Real canonical forms for the adjoint action of the Lie groups \({\mathrm{Sp}}(V,B)\) and \({\mathrm{O}}(V,B)\) on their Lie algebras (Q2419440)

Let \(G(V)\) denote either the real symplectic group \(\mathrm{Sp}(V)\) or the real orthogonal group \(\mathrm{O}(V)\) for a finite-dimensional real vector space \(V\) (with appropriate bilinear form). Let \(\mathfrak{g}(V)\) denote the associated Lie algebra \(\mathfrak{sp}(V)\) or \(\mathfrak{o}(V)\). A classical (and well-studied) problem is to understand the orbits in \({ g}(V)\) under the adjoint action of \(G(V)\). In this paper, the authors present a new approach to this problem from an elementary linear algebra perspective. Indeed, only a basic knowledge of linear algebra is required in the arguments. From a matrix perspective, the problem of classifying orbits is equivalent to identifying the real canonical forms for a matrix \(X \in \mathfrak{g}(V)\) under (symplectic) similarity. The main result is a procedure for identifying the canonical form associated to a given \(X\). Given \(X\), one computes its eigenvalues. This gives rise to a decomposition of \(V\) into a direct sum \(\bigoplus V_{\lambda}\) of cyclic subspaces \(V_{\lambda}\) associated to an eigenvalue \(\lambda\). The authors observe that for a non-zero eigenvalue \(\lambda\), \(-\lambda\) is also an eigenvalue, and, moreover, that the subspace \(V_{\lambda}\oplus V_{-\lambda}\) has the same (symplectic or orthogonal) geometry as \(V\). If \(X\) admits a zero eigenvalue, it is shown that the associated cyclic subspace \(V_0\) further decomposes into a set of singletons or pairs having the same geometry as \(V\). The authors describe the canonical form of the restriction of \(X\) to these various subspaces (e.g., on a subspace of the form \(V_{\lambda}\oplus V_{-\lambda}\)) and then show how these can be combined to get the full canonical form for \(X\). As an application, the authors explicitly identify all the possible canonical forms for \(\mathrm{Sp}(2,{\mathbb R})\) and \(\mathrm{Sp}(4,{\mathbb R})\), as well as the associated (nonequivalent) quadratic Hamiltonian forms on \({\mathbb R}^2\) and \({\mathbb R}^4\).