Transformation Groups

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




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.


Weyl Group Simple Root Borel Subgroup Classi Cation Bruhat Order 
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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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