-
On the universal module of p-adic spherical Hecke algebras
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 136, Number 3, June 2014
- pp. 599-652
- 10.1353/ajm.2014.0019
- Article
- Additional Information
- Purchase/rental options available:
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 $\chi:{\frak
H}_V(\widetilde{G},\widetilde{K})\rightarrow B$ of the spherical Hecke
algebra ${\frak H}_V(\widetilde{G},\widetilde{K})={\rm
End}_{R[\widetilde{G}]} {\rm ind}_{\widetilde{K}}^{\widetilde{G}}(V)$ we
consider the specialization $$ M_{\chi}(V)={\rm
ind}_{\widetilde{K}}^{\widetilde{G}}V\otimes_{{\frak
H}_V(\widetilde{G},\widetilde{K}),\chi}B $$ of the universal ${\frak
H}_V(\widetilde{G},\widetilde{K})$-module ${\rm
ind}_{\widetilde{K}}^{\widetilde{G}}(V)$. For large classes of $R$
(including $\cal{O}_F$ and $\overline{\Bbb{F}}_p$), $V$, $B$ 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
${\frak N}$-equivariant $R$-module complexes, where ${\frak N}$ is the
group of $\cal{O}_F$-valued points of the unipotent radical of a Borel
subgroup in $\widetilde{G}$. We verify it if $F={\Bbb Q}_p$ and if $V$ is
an irreducible $\overline{\Bbb{F}}_p[\widetilde{K}]$-representation with
highest weight in the (closed) bottom $p$-alcove, or a lift of it to
$\cal{O}_F$. We use this to construct $p$-adic integral structures in
certain locally algebraic representations of $\widetilde{G}$.