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

2012/08/13 by Karl-Hermann Neeb, Neeb, Karl-Hermann
Mathematics · Physics and Astronomy · #22E45 #22E66. 22D10 #43A65 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Representation Theory (math.RT) #math-ph #math.MP #math.RT #msc:22D10 #msc:22E45 #msc:22E66. #msc:43A65

paper · pdf · doi:10.48550/arxiv.1208.2511

22 pages

arxiv created 2012/08/13 · arxiv updated 2012/08/14

Abstract

Let G and T be topological groups, α: T → \Aut(G) a homomorphism defining a continuous action of T on G and G^\sharp := G \rtimes_αT the corresponding semidirect product group. In this paper we address several issues concerning irreducible continuous unitary representations (π^\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 := π^\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.

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