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International Mathematics Research Notices Advance Access published online on September 23, 2009
International Mathematics Research Notices, doi:10.1093/imrn/rnp145
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© The Author 2009. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org

Actions of the Derived Group of a Maximal Unipotent Subgroup on G-Varieties

Dmitri I. Panyushev

Independent University of Moscow, Bol’shoi Vlasevskii per. 11, 119002 Moscow, Russia

Correspondence: Correspondence to be sent to: panyush{at}mccme.ru

Let U be a maximal unipotent subgroup of a connected semisimple group G and U' the derived group of U. We study actions of U' on affine G-varieties. First, we consider the algebra of U' invariants on G/ U. We prove that Formula is a polynomial algebra of Krull dimension 2r, where r = rk(G). A related result is that, for any simple finite-dimensional G-module V, Formula is a cyclic U/ U'-module. Second, we study "symmetries" of Poincare series for U'-invariants on affine conical G-varieties. The results we obtain are very similar to those for the algebras of U-invariants. Third, we obtain a classification of simple G-modules V with polynomial algebras of U'-invariants (for G simple).

Communicated by Corrado De Concini


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