Ramified tori and Gelfand--Kirillov dimension of mod $p$ representations
Let $K/\mathbb{Q}_p$ be a finite extension of degree $n$ with $p\geq3$, and let $D$ be the non-split quaternion algebra over $K$. We prove that every infinite-dimensional admissible smooth representation of $\mathrm{GL}_2(K)$ or $D^\times$ with a central character and satisfying a Casimir annihilation condition has Gelfand--Kirillov dimension $n$. As a corollary, every infinite-dimensional admissible Banach representation whose locally analytic vectors have an infinitesimal character has dimension $n$, strengthening Sorgdrager's bound to an equality. An appendix by Yongquan Hu proves that, when $K\ne\mathbb{Q}_p$ and the coefficient field is algebraically closed, the canonical diagram of an admissible supersingular representation satisfying the same Casimir annihilation condition is $(π,π,\mathrm{id})$, as in the case of local fields of characteristic $p$.