An inclusion between sets of orbits and surjectivity of the restriction map of rings of invariants

Abstract

Let $V$ be a finite dimensional vector space over the complex number field $\mathbb{C}$. Suppose that, by the adjoint action, a reductive subgroup $\tilde G$ of $GL(V)$ acts on a subspace $\tilde L$ of End$(V)$ and a closed subgroup $G$ of $\tilde G$ acts on a subspace $L$ of $\tilde L$. In this paper, we give a sufficient condition on the inclusion $(G,L) \hookrightarrow (\tilde G, \tilde L)$ for which the orbits correspondence $L/G \to \tilde L/ \tilde G (O \mapsto \tilde O := $Ad$(\tilde G) \dot O)$ is injective. Moreover we show that 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. Then, if the ring $\mathbb{C}[\tilde L]^{\tilde G}|_L$ is normal, the restriction map rest : $\mathbb{C}[\tilde L]^{\tilde G} \to \mathbb{C}[L]^G (f \mapsto f|_L)$ is surjective. By using this, we give some examples for which $L/G \to \tilde L/ \tilde G$ is injective and rest $: \mathbb{C}[\tilde L]^{\tilde G} \to \mathbb{C}[L]^G$ is surjective.