Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter April 1, 2022

Monodromic model for Khovanov–Rozansky homology

  • Roman Bezrukavnikov and Kostiantyn Tolmachov EMAIL logo


We describe a new geometric model for the Hochschild cohomology of Soergel bimodules based on the monodromic Hecke category studied earlier by the first author and Yun. Moreover, we identify the objects representing individual Hochschild cohomology groups (for the zero and the top degree cohomology this reduces to an earlier result of Gorsky, Hogancamp, Mellit and Nakagane). These objects turn out to be closely related to explicit character sheaves corresponding to exterior powers of the reflection representation of the Weyl group. Applying the described functors to the images of braids in the Hecke category of type A we obtain a geometric description for Khovanov–Rozansky knot homology, essentially different from the one considered earlier by Webster and Williamson.

Funding statement: The first author is partially supported by NSF grant DMS-2101507. This work was done while the second author was a Postdoctoral Fellow at the Perimeter Institute of Theoretical Physics. Perimeter Institute is supported in part by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Colleges and Universities.

A Examples

A.1 Type A1

We have R=¯[x], s(x)=-x. Write RR=¯[x,y],


We have


and a morphism of complexes Δs[1](-1)sK𝕊:

which gives a distinguished triangle


From the long exact sequence of cohomology we get


(note that Δs(-1) and s(1) are in the heart of the t-structure, by definition) and, finally,


A.2 Type A2

We omit most of the differentials from notations. Let


Write Bi=Bsi,Bij=BiBj,B121=Bs1s2s1=Bs2s1s2.

We have B1B2B1=B121B1,B2B1B2=B121B2.


We have maps


Taking the cone twice, we see that 0(w0K𝕊)=w0(3),-2(w0K𝕊)=Δw0(-1), and -1 is an extension of shifted Jucys–Murphy complexes




The terms appearing in the corresponding complexes are given their corresponding color.

We get


and w0-1 is an extension of Δ21Δ12(1) and Δ12(1).

Remark A.2.83.

Note that the grading shifts have signs that are opposite to those in Theorem 5.7.66, since the action of the Frobenius is inverted when comparing monodromic categories with Soergel bimodules, cf. Proposition 4.4.32.


We would like to thank the anonymous referees for their helpful suggestions on improving this text. We thank Alexander Braverman for useful discussions, especially for suggesting an idea that helped us establish Theorem 5.6.57 in its present generality. The first author would like to thank Tanmay Deshpande for related discussions. The second author would like to thank Eugene Gorsky for the constant interest in this work, encouragement, many very helpful discussions, and comments on a draft of this paper; and Stefan Dawydiak for reading of the draft and suggestions to improve the presentation. He would also like to acknowledge the role of the WARTHOG workshop “Knot homologies, Hilbert schemes, and Cherednik algebras” of 2016 and AIM workshop “Categorified Hecke algebras, link homology, and Hilbert schemes” of 2018, where he learned about the problem and many related things, and to thank their participants and organizers.


[1] P. N. Achar, A. Henderson, D. Juteau and S. Riche, Weyl group actions on the Springer sheaf, Proc. Lond. Math. Soc. (3) 108 (2014), no. 6, 1501–1528. 10.1112/plms/pdt055Search in Google Scholar

[2] P. N. Achar, S. Riche and C. Vay, Mixed perverse sheaves on flag varieties for Coxeter groups, Canad. J. Math. 72 (2020), no. 1, 1–55. 10.4153/CJM-2018-034-0Search in Google Scholar

[3] S. Arkhipov and R. Bezrukavnikov, Perverse sheaves on affine flags and Langlands dual group, Israel J. Math. 170 (2009), 135–183. 10.1007/s11856-009-0024-ySearch in Google Scholar

[4] A. Beilinson, R. Bezrukavnikov and I. Mirković, Tilting exercises, Mosc. Math. J. 4 (2004), no. 3, 547–557. 10.17323/1609-4514-2004-4-3-547-557Search in Google Scholar

[5] A. Beilinson and V. Ginzburg, Wall-crossing functors and 𝒟-modules, Represent. Theory 3 (1999), 1–31. 10.1090/S1088-4165-99-00063-1Search in Google Scholar

[6] A. Beilinson, V. Ginzburg and W. Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), no. 2, 473–527. 10.1090/S0894-0347-96-00192-0Search in Google Scholar

[7] J. Bernstein and V. Lunts, Equivariant sheaves and functors, Lecture Notes in Math. 1578, Springer, Berlin 2006. Search in Google Scholar

[8] R. Bezrukavnikov, On two geometric realizations of an affine Hecke algebra, Publ. Math. Inst. Hautes Études Sci. 123 (2016), 1–67. 10.1007/s10240-015-0077-xSearch in Google Scholar

[9] R. Bezrukavnikov and T. Deshpande, On the geometric Whittaker model, in preparation. Search in Google Scholar

