND SYMPLECTIC REFLECTION ALGEBRAS 119 + − let the generator 1 ∈ C act, in the basis { f , f }, via the diagonal matrix 1 1 diag(µ + ,µ − ).The n-th cartesian power of this 2-dimensional representation 2 2 n ⊗n gives a Lie algebra representation η : t = C → End F . We consider the Lie algebra homomorphism ⊗n ξ − η : t → D(X, F ), h → ξ ⊗ Id ⊗n − Id ⊗ η(h). F D ⊗n The group Γ acts naturally both on the Lie algebra t and on D(X, F ), and it is ⊗n adt clear that the map ξ − η is Γ -equivariant. It follows in particular that D(X, F ) , ⊗n the centralizer of the image of the map ξ − η in D(X, F ),is a Γ -stable subalgebra. Now, we apply the general construction of algebras of twisted differential opera- × n n tors given in Section 3.1 to the torus T = (C ) acting on X = L and to the repre- reg sentation η defined above. This way, with the notation of Section 3.6, we get an alge- ⊗n adt bra D(X, p,η). By construction, the algebra D(X, p,η) is a quotient of D(X, F ) , and this quotient inherits a natural Γ -action. Thus, wemay form thecorresponding algebra D(X, p,η)Γ . It is straightforward to verify, counting homogeneity degrees of the coefficients, that for any v ∈ L the operator in (4.1.6) is adt-invariant. That is, for each l = 1,..., n, v v ⊗n adt we have Dunkl ∈ D(X, F ) Γ . Therefore, the element Dunkl has a well defined l l image in D(X, p,η)Γ , to be denoted by the same symbol Dunkl and to be called the l-th Dunkl operator associated with v ∈ L. 4.2. Equalizer construction Recall from Section 3.4, the group S ={1,ζ } = Z/2.Thus, S = C ∩ Γ ⊂ GL(L) may be (and will be) viewed either as a subgroup of C or as a subgroup of Γ. We put S := S . This group comes equipped with a natural group imbedding ε : S → Γ ⊂ Γ , such that the image of S is a normal subgroup in Γ ,and also Γ n n with a natural imbedding S → T. In general, let A be an associative algebra equipped with a Γ -action Γ g : a → a , by algebra automorphisms, and also with a homomorphism a : S → A, that is, with a map such that a(1) = 1, and such that a(ss ) = a(s) · a(s ), ∀s, s ∈ S. Assume in addition that the following identities hold (the one on the left says that a is a Γ -equivariant map): g −1 −1 ε (s) (4.2.1) a(s) = a( gsg ), and a(s) · a · a(s ) = a , ∀s ∈ S, g ∈ Γ , a ∈ A. We form the smash product AΓ and introduce the following two homomor- phisms Υ , Υ : S → AΓ , where Υ : s → a(s) 1, Υ : s → 1 ε (s). 1 2 n 1 2 Γ 120 PAVEL ETINGOF, WEE LIANG GAN, VICTOR GINZBURG, ALEXEI OBLOMKOV It is straightforward to verify that equations (4.2.1) imply that the left ideal in the algebra AΓ generated by the set {Υ (s) − Υ (s), s ∈ S} is in effect a two-sided n 1 2 ideal. We let A Γ be the quotient of A Γ by that two-sided ideal, to be S n n called equalizer smash product algebra. Dunkl 4.3. The homomorphism θ Let I ={( , ..., ) | = 0 or 1 for all m}⊂ Z . 1 n m ⊗n Let = ( , ..., ) ∈ I,and write F for the one dimensional subspace of F spanned 1 n + − by f ⊗ ··· ⊗ f ,where f = f if = 0,and f = f if = 1 (m = 1, ..., n). 1 n m m m m ⊗n The representation η of t on the vector space F induces an adjoint action of t on ⊗n End(F ). We have a decomposition ⊗n ∗ End(F ) = F ⊗ (F ) , , ∈I where each component in the direct sum is stable under the action of t.Moreover, ∗ ∗ t acts on F ⊗ (F ) by the character − . Therefore, the t-action on F ⊗ (F ) exponentiates to a T-action. Taking the direct sum over various pairs (, ),weobtain ⊗n ⊗n ⊗n ∗ a well defined adjoint T-action on End F = F ⊗ (F ) .Thus, forany t ∈ T,the ⊗n ⊗n adjoint action of t gives an algebra automorphism Ad (t) : End F → End F . The torus T also acts naturally on X = L . The tensor product of the in- reg ⊗n duced T-action on D(X) with the adjoint T-action on End F gives a T action on ⊗n ⊗n D(X, F ) = D(X) ⊗ End F , to be called the adjoint action Ad : T → D⊗F ⊗n Aut(D(X, F )).The map Ad is clearly Γ -equivariant. It is also clear from the D⊗F n construction that the differential of the Ad -action of T is nothing but the adjoint D⊗F ⊗n adt ⊗n Ad T D ⊗F action of the Lie algebra t.Inparticular, we have D(X, F ) = D(X, F ) . Next, we are going to apply the general construction of Section 4.2 in the fol- lowing special case. Let S → D(L , F) = D(L ) ⊗ End F be a homomorphism given reg reg by the assignment 1 → 1 ⊗ Id ,ζ → 1 ⊗ . D F D 0 −1 ⊗n We define a homomorphism a : S → D(X, F ) to be the composite of the n-th cartesian power of the above homomorphism S → D(L , F), followed by the nat- reg n ⊗n ⊗n ural imbedding D((L ) , F ) → D(X, F ). This homomorphism is clearly Γ - reg n ⊗n adt equivariant and the image of a is contained in D(X, F ) . g ⊗n Write a → a for the action of an element g ∈ Γ on a ∈ D(X, F ).One checks by direct computation that the map a is related to the two natural imbeddings F HARISH–CHANDRA HOMOMORPHISMS AND SYMPLECTIC REFLECTION ALGEBRAS 121 ε : S → T and ε : S → Γ via the following identity T Γ n −1 ε (s) ⊗n (4.3.1) (Ad ◦ ε (s))(a) = a (s ) · a · a (s), ∀s ∈ S, a ∈ D(X, F ). D⊗F T F F ⊗n adt ⊗n Ad T D ⊗F It follows from (4.3.1) that, for any a ∈ D(X, F ) = D(X, F ) and s ∈ S, ε (s) −1 one has a = a (s) · a · a (s ). We conclude that both conditions in (4.2.1) hold for F F ⊗n adt the thus obtained homomorphism a : S → A := D(X, F ) . ⊗n adt Further, we have the algebra projection pr : D(X, F ) D(X, p,η),and we let pr ◦ a : S → D(X, p,η) be the composite homomorphism. The Ad -action D⊗F ⊗n of T on D(X, F ) clearly descends to an Ad -action on D(X, p,η). It follows that D⊗F conditions (4.2.1) hold for the map pr ◦ a as well. Thus, we are in a position to form D(X, p,η) Γ , the corresponding equalizer smash product. We let Dunkl(v, l ) de- S n note the image in D(X, p,η) Γ of the element Dunkl ∈ D(X, p,η)Γ . S n n The mainresultof thissection reads Theorem 4.3.2. — Put t = 1/|Γ|. The assignment, given on the generators g ∈ Γ, v ∈ L , l l l = 1, ..., n (where L stands for the l-th direct summand in L ), of the algebra H (Γ ) by the l t,k,c n formulas below extends uniquely to a well defined and injective algebra homomorphism Dunkl θ : H (Γ ) → D(X, p,η) Γ , g → g, v → Dunkl(v, l ). t,k,c n S n l The injectivity statement in the theorem is not difficult; it follows easily from the PBW theorem for the algebra H (Γ ), by considering principal symbols of differen- t,k,c n tial operators. The difficult part is to verify that the assignment of the theorem does define an algebra homomorphism. The proof of this is quite long and involves a lot of explicit computations. That proof will be given later, in Section 9. In the special case n = 1, the proof is less technical and is presented below. 4.4. Proof of Theorem 4.3.2 in the special case: n = 1 Let u , u denote the coordinates in the symplectic vector space (L,ω). 1 2 For n = 1, the assignment of Theorem 4.3.2 reduces to the map L → D(L , F) Γ that reads reg S 2 ∂ (γ v + v) 0 −v Dunkl v θ : v → , where D = + c γ. D 0 |Γ| ∂v ω γ =1,ζ For any v, w ∈ L we are going to compute all 4 entries of the 2 × 2-matrix Dunkl Dunkl representing the operator [θ (v), θ (w)]∈ D(L , F) Γ. First of all, it is easy reg Dunkl Dunkl Dunkl Dunkl to see that [θ (v), θ (w)] =[θ (v), θ (w)] = 0. 