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

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 $\ddot{I}{€}_1,\dots ,\ddot{I}{€}_n$ of $G$, \[ \operatorname{Hom}_G(π_1 \otimes \cdots \otimes π_n, \mathbb{1}) = 0. \] A refined version holds for $B^{+}:=\left(\begin{array}{cc}{p}^{\mathbb{Z}}& {\mathbb{Q}}_p\\ 0& 1\end{array}\right)$-invariant forms when at least one $\ddot{I}{€}_i$ 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 $F/{\mathbb{Q}}_p$.