Test vectors for trilinear forms when at least one representation is not supercuspidal

TL;DR

This paper investigates the existence of explicit non-vanishing vectors for trilinear functionals on three representations of GL2(F), especially when at least one is not supercuspidal, extending previous uniqueness results.

Contribution

It provides explicit test vectors for trilinear forms in cases where not all three representations are supercuspidal, filling a gap in the understanding of these functionals.

Findings

Dimension of invariant trilinear functionals is at most one.

Explicit non-vanishing vectors are constructed when not all representations are supercuspidal.

Results extend previous work on supercuspidal cases to more general representations.

Abstract

Given three irreducible, admissible, infinite dimensional complex representations of GL2(F), with F a local field, the space of trilinear functionals invariant by the group has dimension at most one. When it is one we provide an explicit vector on which the functional does not vanish assuming that not all three representations are supercuspidal.