12 21 122 PAVEL ETINGOF, WEE LIANG GAN, VICTOR GINZBURG, ALEXEI OBLOMKOV ∂ ∂ Next, write eu = u + u for the Euler operator. We compute 1 2 ∂u ∂u 1 2 Dunkl Dunkl ∨ v ∨ w [θ (v), θ (w)] = w D − v D 2 ∂ ∂ c ∨ ∨ ∨ ∨ ∨ ∨ = w − v + (w (γ v + v) − v (γ w + w) )γ |Γ| ∂v ∂w ω γ =1,ζ 2ω(v, w) c ∨ ∨ ∨ ∨ =− eu + (w (γ v) − (γ w) v ))γ |Γ| ω γ =1,ζ = tω(v, w) − eu + c γ . |Γ| γ =1,ζ Dunkl Dunkl One proves similarly that [θ (v), θ (w)] = ω(v, w)( (eu + 2) + |Γ| c γ). Thus, we find γ =1,ζ Dunkl Dunkl [θ (v), θ (w)] 2ω(v, w) −10 00 = eu + + ω(v, w) c γ. 01 02 |Γ| γ =1,ζ Now, in the 1-dimensional Lie algebra t = C, wehavethegenerator 1 which 1 1 acts in F via the matrix diag(µ + ,µ − ). By definition of twisted differential oper- 2 2 1 1 ators, in the algebra D(X, p,η),wehave eu = 1 = diag(µ + ,µ − ). Therefore, in 2 2 the algebra D(X, p,η),weget Dunkl Dunkl [θ (v), θ (w)] 2ω(v, w) −µ − 0 = + + ω(v, w) c γ. 0 µ − 02 |Γ| γ =1,ζ We have 2 −µ − 0 0 µ − 02 |Γ| 0 10 = − (µ + 1) 0 0 −1 |Γ| 10 10 = + c 01 0 −1 |Γ| where in the last equality we have used the definition of µ from (3.6.2). We find Dunkl Dunkl −1 (4.4.1) [θ (v), θ (w)]= ω(v, w) |Γ| + c + c γ . ζ γ 0 −1 γ =1,ζ HARISH–CHANDRA HOMOMORPHISMS AND SYMPLECTIC REFLECTION ALGEBRAS 123 In these formulas, the matrix diag(1, −1) ∈ End F is viewed as an element of D(L , F). The image of this element in D(X, p,η) Γ, the equalizer smash prod- reg S uct algebra, equals 1 ε (ζ), by definition. Hence, from (4.4.1) we deduce Dunkl Dunkl [θ (v), θ (w)]= ω(v, w)(1/|Γ|+ c γ). γ =1 This completes the proof of Theorem 4.3.2 in the special case n = 1. Dunkl 4.5. The map Θ In Section 4.3, for any and and t ∈ T,wehavedefined the adjoint action Ad (t) : Hom(F , F ) → Hom(F , F ).This Ad -action of T descends to an action on F F D(X, p,η, F → F ), the corresponding quotient space. The reader should be alerted that the resulting Ad -action on D(X, p,η, F → F ) that we are considering at the moment is different from the Ad -action of T considered in Section 4.3: the latter D⊗F ⊗n action, comes from the action of T on both factors of the tensor product D(X, F ) = ⊗n D(X) ⊗ End F , while the former comes from the action of T on the second tensor ⊗n factor, End F , only. Lemma 4.5.1. — Let i, j ∈ I. For the algebra D(X, p,η) Γ , as defined in Section 4.3, S n one has: e (D(X, p,η) Γ )e = D(X, p,, N → N ) , i,n−1 S n j,n−1 j i e (D(X, p,η) Γ )e = D(X, p,, N → N ) , i,n−1 S n s i e(D(X, p,η) Γ )e = D(X, p,, N → N ) , S n j,n−1 j s e(D(X, p,η) Γ )e = D(X, p,, N → N ) . S n s s Proof. — We prove the first equality; the rest are similar. ⊗(l−1) ⊗(n−l ) ∗ Note that CΓ e = N ⊗ N ⊗ N = N ,so n j,n−1 j l=1 o o j ⊗(l−1) ⊗(n−l ) (D(X, p,η) Γ )e = D(X, p,η) ⊗ N ⊗ N ⊗ N , S n j,n−1 S j o o 1≤l≤n where on the right hand side, s ∈ S acts on D(X, p,η) by right multiplication by a(s). For any = ( ,..., ) ∈ I,we write s for the character of S whose value at 1 n ζ is (−1) . (l ) ⊗(l−1) ⊗(n−l ) (l ) Suppose S acts on N ⊗ N ⊗ N by the character s ,where o o (l ) ∈ I.If j is a sink in Q,then (l ) = (0, ..., 0),while if j is a source in Q , then (l ) = (0, ..., 1, ..., 0) (where the 1 is in the l-th position). Under the above (l ) right action of S on D(X, p,η) via a,the s -isotypic component of D(X, p,η) is 124 PAVEL ETINGOF, WEE LIANG GAN, VICTOR GINZBURG, ALEXEI OBLOMKOV ⊕ D(X, p,η, F → F ),so ∈I (l ) ⊗(l−1) ⊗(n−l ) D(X,p,η) ⊗ N ⊗ N ⊗ N S j o o = D(X, p,η, F → F ) ⊗ N . (l ) j ∈I Now, the space e (D(X, p,η) Γ )e can be written as i,n−1 S n j,n−1 ⊗(m−1) ∗ ⊗(n−m) N ⊗ N ⊗ N o i o 1≤m,l≤n (⊕ D(X, p,η, F → F ) ⊗ N ) . ∈I (l ) j ⊗(m−1) ∗ ⊗(n−m) (m) The subgroup S ⊂ Γ acts on N ⊗ N ⊗ N by the character s ,where o i o (m) is (0, ..., 0) if i is a sink, or (0,..., 1, ..., 0) (where the 1 is in m-th position) if i is a source. Recall the Ad -action of T on D(X, p,η, F → F ) described before this F (l ) lemma. We have the group imbedding ε : S → T, see Sections 4.2, 4.3. Equa- tion (4.3.1) implies that the restriction, via ε ,ofthe Ad -action of T to the subgroup T F S coincides with the restriction, via ε , of the action of Γ on D(X, p,η, F → F ) Γ n (l ) to its subgroup S.We have ⊗(m−1) ∗ ⊗(n−m) N ⊗ N ⊗ N ⊕ D(X, p,η, F → F ) ⊗ N ∈I (l ) j o i o = N ⊗ D(X, p,η, F → F ) ⊗ N . (l ) (m) j We conclude that e (D(X, p,η) Γ )e i,n−1 S n j,n−1 = N ⊗ D(X, p,η, F → F ) ⊗ N . (l ) (m) j 1≤m,l≤n The last expression is equal to D(X, p,, N → N ) . j i Dunkl Recall the homomorphism θ of Theorem 4.3.2. For any i, j ∈ I ,recall CM the subspace B of H defined in (1.7.2). Using Lemma 4.5.1, we obtain by restricting i, j Dunkl θ to B , a homomorphism i, j Dunkl Γ Γ n n Θ : B → D(X, p,, N → N ) ⊂ D(X, p,) . i, j j i We define the following algebra homomorphism Dunkl Γ Dunkl Θ : B → D(X, p,) , u → Θ (u ), i, j i, j i, j i, j ∀u ∈ B , i, j ∈ I . i, j i, j CM HARISH–CHANDRA HOMOMORPHISMS AND SYMPLECTIC REFLECTION ALGEBRAS 125 Dunkl Quiver 4.6. Computation of Θ ◦ θ Let a be an edge of Q , viewed as an element of the algebra Π .Wewould CM a Dunkl Quiver Γ like to compute D := Θ ◦ θ (a) ∈ D(X, p,) , the image of that element Dunkl Quiver under the composite map Θ ◦ θ . We will freely use the notation from Section 3.7. a n a Proposition 4.6.1. — If a ∈ Q then D =−φ , and D is an n × n-matrix with the entries −1 ∂ (φ ∗ ◦ (γ + Id)) a m a −1 −1 (D ) = 2|Γ| + c γ , and mm γ (∂φ ) ω(γ ; m, m) a m γ =1,ζ ∗ k (φ ◦ γ) a l (D ) =− γ, for l = m. ml 2 ω(γ ; m, l ) γ ∈Γ Proof of Proposition 4.6.1 for n = 1. — In this special case, we have N = CΓe and i i N = e CΓ, with the pairing defined by (e γ, γ e ) = e γγ e ∈ C. i i i i Thus, for n = 1 the formulas of Proposition 4.6.1 read −1 ∂ φ ∗ ◦ (γ + Id) a a −1 −1 D =−φ , D = 2|Γ| + c γ , ∀a ∈ Q . a γ ∂φ ω γ =1,ζ Dunkl To verify these formulas, we write the Dunkl map in the form Θ (v) = d (v)γ, where v ∈ L and d (v) ∈ D(L , F).Weprove the formula for D γ γ reg γ ∈Γ because the formula for D is easier to prove. Quiver We recall the construction of the map θ .Let ∗ ∗ φ ∗ ∈ Hom (N , N ⊗ L) Γ h(a ) t(a ) ∗ ∗ be the element corresponding to φ ∗ ∈ Hom (N , N ⊗ L). Then the map ∗ ∗ a Γ t(a ) h(a ) Hom (N ∗ , N ∗ ⊗ L) Γ h(a ) t(a ) ∗ ∗ Hom (N , N ⊗ L) → e ∗ C[Γ]⊗ Le ∗ ∗ ∗ Γ h(a ) t(a ) t(a ) h(a ) Quiver ˜ ˜ from the construction of Θ is defined by φ → φ (1 · e ∗ ). ∗ ∗ h(a ) a a s s ˜ ˜ ˜ Choose a basis v , v in L.Then φ = φ ⊗ v where φ ∈ ∗ ∗ ∗ 1 2 s a a a s=1 a Γ Hom(N ∗ , N ∗ ).Weconsider D as element of D(L , N ∗ → N ∗ ) .From the h(a ) t(a ) reg h(a ) t(a ) 126 PAVEL ETINGOF, WEE LIANG GAN, VICTOR GINZBURG, ALEXEI OBLOMKOV Quiver construction of θ we find 2 2 a Dunkl s s ˜ ˜ D = Θ (v )φ ∗ = d (v ) γ ◦ φ ∗ s a γ s a s=1 s=1 γ ∈Γ ∂ (γ v + v ) s s −1 = 2|Γ| + c γ ◦ φ ∂φ s=1 γ =1,ζ −1 ∂ γ ◦ φ ∗ ◦ (γ + Id) −1 = 2|Γ| + c . ∂φ ∗ γ =1,ζ We have natural isomorphisms ∗ ∗ ∗ ∼ ∼ ∗ ∗ ∗ Hom(N , N ) = N ⊗ N = Hom(N , N ). ∗ ∗ ∗ h(a ) t(a ) t(a ) h(a ) t(a ) h(a ) We deduce Γ ∗ ∗ Γ ∗ ∗ D(L , N → N ) D(L , N → N ) . reg h(a ) t(a ) reg ∗ ∗ t(a ) h(a ) −1 Under this isomorphism, the element γ φ ◦ (γ + Id) corresponds to φ ∗ ◦ ∂ ∂ −1 −1 (γ + Id) ◦ γ and corresponds to . This completes the proof. ∂φ ∗ ∂φ ∗ a a We omit the proof of Proposition 4.6.1 for n>1;itissimilar to theabove computation in the case n = 1. It is easy to see that for the edge b: s → o,wehave b Dunkl Quiver t D := Θ ◦ θ (b) =−(1, ..., 1) and b Dunkl Quiver ∗ D := Θ ◦ θ (b ) = ν · (1,..., 1). a Dunkl Quiver Thus, for all a ∈ Q ,wehavecomputed theoperators D := Θ ◦ θ (a) CM where D ∈ D(X, p,). Radial Holland Dunkl Quiver Theorem 4.6.2. — For all values of c, k, we have Θ ◦ θ = Θ ◦ θ . In the special case n = 1,for anyedge a ∈ Q,we have −1 ∂ φ ∗ ◦ (γ + Id) Radial Holland ∗ −1 −1 θ θ (a ) = 2|Γ| + c γ ∂φ ∗ ω γ =1,ζ −1 φ ◦ γ −1 −|Γ| . γ =1,ζ HARISH–CHANDRA HOMOMORPHISMS AND SYMPLECTIC REFLECTION ALGEBRAS 127 Radial Radial Therefore, replacing here the map θ by Θ , we find −1 ∂ φ ◦ (γ + Id) Radial Holland ∗ −1 −1 Θ θ (a ) = 2|Γ| + c γ . ∂φ ∗ ω γ =1,ζ When n> 1, itiscompletely similar. 