The Geometry of Fixed Point Varieties on Affine Flag Manifolds

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

Latex2e, 33 pages. To appear in Transactions of the AMS