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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Weight elements of Chevalley groups

Author: N. A. Vavilov
Translated by: the author
Original publication: Algebra i Analiz, tom 20 (2008), nomer 1.
Journal: St. Petersburg Math. J. 20 (2009), 23-57
MSC (2000): Primary 20G15
Published electronically: November 13, 2008
MathSciNet review: 2411968
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Abstract: The paper is devoted to a detailed study of some remarkable semisimple elements of (extended) Chevalley groups that are diagonalizable over the ground field -- the weight elements. These are the conjugates of certain semisimple elements $ h_{\omega}(\varepsilon)$ of extended Chevalley groups $ \overline{G}=\overline{G}(\Phi,K)$, where $ \omega$ is a weight of the dual root system $ \Phi^{\vee}$ and $ \varepsilon\in K^*$. In the adjoint case the $ h_{\omega}(\varepsilon)$'s were defined by Chevalley himself and in the simply connected case they were constructed by Berman and Moody. The conjugates of $ h_{\omega}(\varepsilon)$ are called weight elements of type $ \omega$. Various constructions of weight elements are discussed in the paper, in particular, their action in irreducible rational representations and weight elements induced on a regularly embedded Chevalley subgroup by the conjugation action of a larger Chevalley group. It is proved that for a given $ x\in\overline{G}$ all elements $ x(\varepsilon)=xh_{\omega}(\varepsilon)x^{-1}$, $ \varepsilon\in K^*$, apart maybe from a finite number of them, lie in the same Bruhat coset $ \overline{B}w\overline{B}$, where $ w$ is an involution of the Weyl group $ W=W(\Phi)$. The elements $ h_{\omega}(\varepsilon)$ are particularly important when $ \omega=\varpi_{i}$ is a microweight of $ \Phi^{\vee}$. The main result of the paper is a calculation of the factors of the Bruhat decomposition of microweight elements $ x(\varepsilon)$ for the case where $ \omega=\varpi_{i}$. It turns out that all nontrivial $ x(\varepsilon)$'s lie in the same Bruhat coset $ \overline{B}w\overline B$, where $ w$ is a product of reflections in pairwise strictly orthogonal roots $ \gamma_1,\ldots,\gamma_{r+s}$. Moreover, if among these roots $ r$ are long and $ s$ are short, then $ r+2s$ does not exceed the width of the unipotent radical of the $ i$th maximal parabolic subgroup in $ \overline G$. A version of this result was first announced in a paper by the author in Soviet Mathematics: Doklady in 1988. From a technical viewpoint, this amounts to the determination of Borel orbits of a Levi factor of a parabolic subgroup with Abelian unipotent radical and generalizes some results of Richardson, Röhrle, and Steinberg. These results are instrumental in the description of overgroups of a split maximal torus and in the recent papers by the author and V. Nesterov on the geometry of tori.

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Additional Information

N. A. Vavilov
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskiĭ Prospekt 28, Staryĭ Peterhof, St. Petersburg 198504, Russia

Keywords: Chevalley groups, semisimple elements, Bruhat decomposition, microweights, Borel orbits, parabolic subgroups with Abelian unipotent radical
Received by editor(s): November 8, 2006
Published electronically: November 13, 2008
Article copyright: © Copyright 2008 American Mathematical Society