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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

Long root tori in Chevalley groups


Authors: N. A. Vavilov and A. A. Semenov
Translated by: N. A. Vavilov
Original publication: Algebra i Analiz, tom 24 (2012), nomer 3.
Journal: St. Petersburg Math. J. 24 (2013), 387-430
MSC (2010): Primary 20G15, 20G40, 20G35
Published electronically: March 21, 2013
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Abstract: In this paper we study some remarkable semisimple elements of an (extended) Chevalley group that are diagonalizable over the ground field -- the `weight elements'. In particular, we calculate the Bruhat decomposition of microweight elements. Results of the present paper are crucial for the description of overgroups of split maximal tori in Chevalley groups.


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Additional Information

N. A. Vavilov
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Petrodvorets, St. Petersburg 198904, Russia
Email: nikolai-vavilov@yandex.ru

A. A. Semenov
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Petrodvorets, St. Petersburg 198904, Russia
Email: semenov@math.spbu.ru

DOI: http://dx.doi.org/10.1090/S1061-0022-2013-01245-3
PII: S 1061-0022(2013)01245-3
Keywords: Chevalley groups, semisimple root elements, Bruhat decomposition, Borel orbits, parabolic subgroups with extraspecial unipotent radical
Received by editor(s): September 9, 2011
Published electronically: March 21, 2013
Additional Notes: For the first author, the main motivation to finalize the present paper came from the research within the framework of the RFBR project 10-01-90016 “The study of the structure of forms of reductive groups, and behavior of small unipotent elements in representations of algebraic groups” (SPbGU). Apart from that, at the final stage his work was supported by the RFBR projects 09-01-00762 (Siberian Federal University), 09-01-00784 (POMI RAS), 09-01-00878 (SPbGU), 09-01-91333 (POMI RAS), 09-01-90304 (SPbGU), 10-01-92651 (SPbGU), and 11-01-00756 (RGPU). The work of both authors was supported by the Presidential Grant NSh-5282.2010.1 “Motives, cohomologies, algebraic groups, representations, reciprocity laws, lower and upper bounds of scheme complexity of Boolean functions” and by the State Financed Research Task 6.38.74.2011 at the Saint Petersburg State University “Structure theory and geometry of algebraic groups and their applications in representation theory and algebraic $K$-theory”
Dedicated: To Nikolaĭ Gordeev, a remarkable mathematician, a dear friend, and a generous colleague
Article copyright: © Copyright 2013 American Mathematical Society