4 problems

Assume the hypotheses imposed in the definitions of the representations in,, and, and let $\pi$ be one of these representations with $\pi\ne0$. Let $\mathcal C$ be the stated categ…

Let $K$ be an unramified local field, let $A$ be the coefficient ring, and let $\pi$ be an automorphic smooth representation of ${\rm GL}_2(K)$ over $\mathbb{F}$. Let $D_A(\pi)$ de…

Let $K=F_v$ be unramified, let $\pi$ be an automorphic admissible smooth representation of ${\rm GL}_2(K)$ over $\mathbb{F}$, and let $\overline{r}_v$ be its corresponding local Ga…

Let $K=F_v$ be unramified, let $\mathbb{F}$ be the coefficient field, and let $\pi$ be an admissible smooth representation of ${\rm GL}_2(\mathcal{O}_K)$ over $\mathbb{F}$ among th…