[10] R. Bezrukavnikov, M. Finkelberg and V. Ostrik, Character D-modules via Drinfeld center of Harish-Chandra bimodules, Invent. Math. 188 (2012), no. 3, 589–620. 10.1007/s00222-011-0354-3Search in Google Scholar

[11] R. Bezrukavnikov and S. Riche, A topological approach to Soergel theory, preprint (2018), 10.1007/978-3-030-82007-7_7Search in Google Scholar

[12] R. Bezrukavnikov and A. Yom Din, On parabolic restriction of perverse sheaves, Publ. Res. Inst. Math. Sci. 57 (2021), no. 3–4, 1089–1107. 10.4171/PRIMS/57-3-12Search in Google Scholar

[13] R. Bezrukavnikov and Z. Yun, On Koszul duality for Kac–Moody groups, Represent. Theory 17 (2013), 1–98. 10.1090/S1088-4165-2013-00421-1Search in Google Scholar

[14] W. Borho and R. MacPherson, Représentations des groupes de Weyl et homologie d’intersection pour les variétés nilpotentes, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 15, 707–710. Search in Google Scholar

[15] M. Boyarchenko and V. Drinfeld, Character sheaves on unipotent groups in positive characteristic: Foundations, Selecta Math. (N. S.) 20 (2014), no. 1, 125–235. 10.1007/s00029-013-0133-7Search in Google Scholar

[16] J.-L. Brylinski, Transformations canoniques, dualité projective, théorie de Lefschetz, transformations de Fourier et sommes trigonométriques, Géométrie et analyse microlocales, Astérisque 140, Société Mathématique de France, Paris (1986), 3–134. Search in Google Scholar

[17] T.-H. Chen, On the conjectures of Braverman–Kazhdan, preprint (2019), 10.1090/jams/992Search in Google Scholar

[18] T.-H. Chen, A vanishing conjecture: The GLn case, Selecta Math. (N. S.) 28 (2022), no. 1, Paper No. 13. 10.1007/s00029-021-00726-2Search in Google Scholar

[19] T.-H. Chen and A. Yom Din, A formula for the geometric Jacquet functor and its character sheafanalogue, Geom. Funct. Anal. 27 (2017), no. 4, 772–797. 10.1007/s00039-017-0413-zSearch in Google Scholar

[20] B. Elias and M. Hogancamp, Categorical diagonalization of full twists, preprint (2017), Search in Google Scholar

[21] B. Elias, S. Makisumi, U. Thiel and G. Williamson, Introduction to Soergel bimodules, RSME Springer Ser. 5, Springer, Cham 2020. 10.1007/978-3-030-48826-0Search in Google Scholar

[22] B. Elias and G. Williamson, The Hodge theory of Soergel bimodules, Ann. of Math. (2) 180 (2014), no. 3, 1089–1136. 10.1007/978-3-030-48826-0_18Search in Google Scholar

[23] W. L. Gan and V. Ginzburg, Quantization of Slodowy slices, Int. Math. Res. Not. IMRN 2002 (2002), no. 5, 243–255. 10.1155/S107379280210609XSearch in Google Scholar

[24] V. Ginsburg, Admissible modules on a symmetric space, Orbites unipotentes et représentations, III, Astérisque 173–174, Société Mathématique de France, Paris (1989), 199–255. Search in Google Scholar

[25] V. Ginzburg, Induction and restriction of character sheaves, I. M. Gel’fand Seminar, Adv. Soviet Math. 16, American Mathematical Society, Providence (1993), 149–167. 10.1090/advsov/016.1/05Search in Google Scholar

[26] E. Gorsky, M. Hogancamp, A. Mellit and K. Nakagane, Serre duality for Khovanov–Rozansky homology, Selecta Math. (N. S.) 25 (2019), no. 5, Paper No. 79. 10.1007/s00029-019-0524-5Search in Google Scholar

[27] E. Gorsky, A. Neguţ and J. Rasmussen, Flag Hilbert schemes, colored projectors and Khovanov–Rozansky homology, Adv. Math. 378 (2021), Paper No. 107542. 10.1016/j.aim.2020.107542Search in Google Scholar

[28] I. Grojnowski, Character sheaves on symmetric spaces, Ph.D. thesis, Department of Mathematics, Massachusetts Institute of Technology, 1992. Search in Google Scholar

[29] J. E. Humphreys, Conjugacy classes in semisimple algebraic groups, Math. Surveys Monogr. 43, American Mathematical Society, Providence 2011. 10.1090/surv/043Search in Google Scholar

[30] A. P. Isaev and O. V. Ogievetsky, On representations of Hecke algebras, Czechoslovak J. Phys. 55 (2005), no. 11, 1433–1441. 10.1007/s10582-006-0022-9Search in Google Scholar

[31] V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. (2) 126 (1987), no. 2, 335–388. 10.1142/9789812798329_0003Search in Google Scholar

