Duke Mathematical Journal

Microlocal KZ-functors and rational Cherednik algebras

Kevin McGerty

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Following the work of Kashiwara and Rouquier and of Gan and Ginzburg, we define a family of exact functors from category O for the rational Cherednik algebra in type A to representations of certain colored braid groups and we calculate the dimensions of the representations thus obtained from standard modules. To show that our constructions make sense in a more general context, we also briefly study the case of the rational Cherednik algebra corresponding to complex reflection group Z/lZ.

Article information

Duke Math. J., Volume 161, Number 9 (2012), 1657-1709.

First available in Project Euclid: 6 June 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 16S38: Rings arising from non-commutative algebraic geometry [See also 14A22]
Secondary: 53D55: Deformation quantization, star products 14A22: Noncommutative algebraic geometry [See also 16S38]


McGerty, Kevin. Microlocal $KZ$ -functors and rational Cherednik algebras. Duke Math. J. 161 (2012), no. 9, 1657--1709. doi:10.1215/00127094-1593353. https://projecteuclid.org/euclid.dmj/1338987166

Export citation


  • [BB] A. Beilinson and J. Bernstein, “A proof of Jantzen conjectures” in I. M. Gelfand Seminar, Adv. Soviet Math. 16, Amer. Math. Soc., Providence, 1993, 1–50.
  • [BK] G. Bellamy and T. Kuwabara, On deformation quantizations of hypertoric varieties, preprint, arXiv:1005.4645v2 [math.RT]
  • [BEG] Y. Berest, P. Etingof, and V. Ginzburg, Cherednik algebras and differential operators on quasi-invariants, Duke Math. J. 118 (2003) 279–337.
  • [BE] R. Bezrukavnikov and P. Etingof, Parabolic induction and restriction functors for rational Cherednik algebras, Selecta Math. (N.S.) 14 (2009), 397–425.
  • [BFG] R. Bezrukavnikov, M. Finkelberg, and V. Ginzburg, Cherednik algebras and Hilbert schemes in characteristic p, Represent. Theory 10 (2006), 254–298.
  • [BDK] J.-L. Brylinski, A. S. Dubson, and M. Kashiwara, Formule de l’indice pour modules holonômes et obstruction d’Euler locale, C. R. Math. Acad. Sci. Paris 293 (1981), 573–576.
  • [D] P. Deligne, Les immeubles des groupes de tresses généralisés, Invent. Math. 17 (1972), 273–302.
  • [DJM] R. Dipper, G. James, and A. Mathas, Cyclotomic q-Schur algebras, Math. Z. 229 (1998), 385–416.
  • [DO] C. F. Dunkl and E. M. Opdam, Dunkl operators for complex reflection groups, Proc. Lond. Math. Soc. (3) 86 (2003), 70–108.
  • [ES] G. Ellingsrud and S. A. Stromme, On the homology of the Hilbert scheme of points in the plane, Invent. Math. 87 (1987), 343–352.
  • [EG] P. Etingof and V. Ginzburg, Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism, Invent. Math. 147 (2002), 243–348.
  • [EM] S. Evens and I. Mirkovic, Characteristic cycles for loop Grassmannian and nilpotent orbits, Duke Math. J. 97 (1999), 109–126.
  • [GG] W. L. Gan and V. Ginzburg, Almost-commuting variety, $\mathcal{D}$-modules, and Cherednik algebras, IMRP Int. Math. Res. Pap. 2006, 1–54.
  • [GMV] S. Gelfand, R. MacPherson, and K. Vilonen, Microlocal perverse sheaves, preprint, arXiv:math/0509440v1 [math.AG]
  • [GGS] V. Ginzburg, I. Gordon, and J. T. Stafford, Differential operators and Cherednik algebras, Selecta Math. (N.S.) 14 (2009), 629–666.
