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

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 quot