On the universal module of $p$-adic spherical Hecke algebras

Let $\widetilde{G}$ be a split connected reductive group with connected center $Z$ over a local non-Archimedean field $F$ of residue characteristic $p$, let $\widetilde{K}$ be a hyperspecial maximal compact open subgroup in $\widetilde{G}$. Let $R$ be a commutative ring, let $V$ be a finitely generated $R$-free $R[\widetilde{K}]$-module. For an $R$-algebra $B$ and a character $\chi:{\mathfrak H}_V(\widetilde{G},\widetilde{K})\to B$ of the spherical Hecke algebra ${\mathfrak H}_V(\widetilde{G},\widetilde{K})={\rm End}_{R[\widetilde{G}]}{\rm ind}_{\widetilde{K}}^{\widetilde{G}}(V)$ we consider the specialization $$M_{\chi}(V)={\rm ind}_{\widetilde{K}}^{\widetilde{G}}V\otimes_{{\mathfrak H}_V(\widetilde{G},\widetilde{K}),\chi}B$$ of the universal ${\mathfrak H}_V(\widetilde{G},\widetilde{K})$-module ${\rm ind}_{\widetilde{K}}^{\widetilde{G}}(V)$. For large classes of $R$ (including ${\mathcal O}_F$ and $\overline{\mathbb F}_p$), $V$, $B$ and $\chi$, arguing geometrically on the Bruhat Tits building we give a sufficient criterion for $M_{\chi}(V)$ to be $B$-free and to admit a $\widetilde{G}$-equivariant resolution by a Koszul complex built from finitely many copies of ${\rm ind}_{\widetilde{K}Z}^{\widetilde{G}}(V)$. This criterion is the exactness of certain fairly small and explicit ${\mathfrak N}$-equivariant $R$-module complexes, where ${\mathfrak N}$ is the group of ${\mathcal O}_F$-valued points of the unipotent radical of a Borel subgroup in $\widetilde{G}$. We verify it if $F={\mathbb Q}_p$ and if $V$ is an irreducible $\overline{\mathbb F}_p[\widetilde{K}]$-representation with highest weight in the (closed) bottom $p$-alcove, or a lift of it to ${\mathcal O}_F$. We use this to construct $p$-adic integral structures in certain locally algebraic representations of $\widetilde{G}$.