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 Algebra i Analiz, 2008, Volume 20, Issue 1, Pages 34–85 (Mi aa497)

Research Papers

Weight elements of Chevalley groups

N. Vavilov

St. Petersburg State University, Department of Mathematics and Mechanics

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 Math. 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 description of overgroups of a split maximal torus and in the recent papers by the author and V. Nesterov on the geometry of tori.

Keywords: Chevalley groups, semisimple elements, Bruhat decomposition, microweights, Borel orbits, parabolic subgroups with Abelian unipotent radical

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English version:
St. Petersburg Mathematical Journal, 2009, 20:1, 23–57

Bibliographic databases:

MSC: 20G15

Citation: N. Vavilov, “Weight elements of Chevalley groups”, Algebra i Analiz, 20:1 (2008), 34–85; St. Petersburg Math. J., 20:1 (2009), 23–57

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. N. A. Vavilov, A. Yu. Luzgarev, “The normalizer of Chevalley groups of type $\mathrm{E}_6$”, St. Petersburg Math. J., 19:5 (2008), 699–718
2. N. A. Vavilov, I. M. Pevzner, “Triples of long root subgroups”, J. Math. Sci. (N. Y.), 147:5 (2007), 7005–7020
3. Vavilov N., “An $A_3$-proof of structure theorems for Chevalley groups of types $E_6$ and $E_7$”, Internat. J. Algebra Comput., 17:5-6 (2007), 1283–1298
4. A. Yu. Luzgarev, “Overgroups of $\mathrm{F}_4$ in $\mathrm{E}_6$ over commutative rings”, St. Petersburg Math. J., 20:6 (2009), 955–981
5. N. A. Vavilov, V. V. Nesterov, “Geometriya mikrovesovykh torov”, Vladikavk. matem. zhurn., 10:1 (2008), 10–23
6. Vavilov N.A., Stepanov A.V., “Nadgruppy poluprostykh grupp”, Vestn. Samarskogo gos. un-ta. Estestvennonauchn. ser., 2008, no. 3, 51–95
7. Hazrat R., Petrov V., Vavilov N., “Relative subgroups in Chevalley groups”, J. K-Theory, 5:3 (2010), 603–618
8. N. A. Vavilov, “Stroenie izotropnykh reduktivnykh grupp”, Tr. In-ta matem., 18:1 (2010), 15–27
9. N. A. Vavilov, A. A. Semenov, “Long root tori in Chevalley groups”, St. Petersburg Math. J., 24:3 (2013), 387–430
10. N. A. Vavilov, A. V. Shchegolev, “Overgroups of subsystem subgroups in exceptional groups: levels”, J. Math. Sci. (N. Y.), 192:2 (2013), 164–195
11. Hazrat R. Vavilov N. Zhang Z., “Relative Commutator Calculus in Chevalley Groups”, J. Algebra, 385 (2013), 262–293
12. J. Math. Sci. (N. Y.), 209:6 (2015), 922–934
13. N. A. Vavilov, A. Yu. Luzgarev, “Normaliser of the Chevalley group of type $\mathrm E_7$”, St. Petersburg Math. J., 27:6 (2016), 899–921
14. V. V. Nesterov, “Reduction theorems for triples of short root subgroups in Chevalley groups”, J. Math. Sci. (N. Y.), 222:4 (2017), 437–452
15. V. A. Koibaev, S. K. Kuklina, A. O. Likhacheva, Ya. N. Nuzhin, “Subgroups, of Chevalley Groups over a Locally Finite Field, Defined by a Family of Additive Subgroups”, Math. Notes, 102:6 (2017), 792–798
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