Ramified tori and Gelfand--Kirillov dimension of mod p representations

Let K/Qₚ be a finite extension of degree n with p≥3, and let D be the non-split quaternion algebra over K. We prove that every infinite-dimensional admissible smooth representation of GL₂(K) or D^× 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 Kₚ and the coefficient field is algebraically closed, the canonical diagram of an admissible supersingular representation satisfying the same Casimir annihilation condition is (π,π,id), as in the case of local fields of characteristic p.