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An Approach to the Connectedness of the Left Cells in Affine Weyl Groups

An Approach to the Connectedness of the Left Cells in Affine Weyl Groups AN APPROACH TO THE CONNECTEDNESS OF THE LEFT CELLS IN AFFINE WEYL GROUPS XI NANHUA 1. Introduction Let (W^S^ be an irreducible affine Weyl group and W be the corresponding extended affine Weyl group. Let / be the standard length function on W and let its standard extension to W also be denoted by /. A subset V of W or of W is said to be connected if for any x,ye V there exits a sequence x = x , JC ...,x = y in Ksuch that 0 15 n x = tfX^ with l(t ) = 1 or 0 (1 < i ^ /)• The concepts of a left cell and a two-sided t t cell of W or of W were defined in [3, § 1] and in [6, 3.1 (e)] and Lusztig conjectured that each left cell of W is connected [1, p. 14]. This is equivalent to each left cell of W being connected. Our main result is that each left cell of W has only finitely many connected components. This is equivalent to the following. THEOREM 1.1. Each left cell of W has only finitely many connected components. In this paper we also http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

An Approach to the Connectedness of the Left Cells in Affine Weyl Groups

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/21.6.557
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Abstract

AN APPROACH TO THE CONNECTEDNESS OF THE LEFT CELLS IN AFFINE WEYL GROUPS XI NANHUA 1. Introduction Let (W^S^ be an irreducible affine Weyl group and W be the corresponding extended affine Weyl group. Let / be the standard length function on W and let its standard extension to W also be denoted by /. A subset V of W or of W is said to be connected if for any x,ye V there exits a sequence x = x , JC ...,x = y in Ksuch that 0 15 n x = tfX^ with l(t ) = 1 or 0 (1 < i ^ /)• The concepts of a left cell and a two-sided t t cell of W or of W were defined in [3, § 1] and in [6, 3.1 (e)] and Lusztig conjectured that each left cell of W is connected [1, p. 14]. This is equivalent to each left cell of W being connected. Our main result is that each left cell of W has only finitely many connected components. This is equivalent to the following. THEOREM 1.1. Each left cell of W has only finitely many connected components. In this paper we also

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Nov 1, 1989

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