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Hessenberg varieties, Slodowy slices, and integrable systems

  • Hiraku Abe
  • Peter CrooksEmail author
Article
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Abstract

This work is intended to contextualize and enhance certain well-studied relationships between Hessenberg varieties and the Toda lattice, thereby building on the results of Kostant, Peterson, and others. One such relationship is the fact that every Lagrangian leaf in the Toda lattice is compactified by a suitable choice of Hessenberg variety. It is then natural to imagine the Toda lattice as extending to an appropriate union of Hessenberg varieties. We fix a simply-connected complex semisimple linear algebraic group G and restrict our attention to a particular family of Hessenberg varieties, a family that includes the Peterson variety and all Toda leaf compactifications. The total space of this family, \(X(H_0)\), is shown to be a Poisson variety with a completely integrable system defined in terms of Mishchenko–Fomenko polynomials. This leads to a natural embedding of completely integrable systems from the Toda lattice to \(X(H_0)\). We also show \(X(H_0)\) to have an open dense symplectic leaf isomorphic to \(G/Z \times S_{\text {reg}}\), where Z is the centre of G and \(S_{\text {reg}}\) is a regular Slodowy slice in the Lie algebra of G. This allows us to invoke results about integrable systems on \(G\times S_{\text {reg}}\), as developed by Rayan and the second author. Lastly, we witness some implications of our work for the geometry of regular Hessenberg varieties.

Keywords

Hessenberg variety Integrable system Slodowy slice Toda lattice 

Mathematics Subject Classification

Primary 14L30 Secondary 17B80 53D20 
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Notes

Acknowledgements

We wish to recognize Ana Bălibanu, Megumi Harada, Takashi Otofuji, and Steven Rayan for helpful conversations. The first author is grateful to Mikiya Masuda for his support and encouragement. He is supported in part by a JSPS Research Fellowship for Young Scientists (Postdoctoral Fellow): 16J04761 and a JSPS Grant-in-Aid for Early-Career Scientists: 18K13413. The second author gratefully acknowledges support from the Natural Sciences and Engineering Research Council of Canada [PDF–516638–2018].

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Osaka City University Advanced Mathematical InstituteOsakaJapan
  2. 2.Department of MathematicsNortheastern UniversityBostonUSA

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