Symmetric Spaces Associated to Classical Groups with Even Characteristic

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 described in [SY], assuming that $ch k\ne 2$. The definition of $\theta$ given there, and of the symmetric space arising from $\theta$, make sense even if $ch k=2$. In this paper, we discuss the Springer correspondence for those symmetric spaces with even characteristic. We show, if $N$ is even, that 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, the number of $G^{\theta }$-orbits in the unipotent variety is infinite, and a very similar phenomenon occurs as in the case of exotic symmetric space of higher level, namely of level $r=3$.