Duke Mathematical Journal

Quasisplit Hecke algebras and symmetric spaces

George Lusztig and David A. Vogan Jr.

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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 O¯ of O, the multiplicity space of L' in a cohomology sheaf of the intersection cohomology of 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.

Article information

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

First available in Project Euclid: 26 March 2014

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


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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