5. Harish–Chandra homomorphism −1 Recall that we assume λ · δ = 1, i.e. t =|Γ| .We shall write H for H (Γ ). k,c t,k,c n 5.1. Modified Holland’s map Holland In this subsection, we define a map Θ : e Π e → A , cf. (1.6.2). To this s s χ end, assume for the moment that ν is a formal variable, and the algebras Π, Π , T , A are all defined over C[ν]. χ χ Radial Γ Lemma 5.1.1. — The map gr(θ ) : grA → gr(D(X, p,ρ ) ⊗ C[ν]) is injec- χ s tive. Proof. — This follows from Proposition 7.2.2, Theorem 7.2.3 and Propos- ition 7.2.5. The preceding lemma implies that grA and A are free C[ν]-modules. χ χ We define a homomorphism Holland −1 GL(α) Θ : CQ ⊗ C[ν]→ (D(Q ,α) ⊗ (End N) ⊗ C[ν, ν ]) CM CM by Holland Holland ˜ ˜ Θ (e ) = θ (e ) for all vertices j, j j Holland Holland ∗ ˜ ˜ Θ (a) = θ (a) for any edge a = b , Holland Holland ∗ −1 ∗ ˜ ˜ Θ (b ) = ν θ (b ). Holland Holland It is easy to see that since θ descends to a homomorphism θ : Π → T , Holland the homomorphism Θ descends to a homomorphism Holland −1 Holland −1 Θ : Π → T [ν ], such that Θ : e Π e → A [ν ]. χ s s χ Radial Holland ∗ Suppose that X ∈ CQ . By (3.7.5), θ θ (b Xb) vanishes if we set ν = 0, so by Lemma 5.1.1, Holland ∗ GL(α) θ (b Xb) ∈ (D(Q ,α)(ξ − (λ − ∂ − ne ))(gl(α))) . CM s 128 PAVEL ETINGOF, WEE LIANG GAN, VICTOR GINZBURG, ALEXEI OBLOMKOV Holland Since χ = (λ − ∂ − ne ) − νe + nνe ,we havethat θ (b Xb) belongs to s o s D(Q ,α)(ξ − χ )(gl(α)) + ν · D(Q ,α)(e − ne )(gl(α)). CM CM o s Since GL(α) is a reductive group, we have a projection map GL(α) pr : D(Q ,α) → D(Q ,α) CM CM such that GL(α) pr(D(Q ,α)(ξ − χ )(gl(α))) ⊂ (D(Q ,α)(ξ − χ )(gl(α))) . CM CM Holland Holland ∗ ∗ ˜ ˜ Thus, the element θ (b Xb) = pr(θ (b Xb)) belongs to GL(α) GL(α) (D(Q ,α)(ξ − χ )(gl(α))) + ν · D(Q ,α) . CM CM Holland Therefore, Θ (e Π e ) ⊆ A .Thus, for any ν ∈ C, we have a homomorphism s s χ Holland Θ : e Π e → A . s s χ Theorem 5.1.2. — The following diagram commutes: Quiver eH e e Π e k,c s s Holland Dunkl Θ Θ Radial D(X, p, ) . Proof. — This follows from Theorem 4.6.2. Holland Proposition 5.1.3. — The map Θ : e Π e → A is surjective. s s χ Proof. — The algebra Π has a filtration with deg(a) = 1 for edges a = b, b , and deg(b) = deg(b ) = 0. It suffices to show that the associated graded map Holland gr(Θ ) : e gr(Π )e → gr(A ) s s χ is surjective. Now e gr(Π )e is generated by e and b (e Π(Q )e )b where Π(Q ) is the prepro- s s s o o jective algebra of Q . By [CB, Theorem 3.4], we have ⊗n S GL(nδ) GL(δ) C[Rep (Π(Q ))] = C[Rep (Π(Q ))] . nδ δ HARISH–CHANDRA HOMOMORPHISMS AND SYMPLECTIC REFLECTION ALGEBRAS 129 Denote by GL(δ) CBH δ Θ : Π(Q ) → C Rep (Π(Q )) ⊗ End ⊕ C i∈I the natural morphism defined in [CBH], which gives a morphism CBH GL(δ) Θ : e Π(Q )e → C[Rep (Π(Q ))] . o o This latter morphism is an isomorphism by [CBH, Theorem 8.10]. Now, given an element X ∈ e Π(Q )e ,we claim that o o Holland ∗ ⊗( p−1) CBH ⊗(n−p) (5.1.4) gr(Θ )(b Xb) =− 1 ⊗ Θ (X) ⊗ 1 . p=1 Indeed, we have Radial Holland ∗ gr(θ ) gr(Θ )(b Xb) Dunkl Quiver ∗ = gr(Θ ) gr(Θ )(b Xb) (by Theorem 5.1.2) Radial ⊗( p−1) CBH ⊗(n−p) = gr(θ ) − 1 ⊗ Θ (X) ⊗ 1 (by (6.3.3) below). p=1 Radial Hence, (5.1.4) follows from injectivity of gr(θ ). It follows from (5.1.4) and Holland Lemma 6.3.4 below that gr(Θ ) is surjective. 5.2. — It follows from Theorem 5.1.2, Proposition 5.1.3, and the injectivity of Dunkl Θ that we have a homomorphism Dunkl −1 Radial (Θ ) ◦ Θ : A → eH e. χ k,c Since Rep (Q ) = Rep (Q ) ⊕ C , we have an obvious embedding α CM nδ : D(Q , nδ) → D(Q ,α). CM Definition 5.2.1. — The Harish–Chandra homomorphism Φ is defined to be the compo- k,c sition Dunkl −1 Radial (Θ ) ◦ Θ GL(nδ) (5.2.2) D(Q , nδ) A eH e. χ k,c Following [EG], we define a 1-parameter space of representations V of gl as follows. As a vector space, V is spanned by expressions (x ··· x ) · P,where P is d 1 n 130 PAVEL ETINGOF, WEE LIANG GAN, VICTOR GINZBURG, ALEXEI OBLOMKOV a Laurent polynomial in x , ..., x of total degree 0.The Lie algebra gl has an ac- 1 n n tion on V by formal differentiation, where e acts by x . We restrict this to an d p,q p ∂x sl action. The desired gl action on V is obtained by pulling back the sl action n d n via the natural Lie algebra projection gl → sl , so that the center of gl acts triv- n n ially. Let Fun() denote the vector space of functions on a formal neighborhood of a point of the slice L .Recall that ∂ =−n Tr | (ζ)e .Wehave N i reg i∈I i gl(α) (5.2.3) (Fun(Rep (Q )) ⊗ C ) α CM −χ gl(nδ) = (Fun(Rep (Q ))⊗V ⊗ C ) . ν−1 −λ+e +∂ nδ o The GL(nδ) action on Rep (Q ) induces a Lie algebra map ad : gl(nδ) → nδ D(Q , nδ).Let ad : Ugl(nδ) → D(Q , nδ) be the induced map on the universal en- veloping algebra of gl(nδ). Define the left ideal J := D(Q , nδ) · ad(Ann (V ⊗ C )) ⊂ D(Q , nδ). k,c ν−1 −λ+e +∂ GL(nδ) By (5.2.3), the ideal J is in the kernel of the map in (5.2.2). k,c Theorem 5.2.4. — The Harish–Chandra homomorphism induces an algebra isomorphism GL(nδ) ∼ GL(nδ) Φ : D(Q , nδ) /J eH e. k,c k,c k,c Proof. — By Theorem 7.2.3 and [EG, Theorem 1.3], the associated graded map, gr Φ , is the isomorphism in (7.2.4), hence Φ is itself an isomorphism. k,c k,c 5.3. Proof of Corollary 1.8.3 −1 Given any C = C γ ∈ C[Γ],we let C = C γ . Correspondingly, γ γ γ ∈Γ γ ∈Γ if λ = Tr | (C)e ,then let λ = Tr | (C)e . We have an anti-isomorphism N i N i i i i∈I i∈I −1 n (5.3.1) H H , g → g , u → −1u, ∀g ∈ Γ , u ∈ L . k,c k,c n We also have an isomorphism ∼ σ (5.3.2) H → H ,σ → (−1) σ, g → g, u → u, k,c −k,c n n ∀σ ∈ S , g ∈ Γ , u ∈ L . The isomorphism in (5.3.2) sends e to e . − HARISH–CHANDRA HOMOMORPHISMS AND SYMPLECTIC REFLECTION ALGEBRAS 131 Now, for any i ∈ I set λ := Tr | (t · 1 + c ). We put i i † −1 † c := −c + 2|Γ| γ, and λ := λ e =−λ + 2e . i o γ =1 i∈I The group GL(nδ) acts on det(Rep (Q ) ) by the character 2∂.We have nδ ∗ ∗ (V ⊗ C ) ⊗ det(Rep (Q ) ) V ⊗ C ⊗ C ν−1 −λ+e +∂ nδ −ν λ−e −∂ 2∂ o o = V ⊗ C † . −ν −λ +e +∂ Let i : D(Q , nδ) → D(Q , nδ) be the anti-isomorphism sending a differential operator to its adjoint. Then for any GL(nδ)-module V, ∗ ∗ i(ad(Ann (V))) = ad((Ann (V ⊗ det(Rep (Q ) )))). nδ The proof of the first isomorphism in Corollary 1.8.3 is now completed by the following isomorphisms op eH e (eH −1 † e) using Theorem 5.2.4 k,c 2|Γ| −k,c eH e using (5.3.1) −1 † 2|Γ| −k,c e H e using (5.3.2). − −1 † − k−2|Γ| ,c We will prove the second isomorphism in Corollary 1.8.3 later in Section 6.7. 6. Reflection isomorphisms Except for Section 6.7, this section is independent of the earlier sections. 6.1. —Let Q be an arbitrary quiver (not necessarily of type A, D,or E). Denote by I the set of vertices of Q.Let R = C,and E the vector space with i∈I basis formed by the set of edges {a ∈ Q }.Thus, E is naturally a R-bimodule. The n n path algebra of Q is CQ := T E = T E,where T E = E ⊗ ··· ⊗ E is the R R R n≥0 R R n-fold tensor product. The trivial path for the vertex i is denoted by e ,anidempotent in R. ⊗n Fix a positive integer n.Let R := R .For any ∈[1, n],define the E-bimodules ⊗(−1) ⊗(n−) E := R ⊗ E ⊗ R and E := E . 