On the existence of regular vectors

Abstract

Let G be a locally convex Lie group and π : G → U(H) be a continuous unitary representation. π is called smooth if the space of π-smooth vectors H ∞ ⊂ H is dense. In this article we show that under certain conditions, concerning in particular the structure of the Lie algebra g of G, a continuous unitary representation of G is automatically smooth. As an application, this yields a dense space of smooth vectors for continuous positive energy representations of oscillator groups, double extensions of loop groups and the Virasoro group. Moreover we show the existence of a dense space of analytic vectors for the class of semibounded representations of Banach-Lie groups. Here π is called semibounded, if π is smooth and there exists a non-empty open subset U ⊂ g such that the operators idπ(x) from the derived representation are uniformly bounded from above for x ∈ U .

Positive energy representations of Sobolev diffeomorphism groups of the circle

“…Accordingly they are completely classified by the central charge c and the lowest conformal energy h [KR87]. Related results including reducible representations have been recently obtained in [NS15,Zel17].…”

Ground State Representations of Some Non-Rational Conformal Nets

“…We are currently not able to do this, because the corresponding Lie group Diff c (R) of compactly supported diffeomorphisms of R do not possess any compact subgroup (in contrast to Diff(S 1 ) which contains finite covers of PSL(2, R), which is the key to differentiate representations [Car04, Appendix], cf. [Zel17]).…”

Ground State Representations of Some Non-Rational Conformal Nets

“…We are currently not able to do this, because the corresponding Lie group Diff c (R) of compactly supported diffeomorphisms of R do not possess any compact subgroup (in contrast to Diff(S 1 ) which contains finite covers of PSL(2, R), which is the key to differentiate representations [32] (Appendix), cf. [33]).…”