Singularities of closures of K-orbits on flag manifolds
type: Item
instance of: scholarly article
title: Singularities of closures of K-orbits on flag manifolds
zbMATH Open document ID: 0544.14035
DOI: 10.1007/BF01389103
author: G. Lusztig
author: David A. jun. Vogan
published in: Inventiones Mathematicae
publication date: 1983-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/142995
review text: Let G be a connected reductive group over the field \({\bar {\mathbb{F}}}_ q\) and \(\theta\) an order two automorphism of G. If K is a subgroup of finite index in the group of elements fixed by \(\theta\), one knows that the action of K on the flag variety \({\mathcal B}\) has finitely many orbits. The authors consider the finite set \({\mathcal D}\) of pairs (\({\mathcal O},\gamma) {\mathcal O}\) a K-orbit in \({\mathcal B}\), \(\gamma\) an isomorphism class of K equivariant sheaves of one dimensional \({\mathbb{Q}}_{\ell}\) vector spaces on \({\mathcal O}\) (\(\ell\) = a prime). Since \(\gamma\) determines \({\mathcal O}\) the authors write \(\gamma\) for (\({\mathcal O},\gamma)\) and set \(\ell(\gamma)=\dim {\mathcal O}.\) If \(\gamma \in {\mathcal D}\) they denote by \({\tilde \gamma}\) the Deligne-Goresky-MacPherson extension of \(\gamma\) to \({\mathcal O}\) and by \({\tilde \gamma}^ i\) its i-th cohomology sheaf. - The main result states that if for \(\gamma,\delta \in {\mathcal D}\), [\(\gamma\),\({\tilde \delta}{}^ i]\) denotes the multiplicity of \(\gamma\) in the Jordan- Hölder series for \({\tilde \delta}^ i\), one has that \({\tilde \delta}^ i=0\) for i odd, and the polynomials \(P_{\gamma,\delta}(u)=\sum [\gamma,{\tilde \delta}^{2i}]u^ i\) are uniquely defined in terms of an action of the Hecke algebra of the Weyl group on the free \({\mathbb{Z}}[u,u^{-1}]\) module with basis \({\mathcal D}.\) The results of the paper are strongly inspired by the analogous results for Schubert varieties [\textit{D. Kazhdan} and \textit{G. Lusztig}, Proc. Symp. pure Math. 36, 185-203 (1980; Zbl 0461.14015)], which are in fact equivalent to the special case of the above in which \(G\) is diagonally embedded in \(G\times G\), but the proofs are substantially more complicated using both the purity theorem of Gabber for intersection cohomology [\textit{P. Deligne}, ''Pureté de la cohomologie de MacPherson- Goresky'', d'après un exposé de \textit{O. Gabber}, Inst. Haut. Étu…
Mathematics Subject Classification ID: 14M15
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 20G10
zbMATH DE Number: 3865490
zbMATH Keywords: action of group on flag variety
Wikidata QID: Q56387047
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3865490