1≤≤n ⊗(−1) ⊗(n−) The natural inclusion E → R ⊗ T E⊗ R induces a canonical identification ⊗(−1) ⊗(n−) T E = R ⊗ T E ⊗ R .Given twoelements ε ∈ E and ε ∈ E of the form R R m (6.1.1) ε = e ⊗ e ⊗ ··· ⊗ a ⊗ ··· ⊗ h(b) ⊗ ··· ⊗ e , i i i 1 2 n (6.1.2) ε = e ⊗ e ⊗ ··· ⊗ t(a) ⊗ ··· ⊗ b ⊗ ··· ⊗ e , i i i 1 2 n 132 PAVEL ETINGOF, WEE LIANG GAN, VICTOR GINZBURG, ALEXEI OBLOMKOV where = m, a, b ∈ Q and i , ..., i ∈ I,we define 1 n ε, ε := (e ⊗ ··· ⊗ a ⊗ ··· ⊗ h(b) ⊗ ··· ⊗ e ) i i 1 n × (e ⊗ ··· ⊗ t(a) ⊗ ··· ⊗ b ⊗ ··· ⊗ e ) i i 1 n − (e ⊗ ··· ⊗ h(a) ⊗ ··· ⊗ b ⊗ ··· ⊗ e ) i i 1 n × (e ⊗ ··· ⊗ a ⊗ ··· ⊗ t(b) ⊗ ··· ⊗ e ). i i 1 n Note that ε, ε is an element in T E. Definition 6.1.3 ([GG, Def. 1.2.3]). — For any λ = λ e where λ ∈ C,and i i i i∈I ν ∈ C, define the algebra A (Q ) to be the quotient of T E C[S ] by the following relations. n,λ,ν R n (i) For any i , ..., i ∈ I and ∈[1, n]: 1 n ∗ ∗ e ⊗ ··· ⊗ a · a − a · a − λ e ⊗ ··· ⊗ e i i i i 1 n {a∈Q | h(a)=i } {a∈Q | t(a)=i } = ν (e ⊗ ··· ⊗ e ⊗ ··· ⊗ e )s . i i i j 1 n {j = | i =i } (ii) For any ε, ε of the form (6.1.1)–(6.1.2): ⎪ ν · (e ⊗ ··· ⊗ h(a) ⊗ ··· ⊗ t(a) ⊗ ··· ⊗ e )s i i m 1 n if b ∈ Q , a = b , ε, ε = −ν · (e ⊗ ··· ⊗ h(a) ⊗ ··· ⊗ t(a) ⊗ ··· ⊗ e )s i i m 1 n if a ∈ Q , b = a , 0 else. When n = 1, there is no parameter ν,and A (Q ) is the deformed preprojec- n,λ,ν tive algebra Π (Q ) defined in [CBH]. 6.2. Quiver functors The goal of this section is to put the construction of the functor M → M ex- ploited in Section 2.2 into an appropriate, more general, context. Let T be a nonempty subset of I,and let e := e .In particular, e = 1.Let T i I i∈T Q be a quiver obtained from Q by adjoining a vertex s,and arrows b : s → i for T i i ∈ T.We call s the special vertex. We shall define a functor G from A (Q )-modules n,λ,ν to Π (Q )-modules. λ−νe +nνe T T s Let M be an A (Q )-module. We want to define a Π (Q )-module n,λ,ν λ−νe +nνe T T s ⊗(n−1) ⊗n G(M).For each i ∈ I,let G(M) := e (e ⊗ e )M. Also, let G(M) := e e M. i n−1 i s n T T HARISH–CHANDRA HOMOMORPHISMS AND SYMPLECTIC REFLECTION ALGEBRAS 133 If a is an edge in Q , then define a : G(M) → G(M) to be the map given t(a) h(a) ⊗(n−1) ⊗n by the element a ⊗ e ∈ A (Q ).Wehaveaninclusion G(M) ⊂ e e M = n,λ,ν s n−1 T T G(M) .For i ∈ T, we have a projection map pr : G(M) → G(M) .Define j i j i j∈T j∈T b : G(M) → G(M) to be the restriction of pr to G(M) .Define b : G(M) → i s i i s i G(M) to be −ν · (1 + s + ··· + s ). s 12 1n The following lemma is a generalization of Lemma 2.2.2. Lemma 6.2.1. — With the above actions, G(M) is a Π (Q )-module. λ−νe +nνe T T s Proof. —Itisclear that (1 + s + ··· + s )e = ne . 12 1n n−1 n On G(M), at the special vertex s,we have b b =−nν. i∈T i At a vertex i ∈ I, i ∈ / T,wehave ∗ ∗ aa − a a = λ a∈Q ;h(a)=i a∈Q ;t(a)=i by the relation (i) in Definition 6.1.3. At a vertex i ∈ T,we have ∗ ∗ ∗ aa − a a = λ + ν · pr (s + ··· + s ) = λ − ν − b b , i i 12 1n i i a∈Q ;h(a)=i a∈Q ;t(a)=i using again the relation (i) in Definition 6.1.3. It is clear that the assigment M → G(M) is functorial. We have constructed afunctor (6.2.2) G : A (Q )-mod → Π (Q )-mod. n,λ,ν λ−νe +nνe T T s Recall the symmetrizer e := s. s∈S n! n Definition 6.2.3. — Let U (Q ) := e A (Q )e be the spherical subalgebra in n,λ,ν n n,λ,ν n A (Q ). n,λ,ν ⊗n n n The idempotents e and e := e commute. For M := A (Q )e e , we get n n,λ,ν n T T T n n G(M) = e U e . s n,λ,ν T T In this case, G(M) is an algebra, and the action of e Π (Q )e on G(M) s s λ−νe +nνe T s s T s commutes with right multiplication by the elements of G(M) . Thus, our construction yields an algebra homomorphism n n (6.2.4) G : e Π (Q )e → e U e . s λ−νe +nνe T s n,λ,ν T s T T 134 PAVEL ETINGOF, WEE LIANG GAN, VICTOR GINZBURG, ALEXEI OBLOMKOV 6.3. Modified version The map G is 0 on nonconstant paths when ν = 0. For this reason, we shall need a slight modification of the constructions in the previous subsection. Define Π (Q ) to be the quotient of the path algebra CQ by the fol- T T λ−νe +nνe T s lowing relations: ∗ ∗ ∗ [a, a ]+ ν b b = λ − νe , b b =−ne . i T i s i i a∈Q i∈T i∈T We have an algebra morphism Π (Q ) → Π (Q ) defined on the λ−νe +nνe T T T s λ−νe +nνe T s edges by ∗ ∗ ∗ a → a for a = b , b → νb . i i i This is an isomorphism only when ν = 0. Given a A (Q )-module M, we construct a Π (Q )-module G (M) n,λ,ν T λ−νe +nνe T s analogous to G(M) in the previous subsection, the only difference is that now, we let b : G (M) → G (M) be −(1 + s + ··· + s ). Hence,as above,we obtain a functor i s 12 1n (6.3.1) G : A (Q )-mod → Π (Q )-mod n,λ,ν T λ−νe +nνe T s as well as a morphism n n (6.3.2) G : e Π (Q )e → e U e . s T s n,λ,ν λ−νe +nνe T T T s The algebra Π (Q ) has a filtration with deg(a) = 1 for a = b , b ,and T i λ−νe +nνe i T s deg(b ) = deg(b ) = 0 for i ∈ T.Also, e gr(Π (Q ))e is generated by e and i s T s s i λ−νe +nνe T s ( b Π (Q )b ). 0 i i, j∈T j We shall assume that Q is a connected quiver without edge-loops, and Q is not a finite Dynkin quiver. Then, by [GG, Theorem 2.2.1] and [GG, Remark 2.2.6], gr A (Q ) = n,λ,ν ⊗n Π (Q ) C[S ].Thus, 0 n n n ⊗n gr e U e = (e Π (Q )e ) . n,λ,ν T 0 T T T Now given an element X ∈ e Π (Q )e where i, j ∈ T,we have j 0 i ⊗( p−1) ⊗(n−p) (6.3.3) gr(G )(b Xb ) =− e ⊗ X ⊗ e . j T T p=1 ⊗n S Lemma 6.3.4. — Let A be any associative algebra with unit 1 ∈ A.Then (A ) is generated as an algebra by elements of the form ⊗( p−1) ⊗(n−p) 1 ⊗ X ⊗ 1 , X ∈ A. p=1 HARISH–CHANDRA HOMOMORPHISMS AND SYMPLECTIC REFLECTION ALGEBRAS 135 ⊗n S ⊗n Proof. —Since (A ) is spanned by elements of the form a where a ∈ A,it suffices to show that the lemma is true for A = C[a], but this follows from the main theorem on symmetric functions. Proposition 6.3.5. — The map G in (6.3.2) is surjective. Proof. — It suffices to show that gr(G ) is surjective. This follows from (6.3.3) and the preceding lemma. 6.4. Reflection functors Recall the setting of reflection functors as in (1.8.1). In particular, we have the Weyl group W generated by the simple reflections r for i ∈ I.Wealso haveanon- empty subset T ⊂ I and we fix a vertex i ∈ / T. Let us apply the reflection functor F to the A (Q )-module A (Q )e .By i n,λ,ν n,λ,ν construction, we have n n n n e F A (Q )e = e A (Q )e i n,λ,ν n,λ,ν T T T T n n n n and the left action of e A (Q )e on e F (A (Q )e ) commutes with the right n,r (λ),ν i n,λ,ν T i T T T n n multiplication by e A (Q )e .Hence, for i ∈ / T, we obtain a homomorphism n,λ,ν T T n n n n (6.4.1) F : e A (Q )e → e A (Q )e . i n,r (λ),ν n,λ,ν T i T T T n n n n Note that F (e U (Q )e ) ⊂ e U e . i n,r (λ),ν n,λ,ν T T T T In the special case when n = 1, reflection functors were constructed in [CBH]; let us recall their definition. Since Π (Q ) does not depend on the orientation of Q , we may assume without loss of generality that i is a sink in Q.Let M be a Π (Q )- module, and M = e M for each j ∈ I. For each edge a ∈ Q such that h(a) = i, j j write π : M → M ,µ : M → M a t(ξ) t(a) a t(a) t(ξ) ξ∈Q ;h(ξ)=i ξ∈Q ;h(ξ)=i for the projection map and inclusion map, respectively. Define π : M → M,π := a ◦ π , t(a) i a a∈Q ;h(a)=i a∈Q ;h(a)=i and µ : M → M ,µ := µ ◦ a . i t(a) a a∈Q ;h(a)=i a∈Q ;h(a)=i 136 PAVEL ETINGOF, WEE LIANG GAN, VICTOR GINZBURG, ALEXEI OBLOMKOV Observe that πµ = λ .Let (F (M)) := M if j = i,and let (F (M)) := Ker(π).If i i j j i i a ∈ Q and h(a), t(a) = i,thenlet a : (F (M)) → (F (M)) be the same map as i t(a) i h(a) a : M → M .If a ∈ Q and h(a) = i,then let a : (F (M)) → (F (M)) be the t(a) h(a) i t(a) i i map (−λ + µπ)µ ,and let a : (F (M)) → (F (M)) be the map π restricted to i a i i i t(a) a (F (M)) . Letting i i F (M) =⊕ (F (M)) , i j∈I i j we have defined the functor F : Π (Q )-mod → Π (Q )-mod for any i ∈ I. i λ r (λ) In particular, for the quiver Q ,and for i ∈ I but i ∈ / T,we have (6.4.2) F : Π (Q )-mod → Π (Q )-mod i λ−νe +nνe T r (λ)−νe +nνe T T s i T s Let i ∈ I but i ∈ / T.Wedefine afunctor F : Π (Q )-mod → Π (Q )-mod T T i λ−νe +nνe r (λ)−νe +nνe T s i T s in exactly the same way as F in (6.4.2). It is easy to see from definitions that the diagram (1.8.6) commutes. 