Positive definite superfunctions and unitary representations of lie supergroups

Abstract

Abstract. For a broad class of Fréchet-Lie supergroups G, we prove that there exists a correspondence between positive definite smooth (resp., analytic) superfunctions on G and matrix coefficients of smooth (resp., analytic) unitary representations of the Harish-Chandra pair (G, g) associated to G.As an application, we prove that a smooth positive definite superfunction on G is analytic if and only if it restricts to an analytic function on the underlying manifold of G.When the underlying manifold of G is 1-connected we obtain a necessary and sufficient condition for a linear functional on the universal enveloping algebra U (g C ) to correspond to a matrix coefficient of a unitary representation of (G, g).The class of Lie supergroups for which the aforementioned results hold is characterised by a condition on the convergence of the Trotter product formula. This condition is strictly weaker than assuming that the underlying Lie group of G is a locally exponential Fréchet-Lie group. In particular, our results apply to examples of interest in representation theory such as mapping supergroups and diffeomorphism supergroups.

Crossed product algebras and direct integral decomposition for Lie supergroups

“…Remark 2.2. Here we should clarify that the condition given in Definition 2.1(iv) is identical to the ones given in our previous papers [NeSa2,Def. 4.6.3(iv)] and [NeSa3,Def.…”

“…7.1(iv)]. More precisely, in [NeSa2] and [NeSa3] we tacitly assume that Ad is an extension of the adjoint action of G on g 0 . Let (π, H ) be a unitary representation of a Lie group G. For x ∈ Lie(G) and v ∈ H , we set…”

“…A standard Zorn Lemma argument shows that every unitary representation can be written as a direct sum of representations which are cyclic up to parity change. Furthermore, in [NeSa2,Thm 6.7.5] a GNS construction is given which results in a correspondence between cyclic unitary representations and positive definite superfunctions on (G, g).…”

On universal enveloping algebras in a topological setting

“…The differential calculus on topological groups, involving functions which are smooth along the one-parameter subgroups (Definition 2.3), plays an important role for these extensions of Lie theory and has recently found remarkable applications also to supergroups and their representation theory ([NS13a], [NS13b]). We have merely mentioned here a very few references that are closer related to the topics of our paper.…”

On universal enveloping algebras in a topological setting

“…It is well-known that Lie theory and the related representation theory have been successfully extended much beyond the classical setting of finite-dimensional real Lie groups, and this research area now includes locally compact groups ( [HM07], [HM13]), Lie groups modeled on Banach spaces or even on locally convex spaces ( [KM97], [Bel06], [Ne06]), and some other classes of topological groups which may not be locally compact ( [BCR81], [Glö02b], [HM05]). The differential calculus on topological groups, involving functions which are smooth along the one-parameter subgroups (Definition 2.3), plays an important role for these extensions of Lie theory and has recently found remarkable applications also to supergroups and their representation theory ( [NS13a], [NS13b]). We have merely mentioned here a very few references that are closer related to the topics of our paper.…”