Lower Multiplicity for Irreducible Representations of Nilpotent Locally Compact Groups

Let G be a nilpotent locally compact group. The lower multiplicity ML(π) is defined for every irreducible representation π of G, which does not form an open point in the dual space Ĝ of G. It is shown that ML(π)=1 if either G is connected or π is finite dimensional. Conversely, for G a nilpotent group with small invariant neighbourhoods, ML(π)<∞ implies that π is finite dimensional.