Generalized Springer correspondence for symmetric spaces associated to orthogonal groups

Abstract

Let $G=G{L}_N$ over an algebraically closed field of odd characteristic, and $\theta$ an involutive automorphism on $G$ such that $H=(G^{\theta }{)}^0$ is isomorphic to $S{O}_N$. Then $G^{\iota \theta }=\{g\in G\mid \theta (g)=g^{-1}\}$ is regarded as a symmetric space $G/G^{\theta }$. Let $G_{uni}^{\iota \theta }$ be the set of unipotent elements in $G^{\iota \theta }$. $H$ acts on $G_{uni}^{\iota \theta }$ by the conjugation. As an analogue of the generalized Springer correspondence in the case of reductive groups, we establish in this paper the generalized Springer correspondence between $H$-orbits in $G_{uni}^{\iota \theta }$ and irreducible representations of various symmetric groups.

76 pages. Final version, to appear in Tokyo J. of Mathematics