The Hodge theory of Soergel bimodules. (English) Zbl 1326.20005
In the Hecke algebra of a Coxeter system, Kazhdan-Lusztig polynomials were introduced by D. Kazhdan and G. Lusztig [Invent. Math. 53, 165-184 (1979; Zbl 0499.20035)] to express the Kazhdan-Lusztig basis in terms of standard basis, which were conjectured to have non-negative coefficients. In [J. Inst. Math. Jussieu 6, No. 3, 501-525 (2007; Zbl 1192.20004)], W. Soergel introduced the monoidal category of Soergel bimodules for an arbitrary Coxeter system. The split Grothendieck group of this category is canonically identified with the Hecke algebra. Soergel then conjectured the existence of indecomposable bimodules whose classes coincide with the Kazhdan-Lusztig basis of the Hecke algebra.
In the paper under review, the authors prove Soergel’s conjecture for an arbitrary Coxeter system and then prove the non-negativity of Kazhdan-Lusztig polynomials. The authors’ proof is inspired by two papers of M. A. A. de Cataldo and L. Migliorini [see Ann. Sci. Éc. Norm. Supér. (4) 35, No. 5, 759-772 (2002; Zbl 1021.14004); ibid. 38, No. 5, 693-750 (2005; Zbl 1094.14005)]), which give Hodge-theoretic proofs of the decomposition theorem.
It was proposed by D. Kazhdan and G. Lusztig [loc. cit.] the Kazhdan-Lusztig conjecture, a character formula of the simple highest weight modules for a complex semi-simple Lie algebra in terms of Kazhdan-lusztig polynomials associated to its Weyl group. In the present paper, the authors obtain an algebraic proof of this conjecture. Note that the Kazhdan-Lusztig conjecture was proved by Beilinson-Bernstein and Brylinski-Kashiwara independently in 1981 by using \(D\)-modules and the Riemann-Hilbert correspondence to establish a connection between highest weight representation theory and perverse sheaves [see A. Beilinson and J. Bernstein, C. R. Acad. Sci., Paris, Sér. I 292, 15-18 (1981; Zbl 0476.14019); J. L. Brylinski and M. Kashiwara, Invent. Math. 64, 387-410 (1981; Zbl 0473.22009)] and by W. Soergel in an alternate way in 1990 [J. Am. Math. Soc. 3, No. 2, 421-445 (1990; Zbl 0747.17008)].
In the paper under review, the authors prove Soergel’s conjecture for an arbitrary Coxeter system and then prove the non-negativity of Kazhdan-Lusztig polynomials. The authors’ proof is inspired by two papers of M. A. A. de Cataldo and L. Migliorini [see Ann. Sci. Éc. Norm. Supér. (4) 35, No. 5, 759-772 (2002; Zbl 1021.14004); ibid. 38, No. 5, 693-750 (2005; Zbl 1094.14005)]), which give Hodge-theoretic proofs of the decomposition theorem.
It was proposed by D. Kazhdan and G. Lusztig [loc. cit.] the Kazhdan-Lusztig conjecture, a character formula of the simple highest weight modules for a complex semi-simple Lie algebra in terms of Kazhdan-lusztig polynomials associated to its Weyl group. In the present paper, the authors obtain an algebraic proof of this conjecture. Note that the Kazhdan-Lusztig conjecture was proved by Beilinson-Bernstein and Brylinski-Kashiwara independently in 1981 by using \(D\)-modules and the Riemann-Hilbert correspondence to establish a connection between highest weight representation theory and perverse sheaves [see A. Beilinson and J. Bernstein, C. R. Acad. Sci., Paris, Sér. I 292, 15-18 (1981; Zbl 0476.14019); J. L. Brylinski and M. Kashiwara, Invent. Math. 64, 387-410 (1981; Zbl 0473.22009)] and by W. Soergel in an alternate way in 1990 [J. Am. Math. Soc. 3, No. 2, 421-445 (1990; Zbl 0747.17008)].
