The conjugation representation \(\gamma\) of a locally compact group G is defined by the unitary operators \[ (\gamma (x)f)(y)=\Delta (x)^{1/2}f(x^{-1}yx),\quad x,y\in G,\quad f\in L^ 2(G) \] on \(L^ 2(G)\), where \(\Delta\) is the modular function of G. This representation has been, in comparison with the regular representation of G, very little studied. Surprisingly, it is still not known, which irreducible representations occur in the conjugation representation of a finite group. M. Moskowitz [J. Pure Appl. Algebra 36, 159-165 (1985; Zbl 0558.22005)] has recently shown that, in case G is a compact connected Lie group, \(\gamma\) contains all the irreducible representations which are trivial on the center of G.
The main result of the paper under review is the following theorem: Let G be a \(\sigma\)-compact, locally compact group, and assume that the reduced \(C^*\)-algebra \(C^*_ r(G)\) is nuclear (this is, for instance, the case if G is connected or amenable or type I). Then \(\gamma\) is weakly equivalent to the set of all tensor products \(\pi\) \(\otimes {\bar \pi}\), where \(\pi\) is an irreducible representation of \(C^*_ r(G)\). A number of interesting applications of this result are given. For example, it is shown that the support of \(\gamma\) is discrete, only if \(G=V\times K\) for a vector group V and a compact group K. Moreover, the support of \(\gamma\) is computed for some concrete groups (free groups, SL(2,\({\mathbb{R}}),...)\).