Test vectors for some ramified representations

TL;DR

This paper constructs explicit test vectors for $T$-equivariant linear functionals on certain $GL_2$ representations over $p$-adic fields, addressing ramified cases and revealing connections to the Langlands correspondence in characteristic $p$.

Contribution

It provides a complete solution for constructing test vectors in ramified principal series, Steinberg, and depth zero supercuspidal representations of $GL_2$, using a novel reduction method and modular representation theory.

Findings

Complete solutions for principal series and Steinberg representations.

Identification of obstructions in characteristic $p$ beyond root number criteria.

Discovery of a dichotomy related to Serre weights and Gauss sum signs.

Abstract

We give an explicit construction of test vectors for $T$-equivariant linear functionals on representations $\Pi$ of $GL_2$ of a $p$-adic field $F$, where $T$ is a non-split torus. Of particular interest is the case when both the representations are ramified; we completely solve this problem for principal series and Steinberg representations of $GL_2$, as well as for depth zero supercuspidals over $\mathbf{Q}_p$. A key ingredient is a theorem of Casselman and Silberger, which allows us to quickly reduce almost all cases to that of the principal series, which can be analyzed directly. Our method shows that the only genuinely difficult cases are the characters of $T$ which occur in the primitive part (or "type") of $\Pi$ when $\Pi$ is supercuspidal. The method to handle the depth zero case is based on modular representation theory, motivated by considerations from Deligne-Lusztig theory and the de Rham cohomology of Deligne-Lusztig-Drinfeld curves. The proof also reveals some interesting features related to the Langlands correspondence in characteristic $p$. We show in particular that the test vector problem has an obstruction in characteristic $p$ beyond the root number criterion of Waldspurger and Tunnell, and exhibits an unexpected dichotomy related to the weights in Serre's conjecture and the signs of standard Gauss sums.