Quasisplit Hecke algebras and symmetric spaces
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.
Keywords
Cite
@article{arxiv.1206.0634,
title = {Quasisplit Hecke algebras and symmetric spaces},
author = {G. Lusztig and D. A. Vogan},
journal= {arXiv preprint arXiv:1206.0634},
year = {2014}
}
Comments
46 pages. Version 2 reorganizes the explicit calculation of the Hecke module, includes details about computing \bar, and corrects small misprints. Version 3 adds two pages relating this paper to unitary representation theory, corrects misprints, and displays more equations. Version 4 corrects misprints, and adds two cases previously neglected at the end of 7.5