On special representations of $p$-adic reductive groups

Let $F$ be a non-Archimedean locally compact field, let $G$ be a split connected reductive group over $F$. For a parabolic subgroup $Q\subset G$ and a ring $L$ we consider the $G$-representation on the $L$-module$$(\ast )C^{\mathrm{\infty }}(G\mathrm{/}Q,L)\mathrm{/}\sum _{Q^{\mathrm{\prime }}⊋Q}{C}^{\mathrm{\infty }}(G\mathrm{/}{Q}^{\mathrm{\prime }},L)\mathrm{.}$$Let $I\subset G$ denote an Iwahori subgroup. We define a certain free finite rank $L$-module $\mathfrak{M}$ (depending on $Q$; if $Q$ is a Borel subgroup then $(\ast )$ is the Steinberg representation and ${\mathfrak M}$isofrankone)andconstructan$I$-equivariantembeddingof$()$into$C^{\infty}(I,{\mathfrak M})$\mathrm{.}Thisallowsthecomputationofthe$I$-invariantsin$()$\mathrm{.}Wethenprovethatif$L$ is a field with characteristic equal to the residue characteristic of $F$ and if $G$ is a classical group, then the $G$-representation $(\ast )$ is irreducible. This is the analog of a theorem of Casselman (which says the same for $L=\mathbb{C}$); it had been conjectured by Vign'eras. Herzig (for $G={\mathrm{G}\mathrm{L}}_n(F)$) and Abe (for general $G$) have given classification theorems for irreducible admissible modulo $p$ representations of $G$ in terms of supersingular representations. Some of their arguments rely on the present work.