Unitary Representations of Lie Groups with Reflection Symmetry

Abstract:  c. All three properties, selfsimilarity, order-covariance, and positivity, are satisfied in a natural context when G is semisimple and hermitean; but when G is the (ax + b)-group, or the Heisenberg group, positivity is incompatible with the other two axioms for the infinitedimensional irreducible representations. We describe a class of G, containing the latter two, which admits a classification of the possible spaces K0 ae H(ss) satisfying the axioms of selfsimilarity and order-covariance. 1 Introduction We consider a class of unitary representations of a Lie group G which possess a certain reflection symmetry defined as follows: If ss is a representation of G in some Hilbert space H, we introduce the following three structures: i) o / 2 Aut(G) of period 2; ii) J: H! H is a unitary operator of period 2 such that J ss(g)J \Lambda = ss(o / (g)), g 2 G (this