6.5. Relations in rank 1 In this subsection, the rank n is equal to 1.Let C = (C ) be the generalized ij Cartan matrix of Q . Proposition 6.5.1. — For all λ ∈ R, we have the following. (i) The map F : (1 − e )Π (Q )(1 − e ) → (1 − e )Π (Q )(1 − e ) i i λ i i r (λ) i is an isomorphism, and F = Id. (ii) If C = 0,then ij F ◦ F = F ◦ F : (1 − e − e )Π (Q )(1 − e − e ) i j j i i j λ i j → (1 − e − e )Π (Q )(1 − e − e ). i j r r (λ) i j i j (iii) If C =−1,then ij F ◦ F ◦ F = F ◦ F ◦ F : (1 − e − e )Π (Q )(1 − e − e ) i j i j i j i j λ i j → (1 − e − e )Π (Q )(1 − e − e ). i j r r r (λ) i j i j i HARISH–CHANDRA HOMOMORPHISMS AND SYMPLECTIC REFLECTION ALGEBRAS 137 Proof. —(i) Thealgebra (1 − e )Π (Q )(1 − e ) is generated by edges a ∈ Q with i λ i h(a), t(a) = i,and paths oflengthtwo: a a with h(a ), t(a ) = i and t(a ) = h(a ) = i. 2 1 2 1 2 1 If a ∈ Q and h(a), t(a) = i,then F (a) = a. Now let a a be a path with h(a ), t(a ) = i and t(a ) = h(a ) = i.If a = a 2 1 2 1 2 1 2 ∗ ∗ or a = a ,then F (a a ) = a a .If a = a ,then F (a a ) =−λ e + a a ,and so 1 i 2 1 2 1 2 i 2 1 i t(a ) 2 1 2 1 1 F (F (a a )) =−λ e + λ e + a a = a a . i i 2 1 i t(a ) i t(a ) 2 1 2 1 1 1 (ii) When C = 0, there is no edge joining i and j . In this case, it is clear that ij F F = F F ,so F F = F F . i j j i i j j i (iii) When C =−1, there is precisely one edge in Q joining i and j,say ij a : i → j. The algebra (1 − e − e )Π (Q )(1 − e − e ) is generated by: i j λ i j –edges a ∈ Q with h(a ), t(a ) = i, j ; 1 1 1 –paths a a with t(a ) = h(a ) = i and h(a ), t(a ) = i, j ; 2 1 2 1 2 1 –paths a a with t(a ) = h(a ) = j and h(a ), t(a ) = i, j ; 2 1 2 1 2 1 –paths a a a with a = a, t(a ) = j , h(a ) = i and h(a ), t(a ) = i, j ; 3 2 1 2 3 1 3 1 –paths a a a with a = a , t(a ) = i, h(a ) = j and h(a ), t(a ) = i, j . 3 2 1 2 3 1 3 1 In thefirstcaseabove, wehave F F F (a ) = a = F F F (a ). i j i 1 1 j i j 1 ∗ ∗ In the second case above, when a = a or a = a ,we have 2 1 1 2 F F F (a a ) = a a = F F F (a a ). i j i 2 1 2 1 j i j 2 1 When a = a ,we have F F F (a a ) = F F (−λ + a a ) = F (−λ + a a ) i j i 2 1 i j i 2 1 i i 2 1 =−λ − λ + a a , i j 2 1 since r (r (λ))e = λ ; and on the other hand, since r (λ)e = λ + λ ,wefind j i i j j i i j F F F (a a ) = F F (a a ) = F (−λ − λ + a a ) =−λ − λ + a a . j i j 2 1 j i 2 1 j i j 2 1 i j 2 1 The third case above is similar to the second case. In the fourth and fifth cases above, note that no two of the edges a , a , a are 1 2 3 reverse of the other, so we have F F F (a a a ) = a a a = F F F (a a a ). i j i 3 2 1 3 2 1 j i j 3 2 1 Lemma 6.5.2. —(i) If λ = 0,then Π (Q ) = Π (Q )(1 − e )Π (Q ). λ λ i λ 138 PAVEL ETINGOF, WEE LIANG GAN, VICTOR GINZBURG, ALEXEI OBLOMKOV (ii) If C =−1,and λ = 0, λ = 0, λ + λ = 0,then ij i j i j Π (Q ) = Π (Q )(1 − e − e )Π (Q ). λ λ i j λ Π (Q ) Proof. —(i) As a Π (Q )-module, is zero at all vertices not equal Π (Q )(1−e )Π (Q ) λ i λ to i,soall edges of Q must act by 0. But then it must also be zero at the vertex i ∗ ∗ since λ e = aa − a a. i i a∈Q ;h(a)=i a∈Q ;t(a)=i (ii) There is only one edge in Q joining i and j,say a : i → j.Let V be the Π (Q ) Π (Q )-module .Now V is zero at all vertices not equal to i or j,so Π (Q )(1−e −e )Π (Q ) λ i j λ ∗ ∗ V = V ⊕ V . Suppose V = 0,say V = 0.Then aa = λ e on V implies that a, a are i j j j j j −1 ∗ ∗ nonzero maps, and a has a right inverse λ a .But then a a =−λ e on V implies i i i −1 that a has a left inverse −λ a .Hence, λ =−λ , a contradiction. j i Using Proposition 6.5.1(i), Π (Q )(1 − e ) is a right (1 − e )Π (Q )(1 − e )- r (λ) i i λ i module, and Π (Q )(1 − e − e ) is a right (1 − e − e )Π (Q )(1 − e − e )-module. r (λ) i j i j λ i j Corollary 6.5.3. —(i) If λ = 0,then F (M) = Π (Q )(1 − e ) ⊗ (1 − e )M i r (λ) i (1−e )Π (Q )(1−e ) i i i λ i for any M ∈ Π (Q )-mod. (ii) If C =−1,and λ = 0, λ = 0, λ + λ = 0,then ij i j i j F (M) = Π (Q )(1 − e − e ) ⊗ (1 − e − e )M i r (λ) i j (1−e −e )Π (Q )(1−e −e ) i j i i j λ i j for any M ∈ Π (Q )-mod. Proof. —(i) Let M ∈ Π (Q ) − mod. By Lemma 6.5.2(i), F (M) = Π (Q )(1 − e ) ⊗ (1 − e )F (M) i r (λ) i (1−e )Π (Q )(1−e ) i i i i r (λ) i = Π (Q )(1 − e ) ⊗ (1 − e )M. r (λ) i (1−e )Π (Q )(1−e ) i i i λ i The proof of (ii) is similar, using Lemma 6.5.2(ii). Corollary 6.5.4. —(i) If λ = 0,then F = Id. (ii) If C = 0,then F F = F F . ij i j j i (iii) If C =−1 and λ = 0, λ = 0, λ + λ = 0,then F F F = F F F . ij i j i j i j i j i j Proof. — (ii) is trivial, while (i) and (iii) are immediate from Proposition 6.5.1 and Corollary 6.5.3. Our proof of Corollary 6.5.4 appears to be simpler than earlier proofs, see [CBH, Theorem 5.1] (for (i)), [Na, Remark 3.20], [Na, Theorem 3.4], [Lu2], and [Maf]. HARISH–CHANDRA HOMOMORPHISMS AND SYMPLECTIC REFLECTION ALGEBRAS 139 6.6. Relations in higher rank In this subsection, n is an integer greater than 1. We shall show that the reflec- tion functors F of (1.8.1) satisfy the Weyl group relations when the parameters are generic. We omit the proof of the following proposition since it is completely similar to the proof of Proposition 6.5.1. Proposition 6.6.1. — Let i, j ∈ I. The homomorphisms F of (6.4.1) satisfy the following for any λ ∈ R and ν ∈ C: (i) Let T = I \{i}. Then the map n n n n F : e A (Q )e → e A (Q )e i n,r (λ),ν n,λ,ν T i T T T is an isomorphism, and F ◦ F = Id. i i (ii) Let T = I \{i, j}.If C = 0,then ij n n n n F ◦ F = F ◦ F : e A (Q )e → e A (Q )e . i j j i n,r r (λ),ν n,λ,ν T i j T T T (iii) Let T = I \{i, j}.If C =−1,then ij n n n n F ◦ F ◦ F = F ◦ F ◦ F : e A (Q )e → e A (Q )e . i j i j i j n,r r r (λ),ν n,λ,ν T i j i T T T Next, we have the following generalization of Lemma 6.5.2. Lemma 6.6.2. —(i) Let T = I \{i}.If λ ± pν = 0 for p = 0, 1, ..., n − 1,then A = A e A . n,λ,ν n,λ,ν n,λ,ν (ii) Let T = I \{i, j} and suppose C = 0.If λ ± pν = 0 and λ ± pν = 0 for ij i j p = 0, 1, ..., n − 1,then A = A e A . n,λ,ν n,λ,ν n,λ,ν (iii) Let T = I \{i, j} and suppose C =−1.If λ ± pν = 0, λ ± pν = 0 and ij i j λ + λ ± pν = 0 for p = 0, 1, ..., n − 1,then A = A e A . i j n,λ,ν n,λ,ν n,λ,ν Proof. — The proof is similar to the proof of Lemma 6.5.2. n,λ,ν To prove (i), let V be the A -module where T = I \{i}.For any n,λ,ν A e A n,λ,ν n,λ,ν n-tuple of vertices i , ..., i ,welet V := (e ⊗ ··· ⊗ e )V,so V = V . 1 n i ,...,i i i i ,...,i 1 n 1 n i ,...,i ∈I 1 n 1 n Since e V = 0,we have V = 0 when none of i , ..., i is i. Suppose now that i i ,...,i 1 n T 1 n appears m times in i , ..., i . We shall prove by induction on m that V = 0,so we 1 n i ,...,i 1 n assume that the statement is true whenever i appears less than m times. Without loss of generality, say i = ··· = i = i. Then by the relation (i) in Definition 6.1.3 and 1 m the induction hypothesis, we have (λ + ν s )V = 0. i 1 i ,...,i 1 n =2 140 PAVEL ETINGOF, WEE LIANG GAN, VICTOR GINZBURG, ALEXEI OBLOMKOV By [Ga, Prop. 5.12], the element λ + ν s is invertible in the group algebra i 1 =2 C[S ].Hence, V = 0, and (i) follows by induction. n i ,...,i 1 n The proofs of (ii) and (iii) are similar, using induction. As in the previous subsection, we obtain Corollary 6.6.3. —(i) Let T = I \{i}.If λ ± pν = 0 for p = 0, 1, ..., n − 1,then n n n n F (M) = A e ⊗ e M, ∀M ∈ A -mod. i n,r (λ),ν e A e n,λ,ν i T n,λ,ν T T T (ii) Let T = I \{i, j} and suppose C = 0.If λ ± pν = 0 and λ ± pν = 0 for ij i j p = 0, 1, ..., n − 1,then n n n n F (M) = A e ⊗ e M, ∀M ∈ A -mod. i n,r (λ),ν e A e n,λ,ν i T n,λ,ν T T T (iii) Let T = I \{i, j} and suppose C =−1.If λ ± pν = 0, λ ± pν = 0 and ij i j λ + λ ± pν = 0 for p = 0, 1, ..., n − 1,then i j n n n n F (M) = A e ⊗ e M, ∀M ∈ A -mod. i n,r (λ),ν e A e n,λ,ν i T n,λ,ν T T T Proposition 1.8.2 is immediate from Proposition 6.6.1 and Corollary 6.6.3. 