Characters of Connected Lie Groups by Lajos Pukanszky

By Lajos Pukanszky

This ebook provides to the good 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 team thought and glossy research. The teams studied listed below are the hooked up Lie teams of basic kind (not inevitably nilpotent or semisimple).

Final effects mirror Kirillov's orbit technique; in terms of teams which may be non-algebraic or non-type I, the tactic calls for massive sophistication. equipment used variety from deep sensible research (the idea of C∗-algebras, elements from F. J. Murray and J. von Neumann, and degree idea) to differential geometry (Lie teams and Hamiltonian actions).

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

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

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We note that this implies that it suffices to show that H^/(Gi)h is countably separated. (ii) We write again K = H/M and denote by A the set of all unitary equivalence classes of irreducible ^-representations of K. For A G A we set ip(\) — (u/ (gu(A))~. (iii) Next we introduce a Borel structure on HQ as follows. We denote by a the class of a in H2(K,T) (cf. [AM], p. 25). Let Ka be the corresponding central extension of K by T. We recall that in this situation a representation of Ka is said to be of class one if its restriction to the canonical 1-torus is a multiple of the identity map onto itself.

A) (cf. [Dreg], p. 428). There is on 5 a Polish locally compact structure such that 1) The Borel structure resulting from this topology coincides with the given one; 2) /j, is a Radon measure. B) (cf. [B int], pp. 64-68). The equivalence relation defined by K on S is fi-measurable. From this we are permitted to infer that for each y G S/K there is a positive, cr-finite measure on 7r _1 (y), such that for each positive, Borel measurable function / we have / f(s) • dfA(s) = [ JS JS/K dv'{y) I I \J'x-1(y) f(t) • d»y{t)) .

2 it is of type I, since [Q,Q] + m is isomorphic to an algebraic Lie algebra. Finally, G/H is abelian by \) 2 [$,&]. 1 in [Dreg], p. 427. LEMMA 6. Let K be a separable, locally compact abelian group. We assume that S is a standard left K-space such that S/K is countably separated. 4. APPLICATION TO REGULAR REPRESENTATION OF CONNECTED LIE GROUP 25 be a positive, a-finite measure on S. ™(s). To prove this, let us write n : 5 —• S/K for the canonical projection. (i) We start by noting that S/K is analytic.

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