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- W2949554932 abstract "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 $$M_{chi}(V)={rm ind}_{widetilde{K}}^{widetilde{G}}Votimes_{{mathfrak H}_V(widetilde{G},widetilde{K}),chi}B$$ 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 $widetilde{G}$." @default.
- W2949554932 created "2019-06-27" @default.
- W2949554932 creator A5040373321 @default.
- W2949554932 date "2014-08-14" @default.
- W2949554932 modified "2023-09-27" @default.
- W2949554932 title "On the universal module of $p$-adic spherical Hecke algebras" @default.
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- W2949554932 hasPublicationYear "2014" @default.
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