6.7. Shift functors In this subsection, we return to the case when Q is the affine Dynkin quiver associated to Γ. Let C = (C ) be the generalized Cartan matrix of Q . The affine Weyl group ij I I I W is generated by the simple reflections r for i ∈ I.Itactson C by r : C → C , i i where r (λ) = λ − C λ e . i ij i j j∈I Let Q be the finite Dynkin quiver obtained from Q by deleting the vertex o. The Weyl group W of Q is the subgroup of W generated by the r for i = o.Let C = (C ) be the Cartan matrix of Q .Then W acts on ⊕ Ce by r (λ) = i =o i i ij λ − C λ e .Denote by w ∈ W the longest element of W. i j 0 j =o ij ∗ ∗ If i ∈ I,thenlet i ∈ I be the vertex such that N = N .Recall thatif λ = λ e ,then λ = λ e . i i i i i∈I i∈I Lemma 6.7.1. — For any λ ∈ C with λ · δ = 1, we have w (λ) =−λ + 2e . 0 0 Proof. — The projection C →⊕ Ce is W-equivariant with kernel Ce .We i =o i o write λ = λ e + λ where λ ∈⊕ Ce .Now w (λ) − w (λ ) ∈ Ce and w (λ ) =−λ . o o i =o i 0 0 0 0 It follows that w (λ) =−λ + 2(λ · δ)e . 0 0 HARISH–CHANDRA HOMOMORPHISMS AND SYMPLECTIC REFLECTION ALGEBRAS 141 We will now prove the second isomorphism in Corollary 1.8.3. For each vertex i = o, we have, from (6.4.1), the homomorphism F : e A (Q )e → e A (Q )e i − n,r (λ),ν−1 − − n,λ,ν−1 − which is an isomorphism by Proposition 6.6.1(i). By writing w as a product of simple reflections, we get an isomorphism (6.7.2) F : e A (Q )e e A (Q )e . w − n,w (λ),ν−1 − − n,λ,ν−1 − 0 0 Proposition 6.6.1 implies that this isomorphism does not depend on the choice of pre- sentation of w as a product of simple reflections. ⊗n ⊗n Write H = H (Γ ). By [GG, (3.5.2)], there is an isomorphism f H f = t,k,c t,k,c n t,k,c A (Q ) where f = e .In particular, e H e = e A (Q )e ,and n,λ,ν i − t,k−2t,c − − † − i∈I n,λ ,ν−1 e H e = e A e . By Lemma 6.7.1, λ = w (λ), so by (6.7.2) we have − t,k−2t,c − − n,λ,ν−1 − 0 the isomorphism F : e H e → e H e . w − t,k−2t,c − − t,k−2t,c − This completes the proof of Corollary 1.8.3. Using the isomorphism eH e e H e of Corollary 1.8.3, we can con- t,k,c − t,k−2t,c − sider H e as a (H , eH e)-bimodule. t,k−2t,c − t,k−2t,c t,k,c Definition 6.7.3. — The shift functor is defined to be S : H -mod → H -mod, V → H e ⊗ eV. t,k,c t,k−2t,c t,k−2t,c − eH e t,k,c 7. Extended Dynkin quiver 7.1. Γ-analogue of commuting variety In this subsection, we will prove a generalization of [EG, Theorem 12.1]. Let R(Γ, n) be the space of extensions of the representation CΓ ⊗ C of Γ to arepresentation of T(L) CΓ, i.e., R(Γ, n) := Hom L, End (CΓ ⊗ C ) . Γ C Let Z = Z (Γ, n) be the (not necessarily reduced) subscheme of R(Γ, n) consisting of those representations ρ such that ρ([X, Y]) = 0 for all X, Y ∈ L. Now CΓ =⊕ End(N ).Let p ∈ CΓ be the idempotent element corresponding i∈I i i to the identity element of End(N ).Define Z = Z (Γ, n) to be the (not necessarily i 1 1 reduced) subscheme of R(Γ, n) consisting of those representations ρ such that, for all 142 PAVEL ETINGOF, WEE LIANG GAN, VICTOR GINZBURG, ALEXEI OBLOMKOV X, Y ∈ L,wehave ρ([X, Y]p ) = 0 for i = o,and ∧ ρ([X, Y]p ) = 0.We remark that i o ρ([X, Y]p ) is a n × n-matrix and ∧ ρ([X, Y]p ) = 0 means that all 2 × 2 minors of o o this matrix vanish (so its rank is at most 1). We shall denote by J and J the defining ideals of Z (Γ, n) and Z (Γ, n),respec- 1 1 tively. Thus, Z (Γ, n) = Spec C[R(Γ, n)]/J and Z (Γ, n) = Spec C[R(Γ, n)]/J . 1 1 Let G := Aut (CΓ ⊗ C ). Observe that the group G acts on R(Γ, n), Z (Γ, n),and Z (Γ, n). G G Theorem 7.1.1. — One has: J = J . G G It is clear that J ⊃ J ,so J ⊃ J . To prove Theorem 7.1.1, we have to show G G that J ⊂ J . We need the following lemmas. First, let us fix a basis X, Y for L. G G Lemma 7.1.2. — The ideal J is generated in C[R(Γ, n)] by functions of the form ρ → Tr(ρ(Q [X, Y])),where Q ∈ T(L) CΓ. Proof. — This follows from Weyl’s fundamental theorem of invariant theory. Therefore, it suffices to show that Tr(ρ(Q [X, Y]p )) = 0 (mod J )for all ρ ∈ i 1 R(Γ, n), Q ∈ T(L) CΓ,and i ∈ I. Thisisobviousfor i = o from the defin- ition of J .For i = o,weshall proveit by induction on thedegree of Q.The case deg Q = 0 is clear, so let d> 0 and assume that Tr(ρ(Q [X, Y]p )) = 0 (mod J ) o 1 whenever deg Q< d . Lemma 7.1.3. — Let deg Q = d.If Q = Q [X, Y]Q for some Q , Q ∈ 1 2 1 2 T(L) CΓ,then Tr(ρ(Q [X, Y]p )) = 0 (mod J ). o 1 Proof. —We may replace Q , Q , Q by p Qp , p Q p , p Q p respectively. 1 2 o o o 1 o o 2 o Modulo J , and writing in terms of matrix elements, we have Tr(ρ(Q [X, Y]p )) = Tr(ρ(Q [X, Y]p Q [X, Y]p )) o 1 o 2 o = ρ(Q ) ρ([X, Y]p ) ρ(Q ) ρ([X, Y]p ) 1 lm o mq 2 qr o rl = ρ(Q ) ρ([X, Y]p ) ρ(Q ) ρ([X, Y]p ) 1 lm o ml 2 qr o rq (since ∧ ρ([X, Y]p ) = 0) = Tr(ρ(Q [X, Y]p )) Tr(ρ(Q [X, Y]p )). 1 o 2 o This is equal to zero by induction hypothesis. HARISH–CHANDRA HOMOMORPHISMS AND SYMPLECTIC REFLECTION ALGEBRAS 143 Let ϕ : T(L) CΓ → S(L) CΓ be the quotient map, where S(L) denotes the symmetric algebra on L. The preceding lemma implies the following corollary. Corollary 7.1.4. — If deg Q ≤ d and Q ∈ Ker ϕ,then Tr(ρ(Q [X, Y]p )) = 0 (mod J ). Note that elements of the form (aX + bY) (where a, b ∈ C)spanaset of repre- sentatives of S(L) in T(L). Thus, it remains to show that Tr(ρ((aX + bY) [X, Y]p )) = 0 (mod J ) for any a, b ∈ C and m ≤ d . This is equivalent to showing that Tr(ρ((aX + bY) [X, Y])) = 0 (mod J ). But we have m m+1 Tr(ρ((aX + bY) [X, Y])) = Tr(ρ([(aX + bY) , Y])) = 0. a(m + 1) This completes the proof of Theorem 7.1.1. 7.2. —Let µ : Rep (Q ) → gl(α) be the moment map, and Z = CM CM CM −1 µ (0) the scheme theoretic inverse image of the point 0. It was proved in [GG2, CM Theorem 1.3.1] that Z is a reduced scheme. Now, there are natural algebra mor- CM phisms f g G G G (7.2.1) C[Z ] ← C[Z ] → C[Z ] . 1 CM By Theorem 7.1.1, f is an isomorphism. The following proposition and its proof is a straightforward generalization of [GG2, Proposition 2.8.2], given our Theorem 7.1.1. Proposition 7.2.2. — The morphism g in (7.2.1) is an isomorphism. From Proposition 7.2.2 and [GG2, Theorem 1.3.1], we have the following gen- eralization of [GG2, Theorem 1.2.1]. Theorem 7.2.3. — One has: J = J . red Let Z := Spec C[Rep (Q )]/ J, a closed subvariety of Rep (Q ).Define an nδ nδ embedding j : L → Rep (Q ) by j( u ,..., u ) = (φ (u ), ..., φ (u )) for any a ∈ Q . nδ 1 n a a 1 a n reg Using formulas (8.2.1) from Section 8.2 below, we deduce that the image of j lies red in Z . Pullback of functions gives a morphism ∗ red G n Γ (7.2.4) j : C[Z ] → C[L ] . reg By [CB, Theorem 3.4] and [Kr, Corollary 3.2], we have the following proposition. Proposition 7.2.5. — The map j in (7.2.4) is an isomorphism. 