The Kazhdan-Lusztig Cells in Certain Affine Weyl Groups
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1179
Shi Jian¥i
The Kazhdan-Lusztig Cells in Certain Affine Weyl Groups
Springer-Verlag Berlin Heidelberg New York Tokyo
Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
1179
Shi Jian¥i
The Kazhdan-Lusztig Cells in Certain Affine Weyl Groups
Springer-Verlag Berlin Heidelberg New York Tokyo
Author SHI Jian-Yi Department of Mathematics, East China Normal University Shanghai, The People's Republic of China
Mathematics Subject Classification (1980): 20B27, 20H15 ISBN 3-540-16439-1 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-16439-1 Springer-Verlag New York Heidelberg Berlin Tokyo
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INTRODUCTION Cells of a Coxeter group play an important role not only in the representation theory of the Coxeter group and its associated Hecke algebra, but also in the representation theory of algebraic groups, finite groups of Lie type, Lie algebras and enveloping algebras. The concept of cells originally came from combinatorial theory. Robinson [R1J defined a map from the symmetric group S to the set of pairs (P,O) of standard n Young tableaux of the same shape and of size n. Then Schensted [5chJ proved that this map is bijective. Hence a left cell of 5 corresponds to the set of such pairs n (P,O) with P fixed. A two-sided cell of Sn corresponds to the set of such pairs (P,O) with P,O of fixed shape. 50 there is a 1-1 correspondence between the set of two-sided cells of Sn and the set of partitions of n. This is the prototype of cells which applies to any Coxeter group. Then Joseph [J01] defined the concept of left cells in the Weyl group Win terms of primitive ideals in the enveloping algebra of a complex semisimple Lie algebra. For w E W, let Jw be the annihilator of the irreducible module of the enveloping algebra with highest weight -wp-p • where p is half the sum of positive roots. Then w , w' are said to be in the same left cell precisely when Jw = Jw" Joseph's definition of left cells and the corresponding Weyl group representations involves some deep results about the multiplicities of the composition factors of the Verma modules with highest weight -WP-p. In 1979, Kazhdan and Lusztig [KL1J gave the definition of cells for an arbitrary Coxeter group. Their definition is elementary but it gives rise not only to representations of the Coxeter group, but also of the corresponding Hecke algebra. This makes possible applications of the results on cells to more general representation theory. On the other hand, the definition of Kazh
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