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 $\overline{ r}: \textrm{ Gal}(\overline{ F^+}/F^+) \longrightarrow { }^C\textrm{ U}_{ 1,1}(\overline{ \mathbb{ F}}_p)$ 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 $\textrm{ U}_{ 1,1}(F^+_v)$ associated to $\overline{ r}$ in spaces of mod $p$ automorphic forms have Gelfand--Kirillov dimension $[F^+_v:\mathbb{ Q}_p]$.
Categories: math.NT math.RT