144 PAVEL ETINGOF, WEE LIANG GAN, VICTOR GINZBURG, ALEXEI OBLOMKOV 8. Proof of Proposition 3.7.2 8.1. — The formula of Proposition 3.7.2(i) is clear. Next, we have Holland ∗ a θ (a ) = e ⊗ . q,p ∂t p,q p,q a ∂ n To compute the restriction of e ⊗ to j( L ) at a point u = (u ,..., u ) a 1 n q,p reg ∂t p,q ∈ L , let g (ε) = Id + εB be an element of GL(α) such that p,q p,q reg a n (8.1.1) g (ε) · j( u) + εe = j( u) + εj( w), w ∈ L , p,q p,q reg where we omit terms of higher order in ε.Thenfor a function f ∈ O(χ, N),we have ∂ ∂ a a a −1 (8.1.2) e ⊗ f j( u) + εe = e ⊗ f g (ε) · (j( u) + εj( w)) p,q q,p p,q q,p ∂ε ∂ε a −1 = e ⊗ χ( g (ε))g (ε) f (j( u) + εj( w)) p,q p,q q,p ∂ε ( j ) a a = e ⊗ f (j( u)) + e ⊗ χ Tr B Id − B f (j( u)) j p,q p,q q,p q,p ∂w j∈I ( j ) ( j ) where B is the component of B in gl(α ).We shall write B for j ∈ I as a n × n p,q p,q j p,q ( j ) ( j ) block matrix ⊕ B (, m) where B (, m) ∈ gl(δ ) is the (, m)-th block. Simi- 1≤,m≤n p,q p,q j a a larly, we write e as ⊕ e (, m). By (8.1.1), we need to solve the equations: 1≤,m≤n p,q p,q 0 if = m (h(a)) (t(a)) a B (, m)φ (u ) − φ (u )B (, m) + e (, m) = a m a p,q p,q p,q φ (w ) if = m (o) (s) B (, m) − B = 0. p,q p,q m=1 (s) where 1 ≤ , m ≤ n.Weshall set B = 0. p,q Suppose ( − 1)δ <p ≤ δ and (m − 1)δ <q ≤ mδ where , m ∈[1, n]. h(a) h(a) t(a) t(a) ( j ) If = m,thenweset B ( , m ) = 0 whenever = m and ( , m ) = (, m).If p,q ( j ) = m,then weset B ( , m ) = 0 whenever = m . p,q 8.2. Proof of (3.7.3) First of all, it is immediate from (2.1.1) that (8.2.1) φ ∗ (u)φ (w) = δ ω(w, u)Id , for each source i in Q ; a a i N a∈Q ;t(a)=i φ (w)φ ∗(u) = δ ω(w, u)Id , for each sink j in Q . a a j N a∈Q ;h(a)=j HARISH–CHANDRA HOMOMORPHISMS AND SYMPLECTIC REFLECTION ALGEBRAS 145 Next, we find a collection of operators β ∈ End(N ) such that (8.2.2) φ (u)β − β φ (w) = f a i j a a ∗ ∗ where i = t(a), j = h(a),and f : N → N are given operators. We write the collection i j β as an element β γ of C[Γ].Since γφ (w) = φ (γ w)γ,we get i γ a a −1 β φ (u − γ w)γ = f , β γφ (γ u − w) = f . γ a a γ a a We multiply the first equation above by φ (u) on the left and add over all edges going out from i. Similarly, let us multiply the second equation above by φ (w) on the right and add over all edges going into j . Using formulas (8.2.1), we obtain: ∗ ∗ δ β ω(u,γ w)γ | = φ (u)f , for sources i; i γ N a a a∈Q ;t(a)=i δ β ω(u,γ w)γ | = f φ ∗ (w), for sinks j . j γ N a a a∈Q ;h(a)=j This implies that −1 −1 −1 −1 ∗ ∗ (8.2.3) β = ω(u,γ w) |Γ| Tr | f φ ∗(w)γ + Tr | φ ∗ (u)f γ . γ N a a N a a h(a) t(a) a∈Q ( j ) Hence, if = m,for ⊕ B (, m) we get the expression j∈I p,q −1 −1 a −1 a −1 ∗ ∗ ∗ ∗ (8.2.4) ω(u ,γ u ) |Γ| Tr | e φ (u )γ + Tr | (φ (u )e γ ) γ m N a m N a p,q p,q h(a) t(a) γ ∈Γ (o) (o) and so, for B (, ) =−B (, m) we obtain the expression p,q p,q −1 −1 a −1 − ω(u ,γ u ) |Γ| Tr | e φ ∗ (u )γ m N a m p,q h(a) γ ∈Γ a −1 + Tr | φ (u )e γ . N a p,q t(a) ( j ) Thus, for all j ∈ I,for B (, ) we obtain the expression p,q −1 −1 a −1 ∗ ∗ − ω(u ,γ u ) |Γ| Tr | e φ (u )γ m N a m p,q h(a) γ ∈Γ a −1 ∗ ∗ + Tr | φ (u )e γ Id . N a δ ×δ p,q i i t(a) 146 PAVEL ETINGOF, WEE LIANG GAN, VICTOR GINZBURG, ALEXEI OBLOMKOV It follows from the last formula and from (8.1.2) that for = m,the (m,)-entry Holland ∗ of the radial part of θ (a ) is δ mδ h(a) t(a) ( j ) ( j ) e (m,) χ Tr B (, ) − B (, ) j p,q p,q q,p p=(−1)δ +1 q=(m−1)δ +1 j∈I j∈I h(a) t(a) −1 −1 ∗ ∗ (φ ) · γ + γ (φ ) a m a −1 =|Γ| (1 − χ δ ) j j ω(γ ; , m) −1 k (φ ◦ (Id + γ )) a m, −1 = γ . 2 ω(γ ; , m) Note that, since ζ acts by −1 on L and by 1 on N , h(a) ∗ ∗ (φ ) (φ ) a m a m −1 −1 (γζ) =− γ ω(γζ ; , m) ω(γ ; , m) and so (φ ) a m −1 γ = 0. ω(γ ; , m) Holland ∗ Hence, the (m,)-entry of the radial part of θ (a ) is equal to k (φ ◦ γ) − γ. 2 ω(γ ; m,) Proof of Lemma 3.3.1. —We set f = 0 in (8.2.2). Then from (8.2.3), we have ω(u,γ w)β = 0 for all γ ∈ Γ.Since not all β are zero, we must have ω(u,γ w) = 0 γ γ for some γ . 8.3. Proof of (3.7.4) We need to solve φ (u)β − β φ (u) = f − φ (w). a i j a a a As above, for γ = 1,ζ,we obtain −1 −1 −1 −1 ∗ ∗ β = ω(u,γ u) |Γ| Tr | f φ ∗(u)γ + Tr | φ ∗ (u)f γ . γ N a a N a a h(a) t(a) a∈Q Moreover, multiplying on the right by φ ∗(v) and summing over all incoming edges a ∈ Q at the vertex j,we get −1 δ β ω(γ u − u, v)γ | = f φ ∗ (v) − δ ω(w, v). j γ N a a j a∈Q :h(a)=j Take the trace of both sides this equation and sum up over all sinks j.We have ∗ ⊕δ ⊕ (N ) = C[Γ/S], where S ={1,ζ }. It follows that the trace of γ in the last sink j j HARISH–CHANDRA HOMOMORPHISMS AND SYMPLECTIC REFLECTION ALGEBRAS 147 sum vanishes if γ = 1,ζ.Let β = 0.Then −1 (8.3.1) w = 2|Γ| Tr( f φ ∗ ). a a a∈Q Hence, for = m,we get ( j ) −1 −1 a −1 ∗ ∗ (8.3.2) ⊕ B (m, m) = ω(u ,γ u ) |Γ| Tr | e (φ ) γ j∈I p,q m m N a m p,q h(a) γ =1,ζ a −1 + Tr | (φ ∗ ) e γ (γ − 1). N a m p,q t(a) It follows from (8.1.2), (8.3.1), (8.3.2) and (8.2.4) that the (m, m)-entry of the ra- Holland ∗ dial part of θ (a ) is mδ mδ h(a) t(a) 2 ∂ ( j ) (8.3.3) + e (m, m) χ Tr(B (m, m)) j p,q q,p |Γ| ∂(φ ∗ ) a m j∈I p=(m−1)δ +1 q=(m−1)δ +1 h(a) t(a) δ mδ h(a) t(a) ( j ) ( j ) − B (m, m) − e (m,) B (, m) p,q p,q q,p j∈I =m p=(−1)δ +1 q=(m−1)δ +1 j∈I h(a) t(a) −1 −1 ∗ ∗ 2 ∂ 1 γ · (φ ) + (φ ) · γ a m a m = + − γ + 1 |Γ| ∂(φ ∗ ) |Γ| ω(γ ; m, m) a m γ =1,ζ −1 1 (φ ∗ ◦ (Id + γ )) a m, − χ (δ − Tr | (γ)) − j j N |Γ| ω(γ ; , m) j =m γ −1 2 ∂ 1 (φ ∗ ◦ (γ + Id)) a mm −1 = + −1 +|Γ|c γ |Γ| ∂(φ ) |Γ| ω(γ ; m, m) a m γ =1,ζ 1 (φ ∗ ◦ γ) + . |Γ| ω(γ ; m,) =m γ The last term in (8.3.3) comes from (8.2.4). It is even easier to compute the radial part for the edge b : s → o.Weomit this computation. This completes the proof of Proposition 3.7.2. 9. Proof of Theorem 4.3.2 9.1. — It easy to check that the operators R have the following Γ -equivari- ml ance properties: γ(v) v v v γ R = R γ ,γ R = R γ , m m l l ml ml ml ml v v v v v v s R = R s , s R = R s , s R = R s , ml ml mj mj lj lj ml lm ml jl ml mj Dunkl Dunkl n where j = m, l.It implies that gΘ (v) = Θ g(v), for any g ∈ Γ and v ∈ L . reg 148 PAVEL ETINGOF, WEE LIANG GAN, VICTOR GINZBURG, ALEXEI OBLOMKOV Next, we prove that Dunkl Dunkl (9.1.1) Θ (w ), Θ (v ) i i −1 = ω(w, v) t · 1 + s γ γ + c γ , ij i γ i j =i γ ∈Γ γ ∈Γ{1} where 1 ≤ i ≤ n. First we prove w Dunkl Dunkl v −1 (9.1.2) R , Θ (v) + Θ (w), R = ω(w, v) s γ γ , ij i ij i i ij j j =i γ ∈Γ 1 ≤ i = j ≤ n. ⊗n We prove that (9.1.2) holds if we apply both sides to f , a basis vector in F such that f , f = f , cf. (4.1.4). Indeed, in that case we have i j (γ w) w Dunkl w v Dunkl R , Θ (v) ( f ) = R (D ) + Θ (v) ( f ) = A f ij i ij i 2 ω(1; i, j ) γ ∈Γ (γ v) Dunkl v Dunkl w w Θ (w), R ( f ) = − Θ (w) − R (D ) ( f ) i ij i ij −1 2 ω(γ ; i, j ) γ ∈Γ = Bf , where ∨ ∨ ∨ ∨ v (γ w) w (γ v) 1 1 i j i j −1 −1 A =− s γ γ , resp., B = s γ γ . ij i ij i j j −1 −1 2 ω(γ ; i, j ) 2 ω(γ ; i, j ) γ ∈Γ γ ∈Γ These formulas yield w Dunkl Dunkl v R ,Θ (v) + Θ (w), R f ij i i ij ∨ ∨ ∨ ∨ v (γ w) − w (γ v) i j i j −1 =− s γ γ f ij i −1 2 ω(γ ; i, j ) γ ∈Γ −1 = ω(w, v) s γ γ f . ij i γ ∈Γ − + We consider the case f = f , f = f .Then wehave: i j (γ w) w Dunkl w ∨ ∨ −1 R , Θ (v) f =−R v f = v s γ γ f = 0, ij i ij i ij i i j −1 2 ω(γ ; i, j ) γ ∈Γ HARISH–CHANDRA HOMOMORPHISMS AND SYMPLECTIC REFLECTION ALGEBRAS 149 because in the sum the terms corresponding γ and ζγ mutually cancel. Analogously, (γ v) Dunkl w v ∨ −1 ∨ Θ (w), R f = R w f =− s γ γ w f = 0. ij i i ij ij i j i −1 2 ω(γ ; i, j ) γ ∈Γ −1 For the same reason we also have s γ γ f = 0. ij i γ ∈Γ + − We consider the case f = f , f = f . A similar argument yields i j w Dunkl Dunkl w −1 R , Θ (v) f = Θ (w), R f = s γ γ f = 0. ij i ij i i ij j γ ∈Γ The case f , f = f is analogous to the first case and we have i j ∨ ∨ w (γ v) i j w Dunkl R , Θ (v) f = f , ij i −1 2 ω(γ ; i, j ) γ ∈Γ ∨ ∨ v (γ w) i j Dunkl v −1 Θ (w), R f =− s γ γ f , ij i i ij j −1 2 ω(γ ; i, j ) γ ∈Γ and w Dunkl Dunkl v R , Θ (v) + Θ (w), R f ij i i ij ∨ ∨ ∨ ∨ w (γ v) − v (γ w) i j i j −1 =− s γ γ f ij i −1 2 ω(γ ; i, j ) γ ∈Γ −1 = ω(w, v) s γ γ f . ij i γ ∈Γ w v We remark that for any 1 ≤ j = i = k ≤ n and w, v ∈ L we have R R = ij ik v w R R = 0. Now, (9.1.1) follows from this equation, the n = 1 case of Theorem 4.3.2, ik ij and (9.1.2). 