[32] N. M. Katz and G. Laumon, Transformation de Fourier et majoration de sommes exponentielles, Publ. Math. Inst. Hautes Études Sci. 62 (1985), 361–418. 10.1007/BF02698808Search in Google Scholar

[33] M. Khovanov, Triply-graded link homology and Hochschild homology of Soergel bimodules, Internat. J. Math. 18 (2007), no. 8, 869–885. 10.1142/S0129167X07004400Search in Google Scholar

[34] M. Khovanov and L. Rozansky, Matrix factorizations and link homology, Fund. Math. 199 (2008), no. 1, 1–91. 10.4064/fm199-1-1Search in Google Scholar

[35] Y. Laszlo and M. Olsson, The six operations for sheaves on Artin stacks. I. Finite coefficients, Publ. Math. Inst. Hautes Études Sci. 107 (2008), 109–168. 10.1007/s10240-008-0011-6Search in Google Scholar

[36] Y. Laszlo and M. Olsson, The six operations for sheaves on Artin stacks. II. Adic coefficients, Publ. Math. Inst. Hautes Études Sci. 107 (2008), 169–210. 10.1007/s10240-008-0012-5Search in Google Scholar

[37] G. Lusztig, Character sheaves. I, Adv. Math. 56 (1985), no. 3, 193–237. 10.1016/0001-8708(85)90034-9Search in Google Scholar

[38] G. Lusztig, Parabolic character sheaves. I, Mosc. Math. J. 4 (2004), no. 1, 153–179. 10.17323/1609-4514-2004-4-1-153-179Search in Google Scholar

[39] G. Lusztig, Truncated convolution of character sheaves, Bull. Inst. Math. Acad. Sin. (N. S.) 10 (2015), no. 1, 1–72. Search in Google Scholar

[40] I. Mirković and K. Vilonen, Characteristic varieties of character sheaves, Invent. Math. 93 (1988), no. 2, 405–418. 10.1007/BF01394339Search in Google Scholar

[41] A. Okounkov and A. Vershik, A new approach to representation theory of symmetric groups, Selecta Math. (N. S.) 2 (1996), no. 4, 581–605. 10.1007/BF02433451Search in Google Scholar

[42] S. Riche, Kostant section, universal centralizer, and a modular derived Satake equivalence, Math. Z. 286 (2017), no. 1–2, 223–261. 10.1007/s00209-016-1761-3Search in Google Scholar

[43] L. Rider and A. Russell, Perverse sheaves on the nilpotent cone and Lusztig’s generalized Springer correspondence, Lie algebras, Lie superalgebras, vertex algebras and related topics, Proc. Sympos. Pure Math. 92, American Mathematical Society, Providence (2016), 273–292. Search in Google Scholar

[44] R. Rouquier, Categorification of 𝔰𝔩2 and braid groups, Trends in representation theory of algebras and related topics, Contemp. Math. 406, American Mathematical Society, Providence (2006), 137–167. 10.1090/conm/406/07657Search in Google Scholar

[45] W. Soergel, Kategorie 𝒪, perverse Garben und Moduln über den Koinvarianten zur Weylgruppe, J. Amer. Math. Soc. 3 (1990), no. 2, 421–445. 10.1090/S0894-0347-1990-1029692-5Search in Google Scholar

[46] W. Soergel, Kazhdan–Lusztig–Polynome und unzerlegbare Bimoduln über Polynomringen, J. Inst. Math. Jussieu 6 (2007), no. 3, 501–525. 10.1017/S1474748007000023Search in Google Scholar

[47] K. Tolmachov, Towards a functor between affine and finite Hecke categories in type A, Ph.D. thesis, Massachusetts Institute of Technology, 2018. Search in Google Scholar

[48] B. Webster and G. Williamson, A geometric model for Hochschild homology of Soergel bimodules, Geom. Topol. 12 (2008), no. 2, 1243–1263. 10.2140/gt.2008.12.1243Search in Google Scholar

[49] B. Webster and G. Williamson, The geometry of Markov traces, Duke Math. J. 160 (2011), no. 2, 401–419. 10.1215/00127094-1444268Search in Google Scholar

[50] B. Webster and G. Williamson, A geometric construction of colored HOMFLYPT homology, Geom. Topol. 21 (2017), no. 5, 2557–2600. 10.2140/gt.2017.21.2557Search in Google Scholar

[51] H. Wenzl, Hecke algebras of type An and subfactors, Invent. Math. 92 (1988), no. 2, 349–383. 10.1007/BF01404457Search in Google Scholar

[52] Z. Yun, Weights of mixed tilting sheaves and geometric Ringel duality, Selecta Math. (N. S.) 14 (2009), no. 2, 299–320. 10.1007/s00029-008-0066-8Search in Google Scholar

Received: 2020-10-14
Revised: 2022-01-12
Published Online: 2022-04-01
Published in Print: 2022-06-01

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 7.6.2023 from
Scroll to top button