Symmetric Spaces Associated to Classical Groups with Even Characteristic

Let G=GL(V) for an N-dimensional vector space V over algebraically closed field k, and G? the fixed point subgroup of G under involution ? on G. In case where G?=O(V), generalized Springer correspondence unipotent variety symmetric G/G? was described in [SY], assuming that ch k?2. The definition given there, arising from ?, make sense even if k=2. this paper, we discuss those spaces with characteristic. We show, N is even, reduced to symplectic Lie algebras k=2, which determined by Xue. While odd, number G?-orbits infinite, a very similar phenomenon occurs as exotic higher level, namely level r=3.