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Transformation Groups

, Volume 22, Issue 2, pp 503–524 | Cite as

ON THE ORBITS OF A BOREL SUBGROUP IN ABELIAN IDEALS

  • DMITRI PANYUSHEV
Article

Abstract

Let B be a Borel subgroup of a semisimple algebraic group G, and let \( \mathfrak{a} \) be an abelian ideal of \( \mathfrak{b} \) = Lie (B). The ideal \( \mathfrak{a} \) is determined by a certain subset \( {\varDelta}_{\mathfrak{a}} \) of positive roots, and using \( {\varDelta}_{\mathfrak{a}} \) we give an explicit classification of the B-orbits in \( \mathfrak{a} \) and \( \mathfrak{a} \) *. Our description visibly demonstrates that there are finitely many B-orbits in both cases. Then we describe the Pyasetskii correspondence between the B-orbits in \( \mathfrak{a} \) and \( \mathfrak{a} \) * and the invariant algebras k[\( \mathfrak{a} \)] U and k[\( \mathfrak{a} \)*] U where U = (B, B). As an application, the number of B-orbits in the abelian nilradicals is computed. We also discuss related results of A. Melnikov and others for classical groups and state a general conjecture on the closure and dimension of the B-orbits in the abelian nilradicals, which exploits a relationship between between B-orbits and involutions in the Weyl group.

Keywords

Weyl Group Simple Root Borel Subgroup Classi Cation Bruhat Order 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Institute for Information Transmission ProblemsRussian Academy of SciencesMoscowRussia

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