Lower multiplicity for irreducible representations of nilpotent locally compact groups

Let G be a nilpotent locally compact group. The lower multiplicity M-L(pi) is defined for every irreducible representation pi of G, which does not for m an open point in the dual space (G) over cap of G It is shown that M-L(pi ) = 1 if either G is connected or pi is finite dimensional. Conversely, for G a nilpotent group with small invariant neighbourhoods, M-L(pi)< infinity implies that pi is finite dimensional. (C) 1999 Academic.