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Restrictions to G(𝔽p) and G(r) of rational G-modules

Part of: Representation theory of groups Linear algebraic groups and related topics

Published online by Cambridge University Press:  26 August 2011

Eric M. Friedlander
Eric M. Friedlander*
Affiliation:
Department of Mathematics, University of Southern California, Los Angeles, CA 90089, USA (email: ericmf@usc.edu)
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Abstract

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We fix a prime p and consider a connected reductive algebraic group G over a perfect field k which is defined over 𝔽p. Let M be a finite-dimensional rational G-module M, a comodule for k[G]. We seek to somewhat unravel the relationship between the restriction of M to the finite Chevalley subgroup G(𝔽p)⊂G and the family of restrictions of M to Frobenius kernels G(r) ⊂G. In particular, we confront the conundrum that if M is the Frobenius twist of a rational G-module N,M=N(1), then the restrictions of M and N to G(𝔽p) are equal whereas the restriction of M to G(1) is trivial. Our analysis enables us to compare support varieties (and the finer non-maximal support varieties) for G(𝔽p) and G(r) of a rational G-module M where the choice of r depends explicitly on M.

Keywords

1-parameter subgroupsFrobenius kernelsp-nilpotent degree

MSC classification

Secondary: 20C20: Modular representations and characters 20G40: Linear algebraic groups over finite fields
Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 6 , November 2011 , pp. 1955 - 1978
Copyright
Copyright Š Foundation Compositio Mathematica 2011

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