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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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