Quasisplit Hecke algebras and symmetric spaces

TL;DR

This paper explores the structure of quasisplit Hecke algebras associated with symmetric pairs and their role in understanding the fixed points of local systems on flag manifolds under automorphisms.

Contribution

It introduces a novel module structure over quasisplit Hecke algebras to compute involution eigenspaces related to symmetric pairs and local systems.

Findings

Dimension formulas for involution eigenspaces are derived.

The module structure over quasisplit Hecke algebras is explicitly constructed.

Applications to the representation theory of symmetric spaces are discussed.

Abstract

Let (G,K) be a symmetric pair over an algebraically closed field of characteristic different of 2 and let sigma 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 sigma-stable K-orbit on the flag manifold of G and L is an irreducible K-equivariant local system on O which is "fixed" by sigma. Given two such pairs (O,L), (O',L'), with O' in the closure \bar O of O, the multiplicity space of L' in the a cohomology sheaf of the intersection cohomology of \bar O with coefficients in L (restricted to O') carries an involution induced by sigma 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.