Irreducible Components of Fixed Point Subvarieties of Flag Varieties

Suppose G is a connected reductive algebraic group, P is a parabolic subgroup of G, L is a Levi factor of P , and e is a regular nilpotent element in Lie L. We assume that the characteristic of the underlying field is good for G. Choose a maximal torus, T , and a Borel subgroup, B , of G, so that T⊆ B∩L , B ⊆ P and e ∈ Lie B. Let β be the variety of Borel subgroups of G and let e be the subset of consisting of Borel subgroups whose Lie algebras contain e. Finally, let W be the Weyl group of G with respect to T. For ω ∈ W let ω be the B‐orbit in containing ω B. We consider the intersections ω ∩ e . The main result is that if dim ω ∩ e = dim e , then ω ∩ e is an affine space. Thus, the irreducible components of e are indexed by Weyl group elements. It is also shown that if G is of type A , then this set of Weyl group elements is a right cell in W.