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 θ the fixed point subgroup of G under an involution θ on G. In the case where G θ = O(V ), the generalized Springer correpsondence for the unipotent variety of the symmetric space G/G θ 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 G θ -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.