Generalized Springer Correspondence for Symmetric Spaces Associated to Orthogonal Groups

Abstract

Let $G = GL_N(\mathbf{k})$, where $\mathbf{k}$ is an algebraically closed field of $\operatorname{ch} \mathbf{k} \ne 2$, and $\theta$ an involutive automorphism of $G$ such that $H = (G^{\theta})^0$ is isomorphic to $SO_N(\mathbf{k})$. Then $G^{\iota\theta} = \{ g \in G \mid \theta(g) = g^{-1}\}$ is regarded as a symmetric space $G/G^{\theta}$. Let $G^{\iota\theta}_{\operatorname{uni}}$ be the set of unipotent elements in $G^{\iota\theta}$. $H$ acts on $G^{\iota\theta}_{\operatorname{uni}}$ 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 $G^{\iota\theta}_{\operatorname{uni}}$ and irreducible representations of various symmetric groups.