Test vectors for trilinear forms : the case of two principal series

TL;DR

This paper describes explicit test vectors for tensor products of two principal series representations of GL(2,F), extending previous work on unramified cases to more general principal series scenarios.

Contribution

It provides explicit constructions of test vectors in the case where two of the three representations are principal series, broadening the understanding of invariant linear forms.

Findings

Explicit test vectors are constructed for two principal series representations.

The results extend known cases from unramified to more general principal series.

The dimension of the space of G-invariant linear forms is at most one.

Abstract

Let F be a finite extension of Qp and G be GL(2,F). When V is the tensor product of three admissible, irreducible, finite dimensional representations of G, the space of G-invariant linear forms has dimension at most one. 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 they are all unramified twists of the Steinberg representation. In this paper we decribe explicit test vectors when two of the representations are principal series.