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 workforce conception and smooth research. The teams studied listed below are the attached Lie teams of normal style (not inevitably nilpotent or semisimple).

Final effects mirror Kirillov's orbit procedure; with regards to teams which may be non-algebraic or non-type I, the strategy calls for substantial sophistication. equipment used variety from deep practical research (the concept of C∗-algebras, components 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 during the literature. the result's a powerful 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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Since k \-> kt (k € K; t is fixed such that 7r(£) = y) is Borel, there is on 7r~1(y) a structure of a if-homogeneous space, giving rise to our Borel structure. For reasons of continuity, kjiy = \(k)/j,y (y G N) for all k in if. (viii) We claim that there is m > 0, measurable on S such that m(ks) = x{k)m(s) for each k e K and almost all 5 in 5. In fact, a) Assume that ^((/ci^i)) = ^((^2^2))- Then x(&i) = x(fe)- In fact, ip((ki,ti)) — ^((fe,£2)) is equivalent to £1 = £2 (= ^ say) and k\t\ = £2^2 • Putting A: = k^1 • k\ we have £ = kt, whence, since K is abelian, ku = u (u G kt) and /c/iy = fiy (y G if) giving x(&) — 1 a n d a l s o x(&i) = x(fe) (if ^((fci,*i)) = ^((fc 2,*2))).

Iii) As we saw above, there is an L C S D Gc such that if we put Si = LSo, then S/Si is finite. 1, if G is separable, locally compact, H D N are closed, invariant type I subgroups of G such that N/G is countably separated, and for any r G H, r\N is transitive, then H/G is countably separated if and only if for all uo in TV, S^/G^ is countably separated (5W = {rj-rj eS such that rj\N « cu}). We conclude from this that Si/G^ is countably separated by replacing G by G w , H by Si and N by S 0 . In fact, 1) So = (Hu)o is locally algebraic and hence of type I.

We define gi by ii + UQ. This being so, we have 1) fli + f) = (i + uo) + (i2 + u) = i + u 0 = g. 2) Next we note that g± is an ideal in g. We have, in fact, [gi,g] = [ii + Uo,i + uo] Q [ii,i] + uo £ ii + u 0 = $i or that [gi,g] ^ $i. Thus 91 is an ideal in g. 3) We claim that [gi, f)] C fj n [g,Uo]. ibn[fl,uo]. 4) We recall (cf. [B], p. 65) that the nilpotent radical of a Lie algebra (= 6 , say) is the intersection of the kernels of finite-dimensional simple representations. We have (cf. loc. ) & = [g, g]Huo.