Closures of conjugacy classes in classical real linear Lie groups. II

Abstract:

By a classical group we mean one of the groups $G{L_n}(R)$, $G{L_n}(C)$, $G{L_n}(H)$, $U(p, q)$, ${O_n}(C)$, $O(p, q)$, $S{O^{\ast }}(2n)$, $S{p_{2n}}(C)$, $S{p_{2n}}(R)$, or $Sp(p, q)$. Let $G$ be a classical group and $L$ its Lie algebra. For each $x \in L$ we determine the closure of the orbit $G \cdot x$ (for the adjoint action of $G$ on $L$). The problem is first reduced to the case when $x$ is nilpotent. By using the exponential map we also determine the closures of conjugacy classes of $G$.