Quasisplit Hecke algebras and symmetric spaces. (English) Zbl 1300.20006

Lusztig, George; Vogan, David A. jun.

Duke Math. J. 163, No. 5, 983-1034 (2014).

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 the authors consider the set of pairs \((\mathcal{O,L})\) 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 \((\mathcal{O,L})\), \((\mathcal{O',L'})\), with \(\mathcal O'\) in the closure 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 the authors are interested in computing the dimensions of its \(+1\) and \(-1\) eigenspaces. They 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 \((\mathcal{O,L})\) as above.

Reviewer: Hu Jun (Beijing)

MSC:

20C08	Hecke algebras and their representations
20G40	Linear algebraic groups over finite fields
14M15	Grassmannians, Schubert varieties, flag manifolds
14L30	Group actions on varieties or schemes (quotients)
20G10	Cohomology theory for linear algebraic groups

Keywords:

symmetric spaces; quasisplit Hecke algebras; \(K\)-equivariant local systems; intersection cohomology

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