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Pro-$\lowercase p$ Iwahori–Hecke algebras are Gorenstein

Part of: Homological methods Representation theory of groups

Published online by Cambridge University Press:  28 November 2013

Rachel Ollivier  and
Peter Schneider
Rachel Ollivier
Affiliation:
Columbia University, Department of Mathematics, 2990 Broadway, New York, NY 10027, USA (ollivier@math.columbia.edu)
Peter Schneider
Affiliation:
Universität Münster, Mathematisches Institut, Einsteinstr. 62, 48291 Münster, Germany (pschnei@uni-muenster.de)
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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 } $.

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).

Keywords

pro-p Iwahori–Hecke algebra; Gorenstein ring; cohomological dimension; duality

MSC classification

Secondary: 20C08: Hecke algebras and their representations 16E65: Homological conditions on rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.) 16E10: Homological dimension
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 13 , Issue 4 , October 2014 , pp. 753 - 809
Copyright
©Cambridge University Press 2013 

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