Unitary Representations of Lie Groups with Reflection Symmetry

We consider the following class of unitary representationsπof some (real) Lie groupGwhich has a matched pair of symmetries described as follows: (i) SupposeGhas a period-2 automorphismτ, and that the Hilbert spaceH(π) carries a unitary operatorJsuch thatJπ=(π∘τ)J(i.e.,selfsimilarity). (ii) An added symmetry is implied ifH(π) further contains a closed subspaceK0having a certainorder-covarianceproperty, and satisfying theK0-restricted positivity : ⦠v | Jv⦔⩾0, ∀v∈K0, where ⦠· | ·⦔ is the inner product inH(π). From (i)–(ii), we get an induced dual representation of an associated dual groupGc. All three properties, selfsimilarity, order-covariance, and positivity, are satisfied in a natural context whenGis semisimple and hermitean; but whenGis the (ax+b)-group, or the Heisenberg group, positivity is incompatible with the other two axioms for the infinite-dimensional irreducible representations. We describe a class ofG, containing the latter two, which admits a classification of the possible spacesK0⊂H(π) satisfying the axioms of selfsimilarity and order-covariance.