By Lajos Pukanszky

This publication provides to the nice physique of analysis that extends again to A. Weil and E. P. Wigner at the unitary representations of in the community compact teams and their characters, i.e. the interaction among classical staff idea and sleek research. The teams studied listed here are the hooked up Lie teams of basic style (not inevitably nilpotent or semisimple).

Final effects mirror Kirillov's orbit technique; in relation to teams that could be non-algebraic or non-type I, the tactic calls for huge sophistication. tools used diversity from deep practical research (the idea of C∗-algebras, elements from F. J. Murray and J. von Neumann, and degree conception) to differential geometry (Lie teams and Hamiltonian actions).

This ebook offers for the 1st time a scientific and concise compilation of proofs formerly dispersed in the course of the literature. the result's a powerful instance of the deepness of Pukánszky's work.

Readership: Graduate scholars and learn mathematicians operating in topological teams and Lie teams; theoretical physicists

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**Example text**

H~x G M. In other words, r and /i commute modulo M. (iii) From here, since Gi and H are connected, the same is true for arbitrary elements of Gi and H. Below we assume that m = (iJ). 2, which we have already discussed. Let us suppose that we are given an element uo of M, such that dim (a;) > 1, and such that uj(exp(l)) = exp(i(Z, h))I (I G m c ), I = identity and h is not orthogonal to m c . Then we have Mh = h+(mc)1-. M. We write K = H/N and t \ H -> K will be the canonical projection. We recall that if u/ and uo" are projective extensions, belonging to the same cocycle, of UJ from M to i7, then there exists a x G X(i^) such that uo" — L{X)U'.

In fact, we have by definition Mi = J " 1 • M'. Then M commutes with M' and, by M = SM'S and J " 1 = SJS, M commutes with J - 1 . Conclusion. So far J = M' • M^1, J = M[ • M _ 1 and M commutes with M x , hence M' commutes with M[. (iv) We conclude from (iii) that M' • M^1 = M[ • M " 1 whence ( M ^ ) - 1 • M ; = _1 M • Mi, and that there is a C positive, selfadjoint, invertible, Crj(L(G))c such that M • M f 1 = C, ( M ' ) " 1 • M{ = C, and S C S = C'1. (v) We have J = M' • M f 1 = M 7 • M " 1 • MM'1 = CM' • M~\ 2 (vi) We define L> > 0 by C = £> ; then also S D S = D'1 and J = We write TV = D~XM; this implies AT' = SD"- 1 .

The principal results are easy to state. They fall into two sets. Let U be the centralizer of the connected center of T. We denote by T the set of all those characters of Uc which, when restricted to the canonical torus of T, give rise to the identity map of the latter onto itself. Let x be an element of J-. Then the representation induced by x m I \ V(x) saY> ls a semifinite factor representation of T. It is of type one if and only if the subgroup UCTQ is of finite index in U. Finally, for x a n d x' m 3~, V(x) a n d ^(xO a r e quasi-equivalent if and only if x and x' li e o n the same orbit of T on T\ in this case they are also unitarily equivalent (cf.