Test vectors for trilinear forms, when two representations are unramified

TL;DR

This paper extends the explicit construction of test vectors for trilinear forms on GL(2,F) representations, specifically when two are unramified and the third has ramification, broadening previous unramified cases.

Contribution

It provides an explicit test vector construction for cases where two representations are unramified principal series and the third is ramified, filling a gap in existing literature.

Findings

Explicit test vector constructed for mixed ramification cases.

Generalization of previous unramified test vector results.

Enhances understanding of invariant linear forms in representation theory.

Abstract

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.