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Support varieties for rational representations

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

Published online by Cambridge University Press:  13 November 2014

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

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We introduce support varieties for rational representations of a linear algebraic group $G$ of exponential type over an algebraically closed field $k$ of characteristic $p>0$. These varieties are closed subspaces of the space $V(G)$ of all 1-parameter subgroups of $G$. The functor $M\mapsto V(G)_{M}$ satisfies many of the standard properties of support varieties satisfied by finite groups and other finite group schemes. Furthermore, there is a close relationship between $V(G)_{M}$ and the family of support varieties $V_{r}(G)_{M}$ obtained by restricting the $G$ action to Frobenius kernels $G_{(r)}\subset G$. These support varieties seem particularly appropriate for the investigation of infinite-dimensional rational $G$-modules.

Keywords

support variety1-parameter subgroup

MSC classification

Primary: 20G05: Representation theory 20C20: Modular representations and characters 20G10: Cohomology theory
Type
Research Article
Information
Compositio Mathematica , Volume 151 , Issue 4 , April 2015 , pp. 765 - 792
Copyright
© The Author 2014 

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