Semigroups in 3-graded Lie groups and endomorphisms of standard subspaces

Abstract

Let $\mathtt{V}$ be a standard subspace in the complex Hilbert space $\mathcal{H}$, and let $U:G\to \mathrm{U}(\mathcal{H})$ be a unitary representation of a finite-dimensional Lie group. We assume the existence of an element $h\in \mathfrak{g}$ such that $U(expth)={\mathrm{\Delta }}_{\mathtt{V}}^{it}$ is the modular group of $\mathtt{V}$ and that the modular involution $J_{\mathtt{V}}$ normalizes $U(G)$. We want to determine the semigroup $S_{\mathtt{V}}=\{g\in G:U(g)\mathtt{V}\subseteq \mathtt{V}\}$. In previous work, we have seen that its infinitesimal generators span a Lie algebra on which $\mathrm{ad}h$ defines a 3-grading, and here we completely determine the semigroup $S_{\mathtt{V}}$ under the assumption that $\mathrm{ad}h$ defines a 3-grading on $\mathfrak{g}$. Concretely, we show that the $\mathrm{ad}h$-eigenspaces ${\mathfrak{g}}^{\pm 1}$ contain closed convex cones $C_{\pm }$, such that

$$S_{\mathtt{V}}=exp(C_{+})G_{\mathtt{V}}exp(C_{-}),$$

where $G_{\mathtt{V}}=\{g\in G:U(g)\mathtt{V}=\mathtt{V}\}$ is the stabilizer of $\mathtt{V}$. To obtain this result, we compare several subsemigroups of G specified by the grading and the positive cone $C_U$ of U. In particular, we show that the orbit ${\mathcal{O}}_{\mathtt{V}}=U(G)\mathtt{V}$ with the inclusion order is an ordered symmetric space covering the adjoint orbit ${\mathcal{O}}_h=\mathrm{Ad}(G)h$, endowed with the partial order defined by $C_U$.