Test vectors for trilinear forms, when two representations are unramified

Louise Nyssen

Published 2007-10-25, updated 2009-03-02Version 2

Let F be a finite extension of Qp and G be GL(2,F). When V is the tensor product of three infinite dimensional, irreducible, admissible representations of G, the space of G-invariant linear forms has dimension 0 or 1. When a non-zero linear form exists, one wants to find an element of V which is not in its kernel : this is a test vector. Gross and Prasad found explicit test vectors when the three representations are unramified principal series, and when the three representations are unramified twists of the Steinberg representation. In this paper, we find an explicit test vector when two of the representations are unramified principal series and the third one has ramification at least 1.