Reviewer: Shi Jian-yi (Shanghai)
MSC:
| 20C08 | Hecke algebras and their representations |
| 20F55 | Reflection and Coxeter groups (group-theoretic aspects) |
Keywords:
Coxeter groups; Hecke algebras; Coxeter systems; Hodge theory; Kazhdan-Lusztig polynomials; positivity; Soergel bimodulesCitations:
Zbl 0499.20035; Zbl 1192.20004; Zbl 1021.14004; Zbl 1094.14005; Zbl 0476.14019; Zbl 0473.22009; Zbl 0747.17008References:
| [1] | D. Kazhdan and G. Lusztig, ”Representations of Coxeter groups and Hecke algebras,” Invent. Math., vol. 53, iss. 2, pp. 165-184, 1979. · Zbl 0499.20035 · doi:10.1007/BF01390031 |
| [2] | D. Kazhdan and G. Lusztig, ”Schubert varieties and Poincaré duality,” in Geometry of the Laplace Operator, Providence, R.I.: Amer. Math. Soc., 1980, vol. XXXVI, pp. 185-203. · Zbl 0461.14015 |
| [3] | A. Beuilinson and J. Bernstein, ”Localisation de \(g\)-modules,” C. R. Acad. Sci. Paris Sér. I Math., vol. 292, iss. 1, pp. 15-18, 1981. · Zbl 0476.14019 |
| [4] | J. -L. Brylinski and M. Kashiwara, ”Kazhdan-Lusztig conjecture and holonomic systems,” Invent. Math., vol. 64, iss. 3, pp. 387-410, 1981. · Zbl 0473.22009 · doi:10.1007/BF01389272 |
| [5] | W. Soergel, ”Kategorie \(\mathcal O\), perverse Garben und Moduln über den Koinvarianten zur Weylgruppe,” J. Amer. Math. Soc., vol. 3, iss. 2, pp. 421-445, 1990. · Zbl 0747.17008 · doi:10.2307/1990960 |
| [6] | W. Soergel, ”The combinatorics of Harish-Chandra bimodules,” J. Reine Angew. Math., vol. 429, pp. 49-74, 1992. · Zbl 0745.22014 · doi:10.1515/crll.1992.429.49 |
| [7] | A. A. Beuilinson, J. Bernstein, and P. Deligne, ”Faisceaux pervers,” in Analysis and Topology on Singular Spaces, I, Paris: Soc. Math. France, 1982, vol. 100, pp. 5-171. |
| [8] | M. Härterich, Kazhdan-Lusztig-Basen, unzerlegbare Bimoduln und die Topologie der Fahnenmannigfaltigkeit einer Kac-Moody-Gruppe, 1999. · Zbl 1065.17503 |
| [9] | W. Soergel, ”Kazhdan-Lusztig-Polynome und unzerlegbare Bimoduln über Polynomringen,” J. Inst. Math. Jussieu, vol. 6, iss. 3, pp. 501-525, 2007. · Zbl 1192.20004 · doi:10.1017/S1474748007000023 |
| [10] | P. Fiebig, ”The combinatorics of Coxeter categories,” Trans. Amer. Math. Soc., vol. 360, iss. 8, pp. 4211-4233, 2008. · Zbl 1160.20032 · doi:10.1090/S0002-9947-08-04376-6 |
| [11] | M. A. A. de Cataldo and L. Migliorini, ”The hard Lefschetz theorem and the topology of semismall maps,” Ann. Sci. École Norm. Sup., vol. 35, iss. 5, pp. 759-772, 2002. · Zbl 1021.14004 · doi:10.1016/S0012-9593(02)01108-4 |
| [12] | M. A. A. de Cataldo and L. Migliorini, ”The Hodge theory of algebraic maps,” Ann. Sci. École Norm. Sup., vol. 38, iss. 5, pp. 693-750, 2005. · Zbl 1094.14005 · doi:10.1016/j.ansens.2005.07.001 |
| [13] | W. Soergel, ”Andersen filtration and hard Lefschetz,” Geom. Funct. Anal., vol. 17, iss. 6, pp. 2066-2089, 2008. · Zbl 1143.22012 · doi:10.1007/s00039-007-0640-9 |
| [14] | H. H. Andersen, J. C. Jantzen, and W. Soergel, Representations of Quantum Groups at a \(p\)th Root of Unity and of Semisimple Groups in Characteristic \(p\): Independence of \(p\), Paris: Soc. Math. France, 1994, vol. 220. · Zbl 0802.17009 |
| [15] | P. Fiebig, ”The combinatorics of category \(\mathcal O\) over symmetrizable Kac-Moody algebras,” Transform. Groups, vol. 11, iss. 1, pp. 29-49, 2006. · Zbl 1122.17016 · doi:10.1007/s00031-005-1103-8 |
| [16] | P. Fiebig, ”Sheaves on affine Schubert varieties, modular representations, and Lusztig’s conjecture,” J. Amer. Math. Soc., vol. 24, iss. 1, pp. 133-181, 2011. · Zbl 1270.20053 · doi:10.1090/S0894-0347-2010-00679-0 |
