Gelfand--Kirillov dimension and mod $p$ cohomology for inner forms of $\mathrm{GL}_2$

Abstract

Under standard assumptions, we compute the GK-dimension of Hecke eigenspaces in the mod $p$ cohomology of an inner form $D^\times$ of $\mathrm{GL}_2$ over a totally real field unramified at $p$, allowing $D$ to be a division algebra at $p$. Our arguments also apply when $D$ is a matrix algebra at $p$, in which case they give a simplified proof of a theorem of Breuil--Herzig--Hu--Morra--Schraen.