A generalization of a result of Kazhdan and Lusztig

Authors:
Jeffrey D. Adler and Stephen DeBacker

Journal:
Proc. Amer. Math. Soc. **132** (2004), 1861-1868

MSC (2000):
Primary 22E35, 22E65; Secondary 17B45, 20G05, 20G15, 22E50, 20G25

DOI:
https://doi.org/10.1090/S0002-9939-03-07261-7

Published electronically:
December 1, 2003

MathSciNet review:
2051152

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Kazhdan and Lusztig showed that every topologically nilpotent, regular semisimple orbit in the Lie algebra of a simple, split group over the field is, in some sense, close to a regular nilpotent orbit. We generalize this result to a setting that includes most quasisplit -adic groups.

**1.**J. D. Adler and S. DeBacker,*Murnaghan-Kirillov theory for supercuspidal representations of tame general linear groups*, submitted.**2.**J. D. Adler and S. DeBacker,*Some applications of Bruhat-Tits theory to harmonic analysis on the Lie algebra of a reductive -adic group*, Michigan Math. J.**50**(2002), no. 2, 263-286. MR**2003g:22016****3.**P. Bala and R. W. Carter,*Classes of unipotent elements in simple algebraic groups. II*, Math. Proc. Cambridge Philos. Soc.**80**(1976), no. 1, 1-17. MR**54:5363b****4.**A. Borel,*Linear algebraic groups*, 2nd ed., Graduate Texts in Mathematics, vol. 126, Springer-Verlag, New York, 1991. MR**92d:20001****5.**F. Bruhat and J. Tits,*Groupes réductifs sur un corps local I: Données radicielles valuées*, Inst. Hautes Études Sci. Publ. Math.**41**(1972), 5-251. MR**48:6265****6.**S. DeBacker,*Parametrizing nilpotent orbits via Bruhat-Tits theory*, Ann. Math.**156**(2002), 295-332. MR**2003i:20086****7.**D. Kazhdan and G. Lusztig,*Fixed point varieties on affine flag manifolds*, Israel J. Math.**62**(1988), no. 2, 129-168. MR**89m:14025****8.**B. Kostant,*Lie group representations on polynomial rings*, Amer. J. Math.**85**(1963), 327-404. MR**28:1252****9.**R. E. Kottwitz,*Transfer factors for Lie algebras*, Represent. Theory**3**(1999), 127-138 (electronic). MR**2000g:22028****10.**A. Moy and G. Prasad,*Jacquet functors and unrefined minimal -types*, Comment. Math. Helvetici**71**(1996), 98-121. MR**97c:22021****11.**J.-P. Serre,*Galois cohomology*, Springer-Verlag, Berlin, 1997, Translated from the French by Patrick Ion and revised by the author. MR**98g:12007****12.**T. A. Springer,*The unipotent variety of a semi-simple group*, Algebraic Geometry (Internat. Colloq., Tata Inst. Fund. Res., Bombay, 1968), Oxford Univ. Press, London, 1969, pp. 373-391. MR**41:8429****13.**T. A. Springer,*Linear algebraic groups*, 2nd edition, Birkhäuser, Boston, 1998. MR**99h:20075****14.**R. Steinberg,*Regular elements of semisimple algebraic groups*, Inst. Hautes Études Sci. Publ. Math.**25**(1965), 49-80. MR**31:4788****15.**R. Steinberg,*Torsion in reductive groups*, Advances in Math.**15**(1975), 63-92. MR**50:7369****16.**J.-K. Yu,*Construction of tame supercuspidal representations*, J. Amer. Math. Soc.**14**(2001), no. 3, 579-622. MR**2002f:22033**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
22E35,
22E65,
17B45,
20G05,
20G15,
22E50,
20G25

Retrieve articles in all journals with MSC (2000): 22E35, 22E65, 17B45, 20G05, 20G15, 22E50, 20G25

Additional Information

**Jeffrey D. Adler**

Affiliation:
Department of Theoretical and Applied Mathematics, The University of Akron, Akron, Ohio 44325-4002

Email:
adler@uakron.edu

**Stephen DeBacker**

Affiliation:
Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138

Address at time of publication:
Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1109

Email:
debacker@math.harvard.edu, smdbackr@umich.edu

DOI:
https://doi.org/10.1090/S0002-9939-03-07261-7

Keywords:
$p$-adic,
Lie algebra,
stability

Received by editor(s):
September 23, 2002

Received by editor(s) in revised form:
February 26, 2003

Published electronically:
December 1, 2003

Additional Notes:
The authors were partially supported by the National Security Agency (#MDA904-02-1-0020) and the National Science Foundation (Grant No. 0200542), respectively. This work was begun while the authors were attending workshops in Banff in 2001–2002 sponsored by the Mathematical Sciences Research Institute and the Pacific Institute for the Mathematical Sciences, and completed while the authors were visiting the Institute for Mathematical Sciences (IMS) at the National University of Singapore (NUS) in 2002, visits supported by IMS and NUS

Communicated by:
Rebecca Herb

Article copyright:
© Copyright 2003
American Mathematical Society