给定一个代数群，以及作用在它上面的一个对合自同构，即可定义该代数群关于此对合自同构的不变子群及对称空间。不变子群自然地作用在对称空间上。我们之前的工作，将Lusztig的一般Springer对应理论推广到正交群对应的对称空间，即一般线性群上，不变子群为正交群的对合自同构定义的对称空间。接下来，该项目一方面将继续研究正交群对应的对称空间的相关理论，包括Spaltenstein簇的结构的确定，以及Spaltenstein簇的Lusztig多项式的研究。另一方面，将考虑特征为2的代数闭域上的对称空间，特别是，对称空间中包含无限多个幂幺的不变子群轨道时的Springer对应与一般Springer对应。幂幺轨道个数有限性的消失，将使得与不变子群等变的perverse层，以及Springer对应理论相关的讨论具有更大的难度和研究价值。

For a given algebraic group and an involutive automorphism on it, we can define the invariant subgroup and the symmetric space of the algebraic group with respect to this involutive automorphism. The invariant subgroup acts on the symmetric space naturally. Previously, we apply Lusztig's theory of generalized Springer correspondence to the case of symmetric space associated with orthogonal groups. The present project will continue with the research on the symmetric space associated with orthogonal groups. We will try to determine the structure of the Spaltenstein varieties in this case, and calculate Lusztig's polynomials associated with Spaltenstein varieties. Also, we will consider the symmetric spaces over algebraically closed fields of characteristic 2, in particular, the symmetric spaces in which there are infinitely many unipotent orbits under the action of the invariant subgroup. As the finiteness of the unipotent orbits disappears, the discussion on the invariant group-equivariant perverse sheaves and also on the Springer correspondence will become much more complicated.