Given a semisimple complex linear algebraic group and a lower ideal I in positive roots of G, three objects arise: the ideal arrangement , the regular nilpotent Hessenberg variety , and the regular semisimple Hessenberg variety . We show that a certain graded ring derived from the logarithmic derivation module of is isomorphic to and , the invariants in 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 .
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 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 in types B, C, and G. Such a presentation was already known in type A and when is the Peterson variety. Moreover, we find the volume polynomial of and see that the hard Lefschetz property and the Hodge–Riemann relations hold for , 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.
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.
 H. Abe, L. DeDieu, F. Galetto and M. Harada, Geometry of Hessenberg varieties with applications to Newton–Okounkov bodies, Selecta Math. (N. S.) 24 (2018), no. 3, 2129–2163. 10.1007/s00029-018-0405-3Search in Google Scholar
 H. Abe, M. Harada, T. Horiguchi and M. Masuda, The cohomology rings of regular nilpotent Hessenberg varieties in Lie type A, Int. Math. Res. Not. IMRN (2017), 10.1093/imrn/rnx275. 10.1093/imrn/rnx275Search in Google Scholar
 T. Abe, M. Barakat, M. Cuntz, T. Hoge and H. Terao, The freeness of ideal subarrangements of Weyl arrangements, J. Eur. Math. Soc. (JEMS) 18 (2016), no. 6, 1339–1348. 10.4171/JEMS/615Search in Google Scholar
 A. Białynicki-Birula, Some properties of the decompositions of algebraic varieties determined by actions of a torus, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 24 (1976), no. 9, 667–674. Search in Google Scholar
 P. Brosnan and T. Y. Chow, Unit interval orders and the dot action on the cohomology of regular semisimple Hessenberg varieties, Adv. Math. 329 (2018), 955–1001. 10.1016/j.aim.2018.02.020Search in Google Scholar
 J. B. Carrell, The Bruhat graph of a Coxeter group, a conjecture of Deodhar, and rational smoothness of Schubert varieties, Algebraic groups and their generalizations: Classical methods (University Park 1991), Proc. Sympos. Pure Math. 56, American Mathematical Society, Providence (1994), 53–61. 10.1090/pspum/056.1/1278700Search in Google Scholar
 F. De Mari and M. A. Shayman, Generalized Eulerian numbers and the topology of the Hessenberg variety of a matrix, Acta Appl. Math. 12 (1988), no. 3, 213–235. 10.1007/BF00046881Search in Google Scholar
 W. Fulton, Equivariant cohomology in algebraic geometry. Lecture three: More basics, first examples, Eilenberg Lectures, Columbia University, 2007, https://people.math.osu.edu/anderson.2804/eilenberg/lecture3.pdf. Search in Google Scholar
 M. Goresky, R. Kottwitz and R. MacPherson, Equivariant cohomology, Koszul duality, and the localization theorem, Invent. Math. 131 (1998), no. 1, 25–83. 10.1007/s002220050197Search in Google Scholar
 M. Harada, T. Horiguchi and M. Masuda, The equivariant cohomology rings of Peterson varieties in all Lie types, Canad. Math. Bull. 58 (2015), no. 1, 80–90. 10.4153/CMB-2014-048-0Search in Google Scholar
 T. Harima, T. Maeno, H. Morita, Y. Numata, A. Wachi and J. Watanabe, The Lefschetz properties, Lecture Notes in Math. 2080, Springer, Heidelberg 2013. 10.1007/978-3-642-38206-2Search in Google Scholar
 J. E. Humphreys, Conjugacy classes in semisimple algebraic groups, Math. Surveys Monogr. 43, American Mathematical Society, Providence 1995. Search in Google Scholar
 K. Kaveh, Note on cohomology rings of spherical varieties and volume polynomial, J. Lie Theory 21 (2011), no. 2, 263–283. Search in Google Scholar
 K. Kawakubo, Equivariant Riemann–Roch theorems, localization and formal group law, Osaka J. Math. 17 (1980), no. 3, 531–571. Search in Google Scholar
 G. Malle and D. Testerman, Linear algebraic groups and finite groups of Lie type, Cambridge Stud. Adv. Math. 133, Cambridge University Press, Cambridge 2011. 10.1017/CBO9780511994777Search in Google Scholar
 K. Saito, On the uniformization of complements of discriminant loci, RIMS Kôkyûroku 287 (1977), 117–137. Search in Google Scholar
 K. Saito, Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), no. 2, 265–291. Search in Google Scholar
 K. Saito, Uniformization of the orbifold of a finite reflection group, Frobenius manifolds, Aspects Math. E36, Friedrich Vieweg, Wiesbaden (2004), 265–320. 10.1007/978-3-322-80236-1_11Search in Google Scholar
 T. Shoji, Geometry of orbits and Springer correspondence, Orbites unipotentes et représentations. I. Groupes finis et algèbres de Hecke, Astérisque 168, Société Mathématique de France, Paris (1988), 61–140. Search in Google Scholar
 R. P. Stanley, Combinatorics and commutative algebra, 2nd ed., Progr. Math. 41, Birkhäuser, Boston 1996. Search in Google Scholar
 N. Teff, Representations on Hessenberg varieties and Young’s rule, 23rd international conference on formal power series and algebraic combinatorics—FPSAC 2011, Discrete Math. Theor. Comput. Sci. Proc., The Association, Nancy (2011), 903–914. 10.46298/dmtcs.2963Search in Google Scholar
 H. Terao, Arrangements of hyperplanes and their freeness. I, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), no. 2, 293–312. Search in Google Scholar
 J. S. Tymoczko, Permutation actions on equivariant cohomology of flag varieties, Toric topology, Contemp. Math. 460, American Mathematical Society, Providence (2008), 365–384. 10.1090/conm/460/09030Search in Google Scholar
 J. S. Tymoczko, The geometry and combinatorics of Springer fibers, Around Langlands correspondences, Contemp. Math. 691, American Mathematical Society, Providence (2017), 359–376. 10.1090/conm/691/13903Search in Google Scholar
© 2020 Walter de Gruyter GmbH, Berlin/Boston