On Multilinear Forms for Mod $p$ Representations of $\mathrm{GL}_2(\mathbb{Q}_p)$
Yikun Fan
TL;DR
The paper studies mod $p$ representations of $G=\mathrm{GL}_2(F)$ and establishes a universal vanishing phenomenon for invariant multilinear forms, contrasting with the complex theory. It introduces a reduction framework to $B^+$- and $\mathbb{Z}_p$-representation theory via the notion of vanishing pairs and the difference operator $\Delta$, and constructs explicit vanishing subrepresentations for principal, special, and supersingular types. Consequently, for $F=\mathbb{Q}_p$ and $n\ge1$, if the $\pi_i$ are infinite-dimensional irreducible admissible, then $\mathrm{Hom}_G\left(\bigotimes_{i=1}^n \pi_i, \mathbb{1}\right)=0$, with a refined vanishing for $B^+$-invariant forms when at least one $\pi_i$ is supersingular. Partial extensions to finite extensions $F/\mathbb{Q}_p$ are obtained, highlighting a sharp contrast in the mod $p$ setting relative to the complex theory.
Abstract
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 $F/\mathbb{Q}_p$.