Generalized Springer correspondence for symmetric spaces associated to orthogonal groups

TL;DR

This paper extends the Springer correspondence to symmetric spaces associated with orthogonal groups over fields of odd characteristic, linking unipotent orbits to symmetric group representations.

Contribution

It establishes a generalized Springer correspondence for symmetric spaces related to orthogonal groups, connecting unipotent orbits with symmetric group representations.

Findings

Correspondence between H-orbits in unipotent symmetric space elements and symmetric group representations

Extension of Springer theory to symmetric spaces of orthogonal groups

Framework for analyzing unipotent elements in symmetric spaces

Abstract

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