On the lowest two-sided cell in affine Weyl groups

Guilhot, Jérémie

doi:10.1090/S1088-4165-08-00334-8

On the lowest two-sided cell in affine Weyl groups
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by Jérémie Guilhot

Represent. Theory 12 (2008), 327-345

DOI: https://doi.org/10.1090/S1088-4165-08-00334-8

Published electronically: October 9, 2008

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Abstract:

Bremke and Xi determined the lowest two-sided cell for affine Weyl groups with unequal parameters and showed that it consists of at most $|W_{0}|$ left cells where $W_{0}$ is the associated finite Weyl group. We prove that this bound is exact. Previously, this was known in the equal parameter case and when the parameters were coming from a graph automorphism. Our argument uniformly works for any choice of parameters.

References

Robert Bédard, The lowest two-sided cell for an affine Weyl group, Comm. Algebra 16 (1988), no. 6, 1113–1132. MR 939034, DOI 10.1080/00927878808823622
Kirsten Bremke, On generalized cells in affine Weyl groups, J. Algebra 191 (1997), no. 1, 149–173. MR 1444494, DOI 10.1006/jabr.1996.6889
Meinolf Geck, On the induction of Kazhdan-Lusztig cells, Bull. London Math. Soc. 35 (2003), no. 5, 608–614. MR 1989489, DOI 10.1112/S0024609303002236
Meinolf Geck and Götz Pfeiffer, Characters of finite Coxeter groups and Iwahori-Hecke algebras, London Mathematical Society Monographs. New Series, vol. 21, The Clarendon Press, Oxford University Press, New York, 2000. MR 1778802
George Lusztig, Hecke algebras and Jantzen’s generic decomposition patterns, Adv. in Math. 37 (1980), no. 2, 121–164. MR 591724, DOI 10.1016/0001-8708(80)90031-6
G. Lusztig, Left cells in Weyl groups, Lie group representations, I (College Park, Md., 1982/1983) Lecture Notes in Math., vol. 1024, Springer, Berlin, 1983, pp. 99–111. MR 727851, DOI 10.1007/BFb0071433
G. Lusztig, Hecke algebras with unequal parameters, CRM Monograph Series, vol. 18, American Mathematical Society, Providence, RI, 2003. MR 1974442, DOI 10.1090/crmm/018
Jian Yi Shi, A two-sided cell in an affine Weyl group, J. London Math. Soc. (2) 36 (1987), no. 3, 407–420. MR 918633, DOI 10.1112/jlms/s2-36.3.407
Jian Yi Shi, A two-sided cell in an affine Weyl group. II, J. London Math. Soc. (2) 37 (1988), no. 2, 253–264. MR 928522, DOI 10.1112/jlms/s2-37.2.253
Nan Hua Xi, Representations of affine Hecke algebras, Lecture Notes in Mathematics, vol. 1587, Springer-Verlag, Berlin, 1994. MR 1320509, DOI 10.1007/BFb0074130