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- W2963428690 abstract "Abstract Let $mathfrak{F}$ be a locally compact nonarchimedean field with residue characteristic $p$ , and let $mathrm{G} $ be the group of $mathfrak{F}$ -rational points of a connected split reductive group over $mathfrak{F}$ . For $k$ an arbitrary field of any characteristic, we study the homological properties of the Iwahori–Hecke $k$ -algebra ${mathrm{H} }^{prime } $ and of the pro- $p$ Iwahori–Hecke $k$ -algebra $mathrm{H} $ of $mathrm{G} $ . We prove that both of these algebras are Gorenstein rings with self-injective dimension bounded above by the rank of $mathrm{G} $ . If $mathrm{G} $ is semisimple, we also show that this upper bound is sharp, that both $mathrm{H} $ and ${mathrm{H} }^{prime } $ are Auslander–Gorenstein, and that there is a duality functor on the finite length modules of $mathrm{H} $ (respectively ${mathrm{H} }^{prime } $ ). We obtain the analogous Gorenstein and Auslander–Gorenstein properties for the graded rings associated to $mathrm{H} $ and ${mathrm{H} }^{prime } $ . When $k$ has characteristic $p$ , we prove that in ‘most’ cases $mathrm{H} $ and ${mathrm{H} }^{prime } $ have infinite global dimension. In particular, we deduce that the category of smooth $k$ -representations of $mathrm{G} = {mathrm{PGL} }_{2} ({ mathbb{Q} }_{p} )$ generated by their invariant vectors under the pro- $p$ Iwahori subgroup has infinite global dimension (at least if $k$ is algebraically closed)." @default.
- W2963428690 created "2019-07-30" @default.
- W2963428690 creator A5027666103 @default.
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- W2963428690 date "2013-11-28" @default.
- W2963428690 modified "2023-10-06" @default.
- W2963428690 title "Pro- Iwahori–Hecke algebras are Gorenstein" @default.
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- W2963428690 doi "https://doi.org/10.1017/s1474748013000303" @default.
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