Singularities of closures of K-orbits on flag manifolds
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. Étud. Sci. (1981)] and the representation theoretic interpretation of \([\gamma,{\tilde \delta}^ i]\) due to \textit{A. Beilinson} and \textit{J. Bernstein} [C. R. Acad. Sci., Paris, Sér. I 292, 15-18 (1981; Zbl 0476.14019)]. The part of the proof using representation theory is only sketched and the reader is referred to \textit{D. A. Vogan} [Invent. Math. 71, 381-417 (1983; Zbl 0505.22016)], where an algorithm for the computation of the \(P_{\gamma,\delta}(u)\) is also given together with representation theoretical applications of the above results.
scientific article; zbMATH DE number 3865490