Irreducible subrepresentations of the conjugation representation of finitep-groups

Locally compact groups whose conjugation representations satisfy a Kazhdan type property or have countable support

“…The difficulty in dealing with such questions for y G is that in general very little is known about suppy G . Although there are various classes of groups satisfying suppy G = (G/Z(G)Y (see [32] and [35]), even for finite groups or 2-step nilpotent simply connected Lie groups suppy G can be strictly contained in (G/Z(G))" (compare [19] and [20]).…”

“…For a locally compact group G with left Haar measure and modular function S the conjugation representation y G of G on L 2 (G) is defined by feL 2 (G), x,yeG. y G has been investigated recently (see [19,20,21,24,32,35]). For semi-simple Lie groups, a related representation has been studied in [25].…”

“…Clearly, y G is trivial on the centre Z(G) of G and separates the elements of G/Z(G). However, suppy G is generally strictly contained in (G/Z(G))" and intricate to determine (compare [19,20,32,35]). It is this unpleasant behaviour that raises most of the difficulties.…”

The conjugation representation and inner amenability of discrete groups.

“…As mentioned at the beginning of the paper, even for finite, 2-step nilpotent groups G not necessarily every irreducible representation of G/Z(G) occurs in y G ([8], [13]). On the other band by Theorem 1.8, ((j/G r ) r g suppy G .…”

On 𝐶*-algebras associated to the conjugation representation of a locally compact group

“…However, so far it is much less understood than the left regular representation. The main difficulty arising is that, even for finite groups, the support of yG is generally strictly contained in the dual of G/Z(G) and is intricate to determine (compare [11], [12], [13], [20], and [22]). …”