Symmetric spaces associated to classical groups with even characteristic

TL;DR

This paper explores the Springer correspondence for symmetric spaces associated with classical groups over fields of even characteristic, revealing reductions to symplectic cases and connections to exotic symmetric spaces.

Contribution

It extends the Springer correspondence analysis to even characteristic fields, linking it to symplectic Lie algebras and exotic symmetric spaces, which was not previously established.

Findings

Springer correspondence reduces to symplectic Lie algebra cases when N is even.

For odd N, the phenomenon resembles that of exotic symmetric spaces of level 3.

The definitions and results apply even in characteristic 2 fields.

Abstract

Let $G = GL(V)$ for an N-dimensional vector space $V$ over an algebraically closed field k, and $G^{\theta}$ the fixed point subgroup of $G$ under an involution $\theta$ on $G$. In the case where $G^{\theta} = O(V)$, the generalized Springer correspondence for the unipotent variety of the symmetric space $G/G^{\theta}$ was studied by last two authors, under the assumption that ch k is odd. The definition of $\theta$, and of the associated symmetric space given there make sense even if ch k = 2. In this paper, we discuss the Springer correspondence for those symmetric spaces of even characteristic. We show that if N is even, the Springer correspondence is reduced to that of symplectic Lie algebras in ch k = 2, which was determined by Xue. While if N is odd, we show that a very similar phenomenon as in the case of exotic symmetric space of level 3 appears.