Journal article
Unitary Representations of Lie Groups with Reflection Symmetry
Journal of functional analysis, Vol.158(1), pp.26-88
1998
DOI: 10.1006/jfan.1998.3285
Abstract
We consider the following class of unitary representations π of some (real) Lie group G which has a matched pair of symmetries described as follows: (i) Suppose G has a period-2 automorphism τ , and that the Hilbert space H ( π ) carries a unitary operator J such that Jπ =( π ∘ τ ) J (i.e., selfsimilarity ). (ii) An added symmetry is implied if H ( π ) further contains a closed subspace K 0 having a certain order-covariance property, and satisfying the K 0 -restricted positivity : ⦠ v | Jv ⦔⩾0, ∀ v ∈ K 0 , where ⦠· | ·⦔ is the inner product in H ( π ). 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 K 0 ⊂ H ( π ) satisfying the axioms of selfsimilarity and order-covariance.
Details
- Title: Subtitle
- Unitary Representations of Lie Groups with Reflection Symmetry
- Creators
- Palle E.T JorgensenGestur Ólafsson
- Resource Type
- Journal article
- Publication Details
- Journal of functional analysis, Vol.158(1), pp.26-88
- DOI
- 10.1006/jfan.1998.3285
- ISSN
- 0022-1236
- eISSN
- 1096-0783
- Language
- English
- Date published
- 1998
- Academic Unit
- Mathematics
- Record Identifier
- 9983985823702771
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