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Central elements in affine mod p Hecke algebras via perverse F_p-sheaves

Cass, Robert (2021) Central elements in affine mod p Hecke algebras via perverse F_p-sheaves. Compositio Mathematica, 157 (10). pp. 2215-2241. ISSN 0010-437X. doi:10.1112/s0010437x2100751x.

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Let G be a split connected reductive group over a finite field of characteristic p>2 such that G_(der) is absolutely almost simple. We give a geometric construction of perverse F_p-sheaves on the Iwahori affine flag variety of G which are central with respect to the convolution product. We deduce an explicit formula for an isomorphism from the spherical mod p Hecke algebra to the center of the Iwahori mod p Hecke algebra. We also give a formula for the central integral Bernstein elements in the Iwahori mod p Hecke algebra. To accomplish these goals we construct a nearby cycles functor for perverse F_p-sheaves and we use Frobenius splitting techniques to prove some properties of this functor. We also prove that certain equal characteristic analogues of local models of Shimura varieties are strongly F-regular, and hence they are F-rational and have pseudo-rational singularities.

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Alternate Title:Central elements in affine mod p Hecke algebras via perverse Fp-sheaves
Additional Information:© 2021 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence. Received 15 June 2020, accepted in final form 12 April 2021. Published online by Cambridge University Press: 14 September 2021. It is a pleasure to thank M. Kisin, C. Pépin, and T. Schmidt for several discussions and comments on an earlier version of this paper. I thank the referee for their careful reading and helpful suggestions which improved the exposition and simplified some of the proofs. I also thank K. Koziol, R. Ollivier, M.-F. Vignéras, D. Yang, and X. Zhu for their interest and helpful conversations. Parts of this paper were written while the author visited the University of Paris 13 and the University of Rennes 1, and he would like to thank these institutions for their hospitality. This material is based upon work supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-1144152.
Funding AgencyGrant Number
NSF Graduate Research FellowshipDGE-1144152
Subject Keywords:Hecke algebras, local models, perverse sheaves, F-singularities
Issue or Number:10
Classification Code:2020 Mathematics Subject Classification: 14M15 (primary), 14F10, 20C08 (secondary)
Record Number:CaltechAUTHORS:20211021-164023037
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Official Citation:Cass, R. (2021). Central elements in affine mod p Hecke algebras via perverse F_p-sheaves. Compositio Mathematica, 157(10), 2215-2241. doi:10.1112/S0010437X2100751X
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:111572
Deposited By: Tony Diaz
Deposited On:22 Oct 2021 22:59
Last Modified:26 Oct 2021 19:18

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