Unitary Representations of Lie Groups with Reflection Symmetry

✍ Scribed by Palle E.T. Jorgensen; Gestur Ólafsson

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 696 KB
• Volume: 158
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1998.3285

✦ Synopsis

We consider the following class of unitary representations ? of some (real) Lie group G which has a matched pair of symmetries described as follows: (i) Suppose G has a period-2 automorphism {, and that the Hilbert space H(?) carries a unitary operator J such that J?=(? b {) J (i.e., selfsimilarity). (ii) An added symmetry is implied if H(?) further contains a closed subspace K 0 having a certain ordercovariance property, and satisfying the K 0 -restricted positivity: (v | Jv) 0, \v # K 0 , where ( } | } ) is the inner product in H(?). 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(?) satisfying the axioms of selfsimilarity and order-covariance.

1998 Academic Press (i) { # Aut(G) of period 2;

(ii) J: H Ä H is a unitary operator of period 2 such that J?( g) J*= ?({( g)), g # G (this will hold if ? is of the form ? + Ä? & with ? + and ? & b { article no.