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 Gθ the fixed point subgroup of G under an involution θ on G. In the case where Gθ=O(V), the generalized Springer correspondence for the unipotent variety of the symmetric space G/Gθ was described in [SY], assuming that ch k≠2. The definition of θ given there, and of the symmetric space arising from θ, 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 Gθ-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.