Test vectors for linear forms (Q1191434)

In the study of automorphic representations it often happens that one wants to know whether a certain invariant form given by an integral is identically zero or not. In the paper the local question is treated in two situations. The first one concerns \(K^*\)-invariant linear forms on irreducible representations of \(G=GL(2,F)\times K^*\), where \(K\) is a quadratic extension of \(F\), the second one concerns \(GL(2,F)\)-invariant linear forms on irreducible representations of \(G = GL(2,F)^ 3\). In both cases a criterion is known for the existence of a non-trivial invariant linear form, which is then unique up to a scalar factor. In this paper it is shown that this linear form is non-zero on the line fixed by a certain open compact subgroup of \(G\). Hence any non-zero vector on that line is a test vector for an invariant linear form. Some global applications are given.