Quasisplit Hecke algebras and symmetric spaces

Abstract

Let $(G,K)$ be a symmetric pair over an algebraically closed field of characteristic different from $2$, and let $\sigma$ be an automorphism with square $1$ of $G$ preserving $K$. In this paper we consider the set of pairs $(\mathcal{O},\mathcal{L})$ where $\mathcal{O}$ is a $\sigma$-stable $K$-orbit on the flag manifold of $G$ and $\mathcal{L}$ is an irreducible $K$-equivariant local system on $\mathcal{O}$ which is “fixed” by $\sigma$. Given two such pairs $(\mathcal{O},\mathcal{L})$, $(\mathcal{O}\text{'},\mathcal{L}\text{'})$, with $\mathcal{O}\text{'}$ in the closure $\overline{\mathcal{O}}$ of $\mathcal{O}$, the multiplicity space of $\mathcal{L}\text{'}$ in a cohomology sheaf of the intersection cohomology of $\overline{\mathcal{O}}$ with coefficients in $\mathcal{L}$ (restricted to $\mathcal{O}\text{'}$) 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 $(\mathcal{O},\mathcal{L})$ as above.