The geometry of fixed point varieties

on affine flag manifolds

Author:
Daniel S. Sage

Journal:
Trans. Amer. Math. Soc. **352** (2000), 2087-2119

MSC (1991):
Primary 14L30, 20G25

DOI:
https://doi.org/10.1090/S0002-9947-99-02295-3

Published electronically:
May 3, 1999

MathSciNet review:
1491876

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a semisimple, simply connected, algebraic group over an algebraically closed field with Lie algebra . We study the spaces of parahoric subalgebras of a given type containing a fixed nil-elliptic element of , i.e. fixed point varieties on affine flag manifolds. We define a natural class of -actions on affine flag manifolds, generalizing actions introduced by Lusztig and Smelt. We formulate a condition on a pair consisting of and a -action of the specified type which guarantees that induces an action on the variety of parahoric subalgebras containing .

For the special linear and symplectic groups, we characterize all regular semisimple and nil-elliptic conjugacy classes containing a representative whose fixed point variety admits such an action. We then use these actions to find simple formulas for the Euler characteristics of those varieties for which the -fixed points are finite. We also obtain a combinatorial description of the Euler characteristics of the spaces of parabolic subalgebras containing a given element of certain nilpotent conjugacy classes of .

**[B-B1]**A. Bialynicki-Birula,*On fixed point schemes of actions of multiplicative and additive groups*, Topology**12**(1973), 99-103. MR**47:1816****[B-B2]**A. Bialynicki-Birula,*Some theorems on actions of algebraic groups*, Annals of Math.**98**(1973), 480-497. MR**51:3186****[B-B3]**A. Bialynicki-Birula,*Some properties of the decompositions of algebraic varieties determined by actions of a torus*, Bulletin de L'Académie Polonaise des sciences**24**(1976), 667-674. MR**56:12020****[Bo]**A. Borel,*Linear algebraic groups*, Graduate Texts in Math.**126**, Springer-Verlag, Berlin, 1991. MR**92d:20001****[B1]**N. Bourbaki,*Algèbre*, Chap. IX, Hermann, Paris, 1959.**[B2]**N. Bourbaki,*Algebra II, Chapters 4-7*, Springer-Verlag, Berlin, 1990. MR**91h:00003****[KL]**D. Kazhdan and G. Lusztig,*Fixed point varieties on affine flag manifolds*, Israel J. Math.**62**(1988), 129-168. MR**89m:14025****[LS]**G. Lusztig and J. M. Smelt,*Fixed point varieties on the space of lattices*, Bull. London Math. Soc.**23**(1991), 213-218. MR**93e:14065****[R]**J. Riordan,*Combinatorial identities*, Robert E. Krieger Publishing Company, Huntington, New York, 1979. MR**80k:05001****[S]**D. S. Sage,*The geometry of fixed point varieties on affine flag manifolds*, Ph.D. thesis, University of Chicago, 1995.**[S1]**J. -P. Serre,*Corps locaux*, Hermann, Paris, 1962. MR**27:133****[S2]**J. -P. Serre,*Cohomologie galoisienne*, Lecture Notes in Math.**5**, Springer-Verlag, Berlin, fourth ed., 1973.**[Sp]**N. Spaltenstein,*Polynomials over local fields, nilpotent orbits and conjugacy classes in Weyl groups*, Astérisque**168**(1988), 191-217. MR**90k:20069****[SS]**T. A. Springer and R. Steinberg,*Conjugacy classes*, Seminar on Algebraic Groups and Related Finite Groups, Lecture Notes in Math.**131**, Springer-Verlag, Berlin, 1970, pp. 167-266. MR**42:3091****[St]**R. Steinberg,*Regular elements of semisimple algebraic groups*, Inst. Hautes Études Sci. Publ. Math.**25**(1965), 49-80. MR**31:4788**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
14L30,
20G25

Retrieve articles in all journals with MSC (1991): 14L30, 20G25

Additional Information

**Daniel S. Sage**

Affiliation:
Department of Mathematics, University of Utah, Salt Lake City, Utah 84112

Address at time of publication:
School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540

Email:
sage@ias.edu

DOI:
https://doi.org/10.1090/S0002-9947-99-02295-3

Keywords:
Fixed point varieties on affine flag manifolds,
Iwahori subalgebras,
parahoric subalgebras,
lattices

Received by editor(s):
November 1, 1997

Published electronically:
May 3, 1999

Article copyright:
© Copyright 2000
American Mathematical Society