Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2601.12021 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
Table of Contents:
- Motivated by the study of trilinear forms for complex representations, we investigate the space of $G$-invariant linear forms on tensor products of irreducible admissible representations of $G = \mathrm{GL}_2(\mathbb{Q}_p)$ over $\overline{\mathbb{F}}_p$. Our main result is a complete vanishing theorem: for any $n \ge 1$ and $n$ infinite-dimensional irreducible admissible representations $π_1,\dots,π_n$ of $G$, \[ \operatorname{Hom}_G(π_1 \otimes \cdots \otimes π_n, \mathbb{1}) = 0. \] A refined version holds for $B^+ := \begin{pmatrix} p^{\mathbb{Z}} & \mathbb{Q}_p \\ 0 & 1 \end{pmatrix}$-invariant forms when at least one $π_i$ is supersingular. The proof proceeds by a detailed analysis of certain subgroups, reducing the problem from $G$ to $B^+$ and ultimately to the representation theory of $\mathbb{Z}_p$. We also deduce partial extensions of the result to $\mathrm{GL}_2(F)$ for finite extensions $F/\mathbb{Q}_p$.