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Iwahori–Hecke algebras are Gorenstein

E-ISSN： 1475-3030|13|4|753-809

ISSN： 1474-7480

Source： Journal of the Institute of Mathematics of Jussieu, Vol.13, Iss.4, 2014-10, pp. : 753-809

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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 } $ .