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

This article is cited in 15 scientific papers (total in 15 papers)

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

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MSC: 20G15
Received: 06.11.2006

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
\by N.~Vavilov
\paper Weight elements of Chevalley groups
\jour Algebra i Analiz
\yr 2008
\vol 20
\issue 1
\pages 34--85
\jour St. Petersburg Math. J.
\yr 2009
\vol 20
\issue 1
\pages 23--57

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    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  mathnet  crossref  mathscinet  zmath  isi
    2. N. A. Vavilov, I. M. Pevzner, “Triples of long root subgroups”, J. Math. Sci. (N. Y.), 147:5 (2007), 7005–7020  mathnet  crossref  mathscinet  elib  elib
    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  crossref  mathscinet  zmath  isi  elib
    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  mathnet  crossref  mathscinet  zmath  isi
    5. N. A. Vavilov, V. V. Nesterov, “Geometriya mikrovesovykh torov”, Vladikavk. matem. zhurn., 10:1 (2008), 10–23  mathnet  mathscinet  elib
    6. Vavilov N.A., Stepanov A.V., “Nadgruppy poluprostykh grupp”, Vestn. Samarskogo gos. un-ta. Estestvennonauchn. ser., 2008, no. 3, 51–95  mathscinet  zmath
    7. Hazrat R., Petrov V., Vavilov N., “Relative subgroups in Chevalley groups”, J. K-Theory, 5:3 (2010), 603–618  crossref  mathscinet  zmath  isi  elib  scopus
    8. N. A. Vavilov, “Stroenie izotropnykh reduktivnykh grupp”, Tr. In-ta matem., 18:1 (2010), 15–27  mathnet
    9. N. A. Vavilov, A. A. Semenov, “Long root tori in Chevalley groups”, St. Petersburg Math. J., 24:3 (2013), 387–430  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    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  mathnet  crossref  mathscinet
    11. Hazrat R. Vavilov N. Zhang Z., “Relative Commutator Calculus in Chevalley Groups”, J. Algebra, 385 (2013), 262–293  crossref  mathscinet  zmath  isi  elib  scopus
    12. J. Math. Sci. (N. Y.), 209:6 (2015), 922–934  mathnet  crossref
    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  mathnet  crossref  mathscinet  isi  elib
    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  mathnet  crossref  mathscinet
    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  mathnet  crossref  crossref  isi  elib
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