The Geometry of Fixed Point Varieties on Affine Flag Manifolds

Let G be a semisimple, simply connected, algebraic group over an algebraically closed field k with Lie algebra g. We study the spaces of parahoric subalgebras of a given type containing a fixed nil-elliptic element of g⊗k((π)), 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 ∈ g⊗ k((π)) 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 g.