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 gk((n)), i.e. fixed point varieties on affine flag manifolds. We define a natural class of fc-actions on affine flag manifolds, generalizing actions introduced by Lusztig and Smelt. We formulate a condition on a pair (N, /) consisting of N 6 0 k((tr)) and a fc-action / of the specified type which guarantees that / 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 fc-fixcd 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.