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=(πτ)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 : vJv≥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. © 1998 Academic Press.
Publication Source (Journal or Book title)
Journal of Functional Analysis
Jorgensen, P., & Ólafsson, G. (1998). Unitary Representations of Lie Groups with Reflection Symmetry. Journal of Functional Analysis, 158 (1), 26-88. https://doi.org/10.1006/jfan.1998.3285