9.2. —Next we prove: Dunkl Dunkl −1 (9.2.1) Θ (w ), Θ (v ) =− ω (γ u, v)s γ γ , 1 ≤ i = m ≤ n. i m L im i γ ∈Γ To this end, we rewrite the RHS of (9.1.1) as follows Dunkl Dunkl Dunkl v w Dunkl Θ (w ), Θ (v ) = k Θ (w), R + R Θ (v) i m i mi im m 2 w v w v w v + k R , R + R , R + R , R im mi im mj ij mj j =m,i w v + R , R . ij mi 150 PAVEL ETINGOF, WEE LIANG GAN, VICTOR GINZBURG, ALEXEI OBLOMKOV We first prove that w v w v w v R , R + R , R + R , R = 0, j = m, i. im mj ij mj ij mi For that it is enough to show that w v v w w v v w w v v w (9.2.2) R R − R R + R R = 0, and − R R + R R − R R = 0. im mj mj ij ij mi mj im ij mj mi ij We prove the first equation, the second is proved similarly. Let f be a basis vec- ⊗n + tor in F such that f , f , f = f , cf. (4.1.4). We compute i j m w v v w w v 4 R R − R R + R R ( f ) im mj mj ij ij mi ∨ ∨ (w) (γ v) i j −1 = s s β γ (γβ) im mj i m −1 −1 ω(β ; i, m)ω((γβ) ; i, j ) β,γ ∈Γ ∨ −1 ∨ (βv) (β γ w) j m −1 −1 − s s (β γ) β γ mj ij i m −1 ω(β ; m, j )ω(β; i, m) γ,β∈Γ ∨ ∨ (βw) (βγ v) j j −1 −1 + s s γ (βγ) β f . ij mi m i j −1 −1 ω(β ; i, j )ω((βγ) ; m, j ) γ,β∈Γ We change summation indices at the first term as γ → γβ, β → γ and at the −1 third term as γ → β ,β → γβ.We get ∨ ∨ ∨ ∨ w (γ v) (γ v) (βw) i j j m −1 −1 −1 −1 ω(β ; i, m)ω(β ; i, j ) ω(γ ; m, j )ω(β ; i, m) γ,β∈Γ ∨ ∨ (γβw) γ v j j −1 −1 + · s s β γ (β γ ) f im mj i m j −1 −1 ω(β ; i, j )ω(γ ; m, j ) γ v = 4 · Y f = 0, β,γ,m, j −1 −1 −1 ω(β ; i, m)ω(β ; i, j )ω(γ ; m, j ) γ,β∈Γ where Y is given by the following expression β,γ,m, j ∨ −1 ∨ −1 Y = w ω(γ ; m, j ) − (βw) ω(β ; i, j ) β,γ,m, j i m ∨ −1 −1 −1 + (γβw) ω(β ; i, m) · s s β γ (β γ ) . im mj i m j j HARISH–CHANDRA HOMOMORPHISMS AND SYMPLECTIC REFLECTION ALGEBRAS 151 + − We consider the case f , f = f , f = f : i m j w v v w w v 4 R R − R R + R R ( f ) im mj mj ij ij mi ∨ −1 ∨ (w) β v i i −1 = s s β γ (γβ) im mj i m −1 −1 ω(β ; i, m)ω((γβ) ; i, j ) β,γ ∈Γ ∨ ∨ v w m i −1 −1 − s s (β γ) β γ mj ij i m −1 −1 ω(β ; m, j )ω(γ β; i, m) γ,β∈Γ ∨ ∨ w βγ v i j −1 −1 + s s γ (βγ) β f . ij mi m i j −1 −1 ω(β ; i, j )ω((βγ) ; m, j ) γ,β∈Γ Performing the same change of summation indices as in the previous paragraph, we deduce that the following sum vanishes ∨ −1 ∨ −1 ∨ −1 ∨ −1 w (β v ω(γ ; m, j ) − v ω((γβ) ; i, j ) + γ v ω(β ; i, m)) i i m j −1 −1 −1 ω(β ; i, m)ω((γβ) ; i, j )ω(γ ; m, j ) γ,β∈Γ −1 −1 ×s s β γ (β γ ) f . im mj i m j − − w v v w w v It easy to see that if f = f or f = f then R R f = R R f = R R f = 0. i m im mj mj ij ij mi Thus, we have proved (9.2.2). Now we prove that w v R , R = 0. im mi w v − − + It is easy to see that [R , R ] f = 0 when f = f or f = f .If f , f = f then i m i m im mi w v [R , R ] f = 0 is equivalent to: im mi ∨ ∨ ∨ −1 −1 ∨ w βγ v − v β γ w i m m i −1 (9.2.3) (βγ) (γβ) f = 0. −1 −1 ω(β ; i, m)ω((βγβ) ; i, m) γ,β∈Γ Fix some γ, β ∈ Γ.Let α := βγ , β := γβ. It is easy to see that the coefficient −1 in front of α β f in (9.2.3) is equal to ∨ ∨ ∨ −1 ∨ (9.2.4) w αv − v β w i m m i −1 ω(δβ; i, m)ω(β δβ; i, m) δ∈Z where Z is the notation for the centaralizer of β ∈ Γ. Wenoticethat α is conjugate −1 to β,hence if β = 1 or β = ζ then α = β = β and (9.2.4) is zero. 152 PAVEL ETINGOF, WEE LIANG GAN, VICTOR GINZBURG, ALEXEI OBLOMKOV Thus we can assume that β = 1,ζ.Inthiscase Z is a cyclic group. We denote the order of Z by l and let ρ be a generator of Z . Then we can assume that β = ρ β β for some q, 0< q< l. −1 Let a, b ∈ L be the basis in L such that ρa = a, ρb = b where is some ∗ ∗ primitive lth root of 1.Let x = a and y = b . Wemakechange ofvariables βu → u m m in (9.2.4). Let z = x y /(x y ). Then, (9.2.4) is proportional to mi i m m i l−1 l−1 1 1 1 p q p 2 −p p −p+q p−q ω(u ,δ u )ω(δ u ,δ u ) (x y ) ( − z )( − z ) i m i m m i mi mi p=0 p=0 l−1 1 1 1 = − = 0. q −q 2 2q−2p −2p ( − )(x x ) z − z − m i mi mi p=0 Finally we show that Dunkl v w Dunkl −1 Θ (w), R + R Θ (v) =− ω (γ w, v)s γ γ . L im i i mi im m m γ ∈Γ If f , f = f then the terms in the LHS of the sum below corresponding to γ i m and γζ mutually cancel out, and we deduce −1 − ω (γ w, v)s γ γ f = 0. L im i γ ∈Γ In the case f , f = f we know that i m Dunkl v w Dunkl Θ (w), R f = R , Θ (v) f = 0. i mi im m In the case f = f = f we have i m 1 γ v Dunkl v w i −1 Θ (w), R f = −(D ) s γ γ i mi m i mi i −1 2 ω(γ ; m, i) γ ∈Γ v −1 m γ w −1 + (D ) s γ γ f , m mi m −1 ω(γ ; m, i) 1 w −1 w Dunkl i γ v −1 R , Θ (v) f = − (D ) s γ γ i im i im m m −1 2 ω(γ ; i, m) γ ∈Γ γ w v m −1 + (D ) s γ γ f . m im i −1 ω(γ ; i, m) Fix a conjugacy class C ⊂ Γ. Then the coefficient in front of − c , β ∈ Cat Dunkl v w Dunkl ([Θ (w), R ]+ [R Θ (v)])f is equal to i mi im m HARISH–CHANDRA HOMOMORPHISMS AND SYMPLECTIC REFLECTION ALGEBRAS 153 ∨ ∨ ∨ (βw + w )βγ v i i i −1 s (βγ) (γ ) mi m i −1 −1 ω(β; i, i)ω(γ β ; m, i) γ ∈Γ,β∈C ∨ −1 ∨ −1 ∨ v βγ w + γ w m m m −1 − s γ (βγ ) mi m i −1 ω(γ ; m, i)ω(β; m, m) ∨ ∨ ∨ (βv + v )βγ w m m m −1 − s (βγ) (γ ) im i m −1 −1 ω(β; m, m)ω(γ β ; i, m) ∨ −1 ∨ −1 ∨ w βγ v + γ v i i i −1 + s γ (βγ ) f . im i m −1 ω(γ ; i, m)ω(β; i, i) −1 Fix γ ∈ Γ,β ∈ C. Then the coefficient in front of s (βγ) γ is equal to im m ∨ −1 ∨ −1 −1 ∨ ∨ ∨ ∨ v γ w + γ β w (βw + w )βγ v i i i m m m F = − −1 ω(β; i, i)ω(βγ ; m, i) ω(βγ ; m, i)ω(γ βγ ; m, m) −1 ∨ ∨ −1 ∨ ∨ ∨ ∨ γ βγ v + v γ w w (βγ v + γ v ) m m m i i i − + . −1 −1 ω(γ βγ ; m, m)ω(γ ; i, m) ω(γ ; i, m)ω(β; i, i) We see that F = F(u , u ) is a homogeneous function in two variables, of bide- i m gree (−1, −1),thatis F ∈ H (P × P, O(−1) O(−1)). It could have simple poles −1 along the divisors u ∼ βu , u ∼ γ βγ u , u ∼ βγ u , u ∼ γ u where ∼ stands for i i m m i m i m being proportional. But is easy to check that the residues actually vanish. We deduce that F = 0. −1 Dunkl v The coefficient in front of s γ γ f in the part of ([Θ (w), R ] mi m i i mi w Dunkl +[R Θ (v)])f that does not contain coefficients c , γ ∈ Γ, equals im m ∨ ∨ ∂ γ v v ∂ i m − + −1 −1 −1 ∂w ω(γ ; m, i) ω(γ ; m, i) ∂(γ w) i m ∨ −1 ∨ w ∂ ∂ γ w i m − + . −1 −1 ω(γ ; i, m) ∂(γ v) ∂v ω(γ ; i, m) i m It is easy to show that this expression vanishes since we have ∂ ∂ ∂ ∂ ∨ ∨ −1 ∨ ∨ w − γ v = γ w − v = ω(w, v)eu. i i m m ∨ −1 ∂(γ v) ∂w ∂(γ v) ∂(γ w) i m i m − + w We consider the case f = f , f = f .Thenwe have [R , Θ (v)] f = 0 and i m m im ∨ ∨ ∨ −1 ∨ 1 w v − (γ v) (γ w) i m i m Dunkl v −1 Θ (w), R f = s γ γ f mi m i mi i −1 2 ω(γ ; m, i) γ ∈Γ −1 =− ω(γ w, v)s γ γ f . mi i γ ∈Γ + − The analysis of the case f = f , f = f is similar. i m 154 PAVEL ETINGOF, WEE LIANG GAN, VICTOR GINZBURG, ALEXEI OBLOMKOV REFERENCES [BB] A. BEILINSON and J. BERNSTEIN, A proof of Jantzen conjectures, I. M. Gelfand Seminar,Adv. SovietMath., vol. 16, part 1, pp. 1–50, Amer. Math. 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Published: Jul 21, 2007
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