Remote Access Representation Theory
Green Open Access

Representation Theory

ISSN 1088-4165



Spherical functions of the symmetric space $G(\mathbb{F} _{q^{2}})/G(\mathbb{F} _q)$

Author: Anthony Henderson
Journal: Represent. Theory 5 (2001), 581-614
MSC (2000): Primary 20G40, 20G05; Secondary 20C15, 32C38
Published electronically: November 28, 2001
MathSciNet review: 1870603
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We apply Lusztig's theory of character sheaves to the problem of calculating the spherical functions of $G(\mathbb{F} _{q^{2}})/G(\mathbb{F} _q)$, where $G$ is a connected reductive algebraic group. We obtain the solution for generic spherical functions for any $G$, and for all spherical functions when $G=GL_n$. The proof includes a result about convolution of character sheaves and its interaction with the associated two-sided cells.

References [Enhancements On Off] (What's this?)

  • 1. E. BANNAI, N. KAWANAKA, AND S.-Y. SONG, The character table of the Hecke algebra $\mathcal{H}(GL_{2n}(\mathbb{F} _q),Sp_{2n}(\mathbb{F} _q))$, J. Algebra, 129 (1990), pp. 320-366. MR 91d:20052
  • 2. A. A. BEILINSON, J. BERNSTEIN, AND P. DELIGNE, Faisceaux pervers, Astérisque, 100 (1982). MR 86g:32015
  • 3. J. BERNSTEIN AND P. LUNTS, Equivariant Sheaves and Functors, no. 1578 in Lecture Notes in Math., Springer-Verlag, 1994. MR 95k:55012
  • 4. P. DELIGNE AND G. LUSZTIG, Representations of reductive groups over a finite field, Ann. of Math., 103 (1976), pp. 103-161. MR 52:14076
  • 5. V. GINZBURG, Admissible modules on a symmetric space, Astérisque, 173-174 (1989), pp. 199-255. MR 91c:22030
  • 6. R. GOW, Two multiplicity-free permutations of the general linear group $GL(n,q^{2})$, Math. Z., 188 (1984), pp. 45-54. MR 86a:20008
  • 7. J. A. GREEN, The characters of the finite general linear groups, Trans. Amer. Math. Soc., 80 (1955), pp. 402-447. MR 17:345e
  • 8. I. GROJNOWSKI, Character Sheaves on Symmetric Spaces, PhD thesis, Massachusetts Institute of Technology, 1992.
  • 9. A. HENDERSON, Spherical functions and character sheaves. Available at:
  • 10. R. HOTTA AND T. A. SPRINGER, A specialization theorem for certain Weyl group representations and an application to the Green polynomials of unitary groups, Invent. Math., 41 (1977), pp. 113-127. MR 58:5945
  • 11. G. LAUMON, Faisceaux caractères (d'après Lusztig), Astérisque, 177-178 (1989), pp. 231-260. MR 91m:20062
  • 12. G. LUSZTIG, Characters of Reductive Groups over a Finite Field, vol. 107 of Ann. of Math. Studies, Princeton University Press, 1984. MR 86j:20038
  • 13. -, Character sheaves, I, Adv. Math., 56 (1985), pp. 193-237 MR 87b:20055; II, Adv. Math., 57 (1985), pp. 226-265 MR 87m:20118a; III, Adv. Math., 57 (1985), pp. 266-315 MR 87m:20118a; IV, Adv. Math., 59 (1986), pp. 1-63 MR 87m:20118b; V, Adv. Math., 61 (1986), pp. 103-155. MR 87m:20118c; Erratum, MR 87m:20118d
  • 14. -, Green functions and character sheaves, Ann. of Math., 131 (1990), pp. 355-408. MR 91c:20054
  • 15. -, $G(\mathbb{F} _q)$-invariants in irreducible $G(\mathbb{F} _{q^{2}})$-modules, Represent. Theory, 4 (2000), pp. 446-465. MR 2001j:20067
  • 16. I. G. MACDONALD, Symmetric Functions and Hall Polynomials, Oxford Univ. Press, second ed., 1995. MR 96h:05207
  • 17. J. G. M. MARS AND T. A. SPRINGER, Character sheaves, Astérisque, 173-174 (1989), pp. 111-198. MR 91a:20044
  • 18. T. SHOJI, Character sheaves and almost characters of reductive groups, Adv. Math., 111 (1995), pp. 244-354. MR 95k:20069
  • 19. B. SRINIVASAN, On Macdonald's symmetric functions, Bull. Lond. Math. Soc., 24 (1992), pp. 519-525. MR 94e:05264

Similar Articles

Retrieve articles in Representation Theory of the American Mathematical Society with MSC (2000): 20G40, 20G05, 20C15, 32C38

Retrieve articles in all journals with MSC (2000): 20G40, 20G05, 20C15, 32C38

Additional Information

Anthony Henderson
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Address at time of publication: School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia

Keywords: Algebraic groups, perverse sheaves
Received by editor(s): December 1, 2000
Received by editor(s) in revised form: August 14, 2001
Published electronically: November 28, 2001
Article copyright: © Copyright 2001 American Mathematical Society

American Mathematical Society