## Duke Mathematical Journal

### 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 $(\mathcal{O},\mathcal{L})$, $(\mathcal{O}',\mathcal{L}')$, with $\mathcal{O}'$ in the closure $\overline{\mathcal{O}}$ of $\mathcal{O}$, the multiplicity space of $\mathcal{L}'$ in a cohomology sheaf of the intersection cohomology of $\overline{\mathcal{O}}$ with coefficients in $\mathcal{L}$ (restricted to $\mathcal{O}'$) 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 $(\mathcal{O},\mathcal{L})$ as above.

#### Article information

Source
Duke Math. J., Volume 163, Number 5 (2014), 983-1034.

Dates
First available in Project Euclid: 26 March 2014

https://projecteuclid.org/euclid.dmj/1395856221

Digital Object Identifier
doi:10.1215/00127094-2644684

Mathematical Reviews number (MathSciNet)
MR3189436

Zentralblatt MATH identifier
1300.20006

Subjects
Primary: 20G40: Linear algebraic groups over finite fields
Secondary: 20C08: Hecke algebras and their representations

#### Citation

Lusztig, George; Vogan Jr., David A. Quasisplit Hecke algebras and symmetric spaces. Duke Math. J. 163 (2014), no. 5, 983--1034. doi:10.1215/00127094-2644684. https://projecteuclid.org/euclid.dmj/1395856221

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