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Generically multiple transitive algebraic group actions. (English) Zbl 1135.14038
Mehta, V. B. (ed.), Algebraic groups and homogeneous spaces. Proceedings of the international colloquium, Mumbai, India, January 6–14, 2004. New Delhi: Narosa Publishing House/Published for the Tata Institute of Fundamental Research (ISBN 978-81-7319-802-1/hbk). Studies in Mathematics. Tata Institute of Fundamental Research 19, 481-523 (2007).
This work was motivated by the following question of M. Burger: for which complex connected simple linear algebraic groups $$S$$ with proper maximal parabolic subgroup $$P$$ is there an open $$S$$ orbit on $$S/P \times S/P \times S/P$$? To address this question, the author introduces the notion of a generically transitive action. Precisely, the action of a connected algebraic group $$G$$ on an irreducible algebraic variety $$X$$ is called generically $$n$$-transitive if the diagonal action of $$G$$ on $$X^n$$ is locally transitive (i.e., admits an open $$G$$-orbit). The generic transitivity degree (gtd) of an action is the supremum over all such $$n$$, and the generic transitivity degree of a group $$G$$ (gtd($$G$$)) is defined to be the supremum over all non-trivial $$G$$-actions on irreducible algebraic varieties. For any non-trivial $$G$$, gtd($$G$$) is a well-defined integer between one and the dimension of $$G$$ (inclusively).
The goal of this paper is to compute the generic transitivity degree for connected linear algebraic groups over the complex numbers. A number of general and specific results are obtained, and we mention here only a few of the highlights. If $$G$$ is reductive, then it is isogenous to the product of a torus and a finite number of simple algebraic groups. It is shown that gtd($$G$$) is the maximum of the gtds of the simple factors. For all simple $$G$$, an explicit computation of gtd($$G$$) is made. This is done in part by reducing to a maximal Levi subgroup and computing the gtd of the adjoint action of the Levi subgroup on the Lie algebra of the unipotent radical of a corresponding parabolic subgroup. Computations are also made for solvable and nilpotent groups.
For a connected nonabelian reductive group $$G$$, it is also shown that there exists a proper maximal parabolic subgroup $$P$$ of $$G$$ such that gtd($$G$$) can be realized as the gtd of the natural action of $$G$$ on $$G/P$$. For a simple $$G$$, explicit computations are made of the gtd of the natural action of $$G$$ on each $$G/P$$ for a standard maximal parabolic subgroup $$P$$. In particular, this information allows the author to answer Burger’s original question. As an application of these computations, the author demonstrates how the gtd of a connected simply connected semisimple algebraic group $$G$$ acting on certain $$G/P$$ (for a parabolic $$P$$) gives information on the $$G$$-fixed points of the tensor product of certain simple $$G$$-modules. This topic is explored further in another work of the author [J. Algebra 313, 392–416 (2007; Zbl 1158.20021)].
For the entire collection see [Zbl 1128.14002].

##### MSC:
 14L30 Group actions on varieties or schemes (quotients) 14M17 Homogeneous spaces and generalizations 20G05 Representation theory for linear algebraic groups
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