Gelfand-Kirillov dimension for mod $p$ representations of $p$-adic unitary groups of rank 2

Abstract

Let $p$ be a prime number and $F/F^{+}$ a CM extension of a totally real field such that every place of $F^{+}$ above $p$ is unramified and inert in $F$. We fix a finite place $v$ of $F^{+}$ above $p$, and let $\overline{r}:\text{Gal}(\overline{F^{+}}/F^{+})⟶{}^C{\text{U}}_{1,1}({\overline{\mathbb{F}}}_p)$ be a modular $L$-parameter valued in the $C$-group of a rank 2 unitary group associated to $F/F^{+}$. We assume $\overline{r}$ is semisimple and sufficiently generic at $v$. Using recent results of Breuil--Herzig--Hu--Morra--Schraen along with our previous work, we prove that certain admissible smooth ${\overline{\mathbb{F}}}_p$-representations of the $p$-adic unitary group ${\text{U}}_{1,1}(F_v^{+})$ associated to $\overline{r}$ in spaces of mod $p$ automorphic forms have Gelfand--Kirillov dimension $[F_v^{+}:{\mathbb{Q}}_p]$.

66 pages. v2: minor edits following referee report. To appear in Representation Theory