On Multilinear Forms for Mod $p$ Representations of $\mathrm{GL}_2(\mathbb{Q}_p)$

TL;DR

This paper proves a complete vanishing theorem for G-invariant multilinear forms on tensor products of irreducible admissible representations of GL2(Qp), revealing new restrictions on such forms in mod p representation theory.

Contribution

It establishes a comprehensive vanishing result for invariant forms, extending the understanding of multilinear forms in mod p representations of GL2 over p-adic fields.

Findings

Hom_G vanishes for tensor products of n ≥ 1 infinite-dimensional irreducible admissible representations.

Refined vanishing for B^+-invariant forms when at least one representation is supersingular.

Partial extension of results to GL2 over finite extensions of Qp.

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 $\pi_1,\dots,\pi_n$ of $G$, \[ \operatorname{Hom}_G(\pi_1 \otimes \cdots \otimes \pi_n, \mathbb{1}) = 0. \] A refined version holds for $B^+ := \begin{pmatrix} p^{\mathbb{Z}} & \mathbb{Q}_p \\ 0 & 1 \end{pmatrix}$-invariant forms when at least one $\pi_i$ 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 $F/\mathbb{Q}_p$.