A UNICITY THEOREM FOR COHERENT FAMILIES ON SEMISIMPLE LIE-GROUPS

Let G be a real semi-simple Lie group with Lie algebra g and K be a maximal compact subgroup of G. Let F be a generalised function on G. If F is an invariant eigendistribution, then F admits a restriction to K. Without additional hypotheses, the restriction F\(K) does not determine F. Here we consider a coherent family F(lambda) of generalised functions on G in the sense of W. Schmid and prove that if F(lambda)\(K)=0 for all lambda then F(lambda)=0 for all lambda.