On multilinear forms for mod $p$ representations of $\operatorname{GL}_2$ over a $p$-adic field

Abstract

Let $F$ be a finite extension of $\mathbb Q_p$, for any prime $p$, and let $L/F$ be a ramified quadratic extension. We compute the spaces of $\mathcal O_L^\times$-equivariant linear forms on the representation $\operatorname{ind}^G_{KZ}σ/(Φ-λ)$. The only nonzero cases arise from one-dimensional quotients. Consequently, the restriction of an infinite-dimensional irreducible smooth mod $p$ representation of $\operatorname{GL}_2(F)$ with a central character to $\mathcal O_L^\times$ has no character quotient. We apply this vanishing result to show that a finite tensor product of smooth mod $p$ representations of $\operatorname{GL}_2(F)$ admits no nonzero invariant linear form whenever at least one factor is infinite-dimensional and irreducible with a central character.