Let be a symmetric pair over an algebraically closed field of characteristic different from , and let be an automorphism with square of preserving . In this paper we consider the set of pairs where is a -stable -orbit on the flag manifold of and is an irreducible -equivariant local system on which is “fixed” by . Given two such pairs , , with in the closure of , the multiplicity space of in a cohomology sheaf of the intersection cohomology of with coefficients in (restricted to ) carries an involution induced by , and we are interested in computing the dimensions of its and 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 as above.
George Lusztig. David A. Vogan Jr.. "Quasisplit Hecke algebras and symmetric spaces." Duke Math. J. 163 (5) 983 - 1034, 1 April 2014. https://doi.org/10.1215/00127094-2644684