Orbits of strongly solvable spherical subgroups on the flag variety

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Abstract

Let G be a connected reductive complex algebraic group and B a Borel subgroup of G. We consider a subgroup \(H \subset B\) which acts with finitely many orbits on the flag variety G / B, and we classify the H-orbits in G / B in terms of suitable root systems. As well, we study the Weyl group action defined by Knop on the set of H-orbits in G / B, and we give a combinatorial model for this action in terms of weight polytopes.

Keywords

Strongly solvable spherical subgroups Orbits of a Borel subgroup Weight polytopes 

Mathematics Subject Classification

14M10 14M27 

Notes

Acknowledgements

We thank P. Bravi, M. Brion, F. Knop and A. Maffei for useful conversations on the subject, and especially R.S. Avdeev for numerous remarks and suggestions on previous versions of the paper which led to significant improvements. This work originated during a stay of the first named author in Friedrich-Alexander-Universität Erlangen-Nürnberg during the fall of 2012 partially supported by a DAAD fellowship, and he is grateful to F. Knop and to the Emmy Noether Zentrum for hospitality. Both the authors were partially supported by the DFG Schwerpunktprogramm 1388—Darstellungstheorie.

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Authors and Affiliations

  1. 1.Scuola Normale SuperiorePisaItaly
  2. 2.Dipartimento di Matematica SapienzaUniversità di RomaRomaItaly

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