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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
Published electronically: May 3, 1999
MathSciNet review: 1491876
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Abstract: Let $G$ be a semisimple, simply connected, algebraic group over an algebraically closed field $k$ with Lie algebra $\mathfrak{g}$. We study the spaces of parahoric subalgebras of a given type containing a fixed nil-elliptic element of $\mathfrak{g}\otimes k((\pi))$, i.e. fixed point varieties on affine flag manifolds. We define a natural class of $k^*$-actions on affine flag manifolds, generalizing actions introduced by Lusztig and Smelt. We formulate a condition on a pair $(N,f)$ consisting of $N\in \mathfrak{g}\otimes k((\pi))$ and a $k^*$-action $f$ of the specified type which guarantees that $f$ induces an action on the variety of parahoric subalgebras containing $N$.

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 $k^*$-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 $\mathfrak{g}$.


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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: http://dx.doi.org/10.1090/S0002-9947-99-02295-3
PII: S 0002-9947(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