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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)].

MSC:
20C08 Hecke algebras and their representations
20F55 Reflection and Coxeter groups (group-theoretic aspects)
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