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Support varieties for modules over Chevalley groups and classical Lie algebras

Author(s): Jon F. Carlson; Zongzhu Lin; Daniel K. Nakano
Journal: Trans. Amer. Math. Soc. 360 (2008), 1879-1906.
MSC (2000): Primary 17B55, 20Gxx; Secondary 17B50
Posted: November 9, 2007
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Abstract: Let $ G$ be a connected reductive algebraic group over an algebraically closed field of characteristic $ p>0$, $ G_{1}$ be the first Frobenius kernel, and $ G({\mathbb{F}}_{p})$ be the corresponding finite Chevalley group. Let $ M$ be a rational $ G$-module. In this paper we relate the support variety of $ M$ over the first Frobenius kernel with the support variety of $ M$ over the group algebra $ kG({\mathbb{F}}_{p})$. This provides an answer to a question of Parshall. Applications of our new techniques are presented, which allow us to extend results of Alperin-Mason and Janiszczak-Jantzen, and to calculate the dimensions of support varieties for finite Chevalley groups.

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Additional Information:

Jon F. Carlson
Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602
Email: jfc@math.uga.edu

Zongzhu Lin
Affiliation: Department of Mathematics, Kansas State University, Manhattan, Kansas 66506
Email: zlin@math.ksu.edu

Daniel K. Nakano
Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602
Email: nakano@math.uga.edu

DOI: 10.1090/S0002-9947-07-04175-X
PII: S 0002-9947(07)04175-X
Received by editor(s): April 21, 2005
Received by editor(s) in revised form: October 17, 2005
Posted: November 9, 2007
Additional Notes: The research of the first author was supported in part by NSF grant DMS-0100662 and DMS-0401431
The research of the second author was supported in part by NSF grant DMS-0200673
The research of the third author was supported in part by NSF grant DMS-0102225 and DMS-0400548
Copyright of article: Copyright 2007, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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