Unitary Representations of Lie Groups with Reflection Symmetry

Details Unitary Representations of Lie Groups with Reflection Symmetry Unitary Representations of Lie Groups with Reflection Symmetry

1997-07-11

arxiv.org/abs/funct-an/9707001v1

J. Funct. Anal. 158 (1998), 26--88

Mathematics

Functional Analysis

49 pages, LaTeX article style, 11pt size option

Scientific paper

10.1006/jfan.1998.3285

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: $< v \mid Jv > \ge 0$, $\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.

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