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Hessenberg varieties and hyperplane arrangements

Takuro Abe, Tatsuya Horiguchi, Mikiya Masuda, Satoshi Murai and Takashi Sato

Abstract

Given a semisimple complex linear algebraic group G and a lower ideal I in positive roots of G, three objects arise: the ideal arrangement 𝒜I, the regular nilpotent Hessenberg variety Hess(N,I), and the regular semisimple Hessenberg variety Hess(S,I). We show that a certain graded ring derived from the logarithmic derivation module of 𝒜I is isomorphic to H*(Hess(N,I)) and H*(Hess(S,I))W, the invariants in H*(Hess(S,I)) under an action of the Weyl group W of G. This isomorphism is shown for general Lie type, and generalizes Borel’s celebrated theorem showing that the coinvariant algebra of W is isomorphic to the cohomology ring of the flag variety G/B.

This surprising connection between Hessenberg varieties and hyperplane arrangements enables us to produce a number of interesting consequences. For instance, the surjectivity of the restriction map H*(G/B)H*(Hess(N,I)) announced by Dale Peterson and an affirmative answer to a conjecture of Sommers and Tymoczko are immediate consequences. We also give an explicit ring presentation of H*(Hess(N,I)) in types B, C, and G. Such a presentation was already known in type A and when Hess(N,I) is the Peterson variety. Moreover, we find the volume polynomial of Hess(N,I) and see that the hard Lefschetz property and the Hodge–Riemann relations hold for Hess(N,I), despite the fact that it is a singular variety in general.

Funding source: Japan Society for the Promotion of Science

Award Identifier / Grant number: 16H03924

Award Identifier / Grant number: 15J09343

Award Identifier / Grant number: 16K05152

Funding statement: The first author is partially supported by JSPS Grant-in-Aid for Scientific Research (B) 16H03924. The second author was partially supported by JSPS Grant-in-Aid for JSPS Fellows 15J09343. The third author is partially supported by JSPS Grant-in-Aid for Scientific Research (C) 16K05152.

Acknowledgements

We are grateful to Hiraku Abe for fruitful discussions and comments on this paper and Naoki Fujita for his stimulating question. We are grateful to Hiroaki Terao for the discussion on the free basis for height subarrangements of type A. We are also grateful to Megumi Harada for her comments on the paper. Finally, we thank the referee for valuable and concrete suggestions to improve the paper.

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Received: 2018-04-14
Revised: 2018-11-15
Published Online: 2019-01-20
Published in Print: 2020-07-01

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