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FROM THE FUNCTION-SHEAF DICTIONARY TO QUASICHARACTERS OF $p$-ADIC TORI

Part of: Algebraic groups Lie groups (Co)homology theory

Published online by Cambridge University Press:  13 October 2015

Clifton Cunningham  and
David Roe
Clifton Cunningham
Affiliation:
Department of Mathematics and Statistics, University of Calgary, 2500 University Drive Northwest, Calgary, Alberta, CanadaT2N 1N4 (cunning@math.ucalgary.ca)
David Roe
Affiliation:
Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, British Columbia, CanadaV6T 1Z2 (roed.math@gmail.com)
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Abstract

We consider the rigid monoidal category of character sheaves on a smooth commutative group scheme $G$ over a finite field $k$, and expand the scope of the function-sheaf dictionary from connected commutative algebraic groups to this setting. We find the group of isomorphism classes of character sheaves on $G$, and show that it is an extension of the group of characters of $G(k)$ by a cohomology group determined by the component group scheme of $G$. We also classify all morphisms in the category character sheaves on $G$. As an application, we study character sheaves on Greenberg transforms of locally finite type Néron models of algebraic tori over local fields. This provides a geometrization of quasicharacters of $p$-adic tori.

Keywords

character sheavesp-adic toriNéron modelsGreenberg functorgeometrizationquasicharacter sheaves

MSC classification

Primary: 14F05: Sheaves, derived categories of sheaves and related constructions
Secondary: 14L15: Group schemes 22E50: Representations of Lie and linear algebraic groups over local fields
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 17 , Issue 1 , February 2018 , pp. 1 - 37
Copyright
© Cambridge University Press 2015 

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