Remote Access Representation Theory
Green Open Access

Representation Theory

ISSN 1088-4165



Hecke algebra representations related to spherical varieties

Authors: J. G. M. Mars and T. A. Springer
Journal: Represent. Theory 2 (1998), 33-69
MSC (1991): Primary 14M15, 55N33
Published electronically: February 11, 1998
MathSciNet review: 1600804
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $G$ be a connected reductive group over the algebraic closure of a finite field and let $Y$ be a spherical variety for $G$. We consider perverse sheaves on $G$ and on $Y$ which have a weight for the action of a Borel subgroup $B$ and are endowed with an action of Frobenius. This leads to the definition of a ``generalized Hecke algebra'', attached to $G$, and of a module over that algebra, attached to $Y$. The same algebra and the same module can also be defined using constructible sheaves. Comparison of the two definitions gives, in the case of a symmetric variety $Y$ and $B$-equivariant sheaves, a geometric proof of results which Lusztig and Vogan obtained by representation theoretic means.

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

  • [B] N. Bourbaki, Groupes et algèbres de Lie, Chap. 4,5,6. Hermann, Paris, 1968. MR 39:1590
  • [BBD] A. A. Beilinson, J. Bernstein, P. Deligne, Faisceaux pervers. Astérisque, vol. 100, Math. France Soc., 1982. MR 86g:32015
  • [K] F. Knop, On the set of orbits for a Borel subgroup. Comment. Math. Helv. 70 (1995), 285-309. MR 96c:14039
  • [LV] G. Lusztig and D. A. Vogan Jr., Singularities of closures of $K$-orbits on flag manifolds. Invent. Math. 71 (1983), 365-379. MR 84h:14060
  • [MS] J. G. M. Mars and T. A. Springer, Character sheaves. Orbites unipotentes et représentations III, Astérisque 173-174 (1989), pp. 33-69. MR 91a:20044
  • [RS1] R. W. Richardson and T. A. Springer, The Bruhat order on symmetric varieties. Geom. Dedicata. 35 (1990), 389-436. MR 92e:20032
  • [RS2] R. W. Richardson and T. A. Springer, Combinatorics and geometry of $K$-orbits on the flag manifold. Linear algebraic groups and their representations (R. Elman, M. Schacher, V. Varadarajan, eds.) Contemporary Math. 153, Amer. Math. Soc., Providence, 1993, pp. 109-142. MR 94m:14065
  • [S1] T. A. Springer, Linear algebraic groups. Birkhäuser, Boston, 1981 (2nd printing 1983). MR 84i:20002
  • [S2] T. A. Springer, Some results on algebraic groups with involutions. Adv. Stud. Pure Math., vol. 6, North-Holland, 1985, pp. 525-543. MR 87j:20073
  • [S3] T. A. Springer, The classification of involutions of simple algebraic groups. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34 (1987), 655-670. MR 89b:20096
  • [SGA4-1/2] P. Deligne, Séminaire de Géométrie Algébrique du Bois-Marie SGA $4\frac{1}{2}$. Lecture Notes in Math. 569, Springer 1977. MR 57:3132
  • [SGA7] P. Deligne and N. Katz, Séminaire de Géométrie Algébrique du Bois-Marie SGA7 II. Lecture Notes in Math. 340, Springer, 1973. MR 50:7135
  • [V] D. A. Vogan Jr., Irreducible characters of semisimple Lie groups III, Proof of Kazhdan-Lusztig conjecture in the integral case. Invent. Math. 71 (1983), 381-417. MR 84h:22036

Similar Articles

Retrieve articles in Representation Theory of the American Mathematical Society with MSC (1991): 14M15, 55N33

Retrieve articles in all journals with MSC (1991): 14M15, 55N33

Additional Information

J. G. M. Mars
Affiliation: Mathematisch Instituut, Universiteit Utrecht, Budapestlaan 6, 3508 TA Utrecht, Netherlands

T. A. Springer
Affiliation: Mathematisch Instituut, Universiteit Utrecht, Budapestlaan 6, 3508 TA Utrecht, Netherlands

Received by editor(s): April 17, 1997
Received by editor(s) in revised form: November 19, 1997
Published electronically: February 11, 1998
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society