Irreducible Components of Fixed Point Subvarieties of Flag Varieties

✍ Scribed by J. Matthew Douglass

• Publisher: John Wiley and Sons
• Year: 1998
• Tongue: English
• Weight: 804 KB
• Volume: 189
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19981890107

✦ Synopsis

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 C B 17 L, B C P and e E Lie B. Let 8 be the variety of Borel subgroups of G and let Be 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 w 6 W let 0, be the B-orbit in Q containing ,B. We consider the intersections 0, n Be. The main result is that if dim 0, n ' 23, = dim Be, then 0, n ' 23, is an affine space. Thus, the irreducible components of 8, 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.