On a certain representation of a compact group (Q762622)

The study of the action of a group on itself, by conjugation, posed, among other things, the following problem: Which irreducibles of \(G\) occur and with what multiplicities? From the very beginning of this note, the author emphasizes that the analysis we mentioned above is not feasible in general; however, utilising an ingenious method, in the case when \(G\) is a compact analytic group, the author is able to give an interesting answer to the problem, namely the following important result is proved. Let \(G\) be a compact analytic group and \(\alpha\) the strongly continuous unitary representation of \(G\) on \(L_2(G)\) given by conjugation. Then \(\alpha\) is the discrete direct sum of irreducible representations all of which have countably infinite multiplicity. Every irreducible of \(\mathrm{Ad}(G)\) lifted of \(G\) occurs in the decomposition. As an important corollary, the converse of Freudenthal's Theorem for compact connected Lie group \(G\), [\textit{H. freudenthal}, Proc. Am. Math. Soc. 7, 175--176 (1956; Zbl 0071.25504)] is obtained. Corollary: Let \(\rho\) be an irreducible representation of a compact connected Lie group \(G\). Then the following conditions are equivalent: (i) \([\alpha;\rho]>0\), (ii) \(\rho (z)=(1)\), i.e. \(\rho\) is the lift of an irreducible representation of \(\mathrm{Ad}(G)\), (iii) \(\rho\) has a nontrivial zero weight vector.