Acta Mathematica

Local Hodge theory of Soergel bimodules

Geordie Williamson

Full-text: Open access


We prove the local hard Lefschetz theorem and local Hodge–Riemann bilinear relations for Soergel bimodules. Using results of Soergel and Kübel, one may deduce an algebraic proof of the Jantzen conjectures. We observe that the Jantzen filtration may depend on the choice of non-dominant regular deformation direction.


Dedicated to Ben and Yeppie.

Article information

Acta Math., Volume 217, Number 2 (2016), 341-404.

Received: 29 October 2014
Revised: 5 May 2016
First available in Project Euclid: 17 August 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

2016 © Institut Mittag-Leffler


Williamson, Geordie. Local Hodge theory of Soergel bimodules. Acta Math. 217 (2016), no. 2, 341--404. doi:10.1007/s11511-017-0146-8.

Export citation


  • Barbasch D.: Filtrations on Verma modules. Ann. Sci. École Norm. Sup., 16, 489–494 (1983)
  • Beilinson, A. & Bernstein, J., A proof of Jantzen conjectures, in I. M. Gel’fand Seminar, Adv. Soviet Math., 16, pp. 1–50. Amer. Math. Soc., Providence, RI, 1993.
  • Beilinson A., Ginzburg V., Soergel W.: Koszul duality patterns in representation theory. J. Amer. Math. Soc., 9, 473–527 (1996)
  • Bernstein, J. & Lunts, V., Equivariant Sheaves and Functors. Lecture Notes in Mathematics, 1578. Springer, Berlin–Heidelberg, 1994.
  • Bosma W., Cannon J., Playoust C.: The Magma algebra system. I. The user language. J. Symbolic Comput., 24, 235–265 (1997)
  • Braden T., MacPherson R.: From moment graphs to intersection cohomology. Math. Ann., 321, 533–551 (2001)
  • Brion, M., Equivariant cohomology and equivariant intersection theory, in Representation Theories and Algebraic Geometry (Montreal, QC, 1997), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 514, pp. 1–37. Kluwer, Dordrecht, 1998.
  • de Cataldo M.A.A., Migliorini L.: The hard Lefschetz theorem and the topology of semismall maps. Ann. Sci. École Norm. Sup., 35, 759–772 (2002)
  • de Cataldo M.A.A., Migliorini L.: The Hodge theory of algebraic maps. Ann. Sci. École Norm. Sup., 38, 693–750 (2005)
  • Elias, B. & Khovanov, M., Diagrammatics for Soergel categories. Int. J. Math. Math. Sci., 2010 (2010), Art. ID 978635, 58 pp.
  • Elias B., Williamson G.: The Hodge theory of Soergel bimodules. Ann. of Math., 180, 1089–1136 (2014)
  • Elias B., Williamson G.: Soergel calculus. Represent. Theory, 20, 295–374 (2016)
  • Fiebig, P., Kazhdan–Lusztig combinatorics via sheaves on Bruhat graphs, in Algebraic and Geometric Combinatorics, Contemp. Math., 423, pp. 195–204. Amer. Math. Soc., Providence, RI, 2006.
  • Fiebig P.: The combinatorics of Coxeter categories. Trans. Amer. Math. Soc., 360, 4211–4233 (2008)
  • Fiebig P.: An upper bound on the exceptional characteristics for Lusztig’s character formula. J. Reine Angew. Math., 673, 1–31 (2012)
  • Fiebig, P. & Williamson, G., Parity sheaves, moment graphs and the p-smooth locus of Schubert varieties. Ann. Inst. Fourier (Grenoble), 64 (2014), 489–536.
  • Gabber O., Joseph A.: Towards the Kazhdan–Lusztig conjecture. Ann. Sci. École Norm. Sup., 14, 261–302 (1981)
  • He, X. & Williamson, G., Soergel calculus and Schubert calculus. To appear in Bull. Inst. Math. Acad. Sinica.
  • Jantzen, J. C., Moduln mit einem höchsten Gewicht. Lecture Notes in Mathematics, 750. Springer, Berlin–Heidelberg, 1979.
  • Juteau D., Williamson G.: Kumar’s criterion modulo p. Duke Math. J., 163, 2617–2638 (2014)
  • Kostant B., Kumar S.: The nil Hecke ring and cohomology of G/P for a Kac–Moody group G. Adv. in Math., 62, 187–237 (1986)
  • Kübel J.: From Jantzen to Andersen filtration via tilting equivalence. Math. Scand., 110, 161–180 (2012)
  • Kübel J.: Tilting modules in category O and sheaves on moment graphs. J. Algebra, 371, 559–576 (2012)
  • Kumar S.: The nil Hecke ring and singularity of Schubert varieties. Invent. Math., 123, 471–506 (1996)
  • Soergel, W., Kategorie O, perverse Garben und Moduln über den Koinvarianten zur Weylgruppe. J. Amer. Math. Soc., 3 (1990), 421–445.
  • Soergel W.: Character formulas for tilting modules over Kac–Moody algebras. Represent. Theory, 2, 432–448 (1998)
  • Soergel, W., Langlands’ philosophy and Koszul duality, in Algebra—Representation Theory (Constanta, 2000), NATO Sci. Ser. II Math. Phys. Chem., 28, pp. 379–414. Kluwer, Dordrecht, 2001.
  • Soergel W.: Kazhdan–Lusztig-Polynome und unzerlegbare Bimoduln über Polynomringen. J. Inst. Math. Jussieu, 6, 501–525 (2007)
  • Soergel W.: Andersen filtration and hard Lefschetz. Geom. Funct. Anal., 17, 2066–2089 (2008)