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. ©2000 American Mathematical Society.
Publication Source (Journal or Book title)
Transactions of the American Mathematical Society
Sage, D. (2000). The geometry of fixed point varieties on affine flag manifolds. Transactions of the American Mathematical Society, 352 (5), 2087-2119. https://doi.org/10.1090/s0002-9947-99-02295-3