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

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 $G={\mathrm{G}\mathrm{L}}_2({\mathbb{Q}}_p)$ over ${\bar{\mathbb{F}}}_p$. 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 $G$, $${\mathrm{Hom}}_G({\pi }_1\otimes \cdots \otimes {\pi }_n,\mathbb{1})=0.$$ 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 ${\pi }_i$ 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 ${\mathrm{G}\mathrm{L}}_2(F)$ for finite extensions $F\mathrm{/}{\mathbb{Q}}_p$.