| [17] | P. Bressler and V. A. Lunts, ”Intersection cohomology on nonrational polytopes,” Compositio Math., vol. 135, iss. 3, pp. 245-278, 2003. · Zbl 1024.52005 · doi:10.1023/A:1022232232018 |
| [18] | K. Karu, ”Hard Lefschetz theorem for nonrational polytopes,” Invent. Math., vol. 157, iss. 2, pp. 419-447, 2004. · Zbl 1077.14071 · doi:10.1007/s00222-004-0358-3 |
| [19] | G. Barthel, L. Kaup, J. -P. Brasselet, and K. Fieseler, ”Hodge-Riemann relations for polytopes: a geometric approach,” in Singularity Theory, Hackensack, NJ: World Sci. Publ., 2007, pp. 379-410. · Zbl 1118.14025 · doi:10.1142/9789812707499_0014 |
| [20] | M. J. Dyer, Kazhdan-Lusztig-Stanley polynomials and quadratic algebras I. · Zbl 0809.20029 |
| [21] | M. J. Dyer, Modules for the dual nil Hecke ring. |
| [22] | T. Maeno, Y. Numata, and A. Wachi, ”Strong Lefschetz elements of the coinvariant rings of finite Coxeter groups,” Algebr. Represent. Theory, vol. 14, iss. 4, pp. 625-638, 2011. · Zbl 1250.13005 · doi:10.1007/s10468-010-9207-9 |
| [23] | Y. Numata and A. Wachi, ”The strong Lefschetz property of the coinvariant ring of the Coxeter group of type \(H_4\),” J. Algebra, vol. 318, iss. 2, pp. 1032-1038, 2007. · Zbl 1134.13003 · doi:10.1016/j.jalgebra.2007.06.016 |
| [24] | C. McDaniel, ”The strong Lefschetz property for coinvariant rings of finite reflection groups,” J. Algebra, vol. 331, pp. 68-95, 2011. · Zbl 1244.13005 · doi:10.1016/j.jalgebra.2010.11.007 |
| [25] | B. Elias and G. Williamson, Diagrammatics for Coxeter groups and their braid groups. · Zbl 1383.20021 |
| [26] | B. Elias and G. Williamson, Soergel calculus. · Zbl 1427.20006 |
| [27] | N. Libedinsky, ”Équivalences entre conjectures de Soergel,” J. Algebra, vol. 320, iss. 7, pp. 2695-2705, 2008. · Zbl 1197.20004 · doi:10.1016/j.jalgebra.2008.05.030 |
| [28] | N. Bourbaki, ”Éléments de mathématique,” in Groupes et Algèbres de Lie. Chapitres 4, 5 et 6. [Lie Groups and Lie Algebras. Chapters 4, 5 and 6], Paris: Masson, 1981, p. 290. · Zbl 0483.22001 |
| [29] | J. E. Humphreys, Reflection groups and Coxeter groups, Cambridge Univ. Press, Cambridge, 1990, vol. 29. · Zbl 0725.20028 · doi:10.1017/CBO9780511623646 |
| [30] | W. Soergel, ”Kazhdan-Lusztig polynomials and a combinatoric for tilting modules,” Represent. Theory, vol. 1, pp. 83-114, 1997. · Zbl 0886.05123 · doi:10.1090/S1088-4165-97-00021-6 |
| [31] | H. Krause, Krull-Remak-Schmidt categories and projective covers. · Zbl 1353.18011 · doi:10.1016/j.exmath.2015.10.001 |
| [32] | R. Rouquier, ”Categorification of \({\mathfraks\mathfrakl}_2\) and braid groups,” in Trends in Representation Theory of Algebras and Related Topics, Amer. Math. Soc., Providence, RI, 2006, vol. 406, pp. 137-167. · doi:10.1090/conm/406/07657 |
| [33] | K. J. Carlin, ”Extensions of Verma modules,” Trans. Amer. Math. Soc., vol. 294, iss. 1, pp. 29-43, 1986. · Zbl 0595.22015 · doi:10.2307/2000116 |
| [34] | J. Rickard, ”Translation functors and equivalences of derived categories for blocks of algebraic groups,” in Finite-dimensional algebras and related topics, Dordrecht: Kluwer Acad. Publ., 1994, vol. 424, pp. 255-264. · Zbl 0829.20023 |
| [35] | R. Rouquier, ”Derived equivalences and finite dimensional algebras,” in International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 191-221. · Zbl 1108.20006 |
| [36] | N. Libedinsky and G. Williamson, Standard objects in 2-braid groups. · Zbl 1316.20046 · doi:10.1112/plms/pdu022 |
| [37] | G. Williamson, ”Singular Soergel bimodules,” Int. Math. Res. Not., vol. 2011, iss. 20, pp. 4555-4632, 2011. · Zbl 1236.18009 · doi:10.1093/imrn/rnq263 |
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