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...
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| Vydáno v: | Journal of functional analysis Ročník 158; číslo 1; s. 26 - 88 |
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| Hlavní autoři: | , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Elsevier Inc
10.09.1998
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| ISSN: | 0022-1236, 1096-0783 |
| On-line přístup: | Získat plný text |
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| Shrnutí: | 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. |
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| ISSN: | 0022-1236 1096-0783 |
| DOI: | 10.1006/jfan.1998.3285 |