An inclusion between sets of orbits and surjectivity of the restriction map of rings of invariants
Let \(V\) be a finite-dimensional complex vector space and \(\sigma:\text{End}(V)\to \text{End}(V)\) be a \({\mathbb C}\)-linear anti-automorphism. Let \(\tilde G\) be a reductive subgroup of \(GL(V)\) acting on a subvector space \(\tilde L\) of \(\text{End}(V)\) by the adjoint action. Suppose that \(\tilde G\) is invariant under \(\sigma\) , and that the restriction of \(\sigma\) to \(\tilde G\) is an involution. Consider a closed subgroup \(G\) of \(\tilde G\) acting on a subspace \(L\) of \(\tilde L\). In this article, a sufficient condition on the inclusion \((G,L)\hookrightarrow (\tilde G, \tilde L)\) is given for which the orbit correspondence \(L/G\mapsto \tilde L/\tilde G\) (\({\mathcal O}\mapsto Ad(\tilde G)\cdot{\mathcal O}\)) is injective. Moreover, under these conditions, the ring \({\mathbb C}[L]^G\) of \(G\)-invariants on \(L\) is the integral closure of \({\mathbb C}[\tilde L]^{\tilde G}|_L\) in its quotient field. In particular, if \({\mathbb C}[\tilde L]^{\tilde G}|_L\) is normal, then the restriction map on the invariants is surjective. Examples are given for which the conditions are satisfied. It is shown, for example, that one can deduce the First Fundamental Theorem of Invariant Theory for the orthogonal and symplectic groups from that of the general linear group.
first fundamental theorem of invariant theory
inclusion theorems between sets of orbits
scientific article; zbMATH DE number 5354266