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 Free Access

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 .

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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