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