Positive energy representations and continuity of projective representations for general topological groups

TL;DR

This paper investigates the continuity and structure of positive energy representations of semidirect product groups, providing criteria for projective representation continuity and characterizing positive definite functions extending to the larger group.

Contribution

It establishes that irreducible positive energy representations of semidirect products with real parameters remain irreducible when restricted, and offers new criteria for the continuity of projective unitary representations.

Findings

Irreducible positive energy representations of $G^lat$ are irreducible when restricted to $G$.

Derived effective criteria for the continuity of projective unitary representations.

Characterized continuous positive definite functions on $G$ extending to $G^lat$.

Abstract

Let $G$ and $T$ be topological groups, $\alpha : T \to \Aut(G)$ a homomorphism defining a continuous action of $T$ on $G$ and $G^\sharp := G \rtimes_\alpha T$ the corresponding semidirect product group. In this paper we address several issues concerning irreducible continuous unitary representations $(\pi^\sharp, \cH)$ of $G^\sharp$ whose restriction to $G$ remains irreducible. First we prove that, for $T = \R$, this is the case for any irreducible positive energy representation of $G^\sharp$, i.e., for which the one-parameter group $U_t := \pi^\sharp(\1,t)$ has non-negative spectrum. The passage from irreducible unitary representations of $G$ to representations of $G^\sharp$ requires that certain projective unitary representations are continuous. To facilitate this verification, we derive various effective criteria for the continuity of projective unitary representations. Based on results on Borchers for $W^*$-dynamical systems, we also derive a characterization of the continuous positive definite functions on $G$ that extend to a $G^\sharp$.