Involutions in Weyl groups

Author:
Robert E. Kottwitz

Journal:
Represent. Theory **4** (2000), 1-15

MSC (2000):
Primary 20F55; Secondary 22E50

DOI:
https://doi.org/10.1090/S1088-4165-00-00050-9

Published electronically:
February 1, 2000

MathSciNet review:
1740177

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a split real group with Weyl group . Let be an irreducible representation of . Let be the stable Lie algebra version of the coherent continuation representation of . The main result of this paper is a formula for the multiplicity of in . The formula involves the position of in Lusztig's set . The paper treats all quasi-split groups as well.

**[Ass98]**M. Assem,*On stability and endoscopic transfer of unipotent orbital integrals on -adic symplectic groups*, Mem. Amer. Math. Soc.**635**(1998). MR**98m:22013****[Bar91]**D. Barbasch,*Unipotent representations for real reductive groups*, Proceedings of the International Congress of Mathematicians, Kyoto, Japan, 1990, The Mathematical Society of Japan, 1991, pp. 769-777. MR**93m:22012****[BL78]**W. M. Beynon and G. Lusztig,*Some numerical results on the characters of exceptional Weyl groups*, Math. Proc. Cambridge Philos. Soc.**84**(1978), 417-426. MR**80a:2001****[BV82a]**D. Barbasch and D. Vogan,*Primitive ideals and orbital integrals in complex classical groups*, Math. Ann.**259**(1982), 153-199. MR**83m:22026****[BV82b]**D. Barbasch and D. Vogan,*Weyl group representations and nilpotent orbits*, Representation Theory of Reductive Groups (P. C. Trombi, ed.), Birkhäuser, 1982, pp. 21-33. MR**85g:22025****[Cas98]**B. Casselman,*Verifying Kottwitz' conjecture by computer*, Represent. Theory**4**(2000), 32-45.**[Ful97]**W. Fulton,*Young tableaux: with applications to representation theory and geometry*, London Mathematical Society Students Texts 35, Cambridge University Press, 1997. MR**99f:05119****[KL79]**D. Kazhdan and G. Lusztig,*Representations of Coxeter groups and Hecke algebras*, Invent. Math.**53**(1979), 165-184. MR**81j:20066****[Kot98]**R. Kottwitz,*Stable nilpotent orbital integrals on real reductive Lie algebras*, Represent. Theory**4**(2000), 16-31.**[Lus79]**G. Lusztig,*Unipotent representations of a finite Chevalley group of type*, Quart. J. Math. Oxford Ser. (2)**30**(1979), 315-338. MR**80j:20041****[Lus84]**G. Lusztig,*Characters of reductive groups over a finite field*, Ann. of Math. Studies, 107, Princeton University Press, 1984. MR**86j:20038****[McG98]**W. M. McGovern,*Cells of Harish-Chandra modules for real classical groups*, Amer. J. Math.**120**(1998), 211-228. MR**98j:22022****[Ros90]**W. Rossmann,*Nilpotent orbital integrals in a real semisimple Lie algebra and representations of Weyl groups*, Operator Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory, Progr. Math., 92, Birkhäuser, Boston, 1990, pp. 263-287. MR**92c:22022****[Tan85]**T. Tanisaki,*Holonomic systems on a flag variety associated to Harish-Chandra modules and representations of a Weyl group*, Algebraic Groups and Related Topics, Adv. Studies in Pure Math., 6, North-Holland, 1985, pp. 139-154. MR**87b:22033****[Tho80]**J. G. Thompson,*Fixed point free involutions and finite projective planes*, Finite Simple Groups II, Proc. Sympos., Univ. Durham 1978, Academic Press, 1980, pp. 321-337.**[Wal99]**J.-L. Waldspurger,*Intégrales orbitales nilpotentes et endoscopie pour les groupes classiques non ramifiés*, preprint, 1999.

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Additional Information

**Robert E. Kottwitz**

Affiliation:
Department of Mathematics, University of Chicago, 5734 University Avenue, Chicago, Illinois 60637

Email:
kottwitz@math.uchicago.edu

DOI:
https://doi.org/10.1090/S1088-4165-00-00050-9

Received by editor(s):
May 14, 1998

Received by editor(s) in revised form:
August 25, 1999

Published electronically:
February 1, 2000

Article copyright:
© Copyright 2000
American Mathematical Society