Unitary Representations of Lie Groups with Reflection Symmetry

We consider the following class of unitary representations $\pi$ of some (real) Lie group $G$ which has a matched pair of symmetries described as follows: (i) Suppose $G$ has a period-2 automorphism $\tau$, and that the Hilbert space $\mathbf{H}(\pi )$ carries a unitary operator $J$ such that $J\pi =(\pi \circ \tau)J$ (i.e., selfsimilarity). (ii) An added symmetry is implied if $\mathbf{H}(\pi )$ further contains a closed subspace ${\mathbf{K}}_0$ having a certain order-covariance property, and satisfying the ${\mathbf{K}}_0$-restricted positivity: $<v\mid Jv>\ge 0$, $\mathrm{\forall }v\in {\mathbf{K}}_0$, where $<\cdot \mid \cdot >$ is the inner product in $\mathbf{H}(\pi )$. From (i)–(ii), we get an induced dual representation of an associated dual group $G^c$. 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 infinite-dimensional irreducible representations. We describe a class of $G$, containing the latter two, which admits a classification of the possible spaces $$\mathbf{K}$_0 \subset $\mathbf{H}$ (\pi)$ satisfying the axioms of selfsimilarity and order-covariance.