Centralizers of elements in the orthogonal, symplectic and unitary groups

Abstract

We explicitly compute and parametrize the centralizers of elements, encompassing dege\-ne\-rate and unipotent ones, in the orthogonal, symplectic, and unitary gro\-ups over the real and complex fields, including the indefinite cases. The same approach simultaneously yields the isotropy groups of $H$-skew-symme\-tric, Hamiltonian, and $H$-Hermitian matrices under the adjoint representations of the corresponding classical groups. The resulting groups are conjugate to subgroups of nonsingular block matrices whose blocks exhibit a rectangular block Toeplitz structure.