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

TL;DR

This paper characterizes the semigroup of group elements that preserve a standard subspace in a Hilbert space, using 3-gradings in Lie algebras and convex cones, extending understanding of modular structures in Lie group representations.

Contribution

It provides a complete description of the semigroup associated with a standard subspace in terms of 3-gradings and convex cones, building on previous infinitesimal analysis.

Findings

Semigroup S_V is expressed as exp(C_+) G_V exp(C_-)

The eigenspaces of ad h contain closed convex cones C_±

The orbit U(G)V forms an ordered symmetric space

Abstract

Let V be a standard subspace in the complex Hilbert space H and U : G \to U(H) be a unitary representation of a finite dimensional Lie group. We assume the existence of an element h in the Lie algebra of G such that U(exp th) is the modular group of V and that the modular involution J_V normalizes U(G). We want to determine the semigroup $S_V = \{ g\in G : U(g)V \subseteq V\}.$ In previous work we have seen that its infinitesimal generators span a Lie algebra on which ad h defines a 3-grading, and here we completely determine the semigroup S_V under the assumption that ad h defines a 3-grading. Concretely, we show that the ad h-eigenspaces for the eigenvalue $\pm 1$ contain closed convex cones $C_\pm$, such that $S_V = exp(C_+) G_V exp(C_-)$, where $G_V$ is the stabilizer of V in G. 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 U(G)V, endowed with the inclusion order, is an ordered symmetric space covering the adjoint orbit $Ad(G)h$, endowed with the partial order defined by~$C_U$.