Quasisplit Hecke algebras and symmetric spaces

• Main Authors: Lusztig, George (Contributor), Vogan, David A. (Contributor)
• Other Authors: Massachusetts Institute of Technology. Department of Mathematics (Contributor)
• Format: Article
• Language: English
• Published: Duke University Press, 2015-01-20T18:31:37Z.
• Subjects: Article
• Online Access: Get fulltext

 Let (G,K) be a symmetric pair over an algebraically closed field of characteristic different from 2, and let σ be an automorphism with square 1 of G preserving K. In this paper we consider the set of pairs (O,L) where O is a σ-stable K-orbit on the flag manifold of G and L is an irreducible K-equivariant local system on O which is "fixed" by σ. Given two such pairs (O,L), (O',L'), with O' in the closure [bar over O] of O, the multiplicity space of L' in a cohomology sheaf of the intersection cohomology of [bar over O] with coefficients in L (restricted to O') carries an involution induced by σ, and we are interested in computing the dimensions of its +1 and −1 eigenspaces. We show that this computation can be done in terms of a certain module structure over a quasisplit Hecke algebra on a space spanned by the pairs (O,L) as above.