Support varieties for modules over Chevalley groups and classical Lie algebras

Carlson, Jon; Lin, Zongzhu; Nakano, Daniel

doi:10.1090/S0002-9947-07-04175-X

Support varieties for modules over Chevalley groups and classical Lie algebras
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by Jon F. Carlson, Zongzhu Lin and Daniel K. Nakano PDF

Trans. Amer. Math. Soc. 360 (2008), 1879-1906 Request permission

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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