  • [GGOR] V. Ginzburg, N. Guay, E. Opdam, and R. Rouquier, On the category $\mathcal{O}$ for rational Cherednik algebras, Invent. Math. 154 (2003), 617–651.
  • [GS] I. Gordon and J. T. Stafford, Rational Cherednik algebras and Hilbert schemes, II: Representations and sheaves, Duke Math. J. 132 (2006), 73–135.
  • [Gr] I. Grojnowski, Instantons and affine algebras, I: The Hilbert scheme and vertex operators, Math. Res. Lett. 3 (1996), 275–291.
  • [Gu] N. Guay, Projective modules in the category $\mathcal{O}$ for the Cherednik algebra, J. Pure Appl. Algebra 182 (2003), 209–221.
  • [H] M. P. Holland, Quantization of the Marsden-Weinstein reduction for extended Dynkin quivers, Ann. Sci. Éc. Norm. Supér. (4) 32 (1999), 813–834.
  • [HK] R. Hotta and M. Kashiwara, The invariant holonomic system on a semisimple Lie algebra, Invent. Math. 75 (1984), 327–258.
  • [HS] M. Hunziker and G. W. Schwarz, A homomorphism of Harish-Chandra and direct images of $\mathcal{D}$-modules, Proc. Amer. Math. Soc. 129 (2001), 3485–3493.
  • [K1] M. Kashiwara, Index theorem for a maximally overdetermined system of linear differential equations, Proc. Japan Acad. 49 (1973), 803–804.
  • [K2] M. Kashiwara, Systems of Microdifferential Equations, Progr. Math. 34, Birkhäuser, Boston, 1983.
  • [K3] M. Kashiwara, “Vanishing cycle sheaves and holonomic systems of differential equations” in Algebraic Geometry (Tokyo/Kyoto, 1982), Lecture Notes in Math. 1016, Springer, Berlin, 1983, 134–142.
  • [K4] M. Kashiwara, Introduction to microlocal analysis, Enseign. Math. (2) 32 (1986), 227–259.
  • [K6] M. Kashiwara, D-modules and Microlocal Calculus, Transl. Math. Monogr. 217, Iwanami Ser. Mod. Math., Amer. Math. Soc., Providence, 2003.
  • [K7] M. Kashiwara, “Equivariant derived category and representation of real semisimple Lie groups” in Representation Theory and Complex Analysis (Venice, 2004), Lecture Notes in Math. 1931, Springer, Berlin, 2008, 137–234.
  • [KK1] M. Kashiwara and T. Kawai, On holonomic systems of microdifferential equations, III: Systems with regular singularities, Publ. Res. Inst. Math. Sci. 17 (1981), 813–979.
  • [KR] M. Kashiwara and R. Rouquier, Microlocalization of rational Cherednik algebras, Duke Math. J. 144 (2008), 525–573.
  • [KS] M. Kashiwara and P. Schapira, Sheaves on Manifolds, Grundlehren Math. Wiss. 292, Springer, Berlin, 1990.
  • [KV] M. Kashiwara and K. Vilonen, On the codimension-three conjecture, Proc. Japan Acad. Ser. A Math. Sci. 86 (2010), 154–158.
  • [Ku] T. Kuwabara, Representation theory of the rational Cherednik algebras of type ℤ/lℤ via microlocal analysis, preprint, arXiv:1003.3407v2 [math.RT]
  • [L] G. Lusztig, Green polynomials and singularities of unipotent classes, Adv. Math. 42, (1981), 169–178.
  • [M] R. D. MacPherson, Chern classes for singular algebraic varieties, Ann. of Math. (2) 100 (1974), 423–432.
  • [MV1] R. MacPherson and K. Vilonen, Elementary construction of perverse sheaves, Invent. Math. 84 (1986), 403–435.
  • [N] H. Nakajima, Lectures on Hilbert Schemes of Points on Surfaces, Univ. Lecture Ser. 18, Amer. Math. Soc., Providence, 1999.
  • [R] R. Rouquier, q-Schur algebras and complex reflection groups, Mosc. Math. J. 8 (2008), 119–158.
  • [W1] I. Waschkies, The stack of microlocal perverse sheaves, Bull. Soc. Math. France 132 (2004), 397–462.
  • [W2] I. Waschkies, Microlocal Riemann-Hilbert correspondence, Publ. Res. Inst. Math. Sci. 41 (2005), 37–72.