## Duke Mathematical Journal

### Rational Cherednik algebras and Hilbert schemes, II: Representations and sheaves

#### Abstract

Let $H_c$ be the rational Cherednik algebra of type $A_{n-1}$ with spherical subalgebra $U_c=eH_ce$. Then $U_c$ is filtered by order of differential operators with associated graded ring ${\rm gr}U_c=\mathbb{C}[\mathfrak{h}\oplus\mathfrak{h}^*]^{W}$, where $W$ is the nth symmetric group. Using the $\mathbb{Z}$-algebra construction from [GS], it is also possible to associate to a filtered $H_c$- or $U_c$-module $M$ a coherent sheaf $\widehat{\Phi}(M)$ on the Hilbert scheme Hilb(n). Using this technique, we study the representation theory of $U_c$ and $H_c$, and we relate it to Hilb(n) and to the resolution of singularities $\tau : {\rm Hilb(n)}\to \mathfrak{h}\oplus\mathfrak{h}^*/W$. For example, we prove the following.

• If $c=1/n$ so that $L_c(\ttt{triv})$ is the unique one-dimensional simple $H_c$-module, then $\widehat{\Phi} (eL_c(\ttt{triv})) \cong \mathcal{O}_{Z_n}$, where $Z_n=\tau^{-1}(0)$ is the punctual Hilbert scheme.

• If $c=1/n+k$ for $k\in \mathbb{N}$, then under a canonical filtration on the finite-dimensional module $L_c(\ttt{triv})$, ${\rm gr} eL_{c}(\ttt{triv})$ has a natural bigraded structure that coincides with that on $\mathrm{H}^0(Z_n, \mathcal{L}^k)$, where $\mathcal{L}\cong\mathcal{O}_{{\rm Hilb(n)}}(1)$; this confirms conjectures of Berest, Etingof, and Ginzburg [BEG2, Conjectures 7.2, 7.3].

• Under mild restrictions on $c$, the characteristic cycle of $\widehat{\Phi} (e\Delta_c(\mu))$ equals $\sum_{\lambda}K_{\mu\lambda}[Z_\lambda]$, where $K_{\mu\lambda}$ are Kostka numbers and the $Z_{\lambda }$ are (known) irreducible components of $\tau^{-1}(\mathfrak{h}/W)$

#### Article information

Source
Duke Math. J., Volume 132, Number 1 (2006), 73-135.

Dates
First available in Project Euclid: 28 February 2006

https://projecteuclid.org/euclid.dmj/1141136437

Digital Object Identifier
doi:10.1215/S0012-7094-06-13213-1

Mathematical Reviews number (MathSciNet)
MR2219255

Zentralblatt MATH identifier
1096.14003

#### Citation

Gordon, I.; Stafford, J. T. Rational Cherednik algebras and Hilbert schemes, II: Representations and sheaves. Duke Math. J. 132 (2006), no. 1, 73--135. doi:10.1215/S0012-7094-06-13213-1. https://projecteuclid.org/euclid.dmj/1141136437

#### References

• Y. Berest, P. Etingof, and V. Ginzburg, Cherednik algebras and differential operators on quasi-invariants, Duke Math. J. 118 (2003), 279--337.
• —, Finite-dimensional representations of rational Cherednik algebras, Internat. Math. Res. Notices 2003, no. 19, 1053--1088.
• J.-E. BjöRk, Rings of Differential Operators, North-Holland Math. Library 21, North-Holland, Amsterdam, 1979.
• —, Filtered noetherian rings'' in Noetherian Rings and Their Applications (Oberwolfach, West Germany, 1983), Math. Surveys Monogr. 24, Amer. Math. Soc., Providence, 1987, 59--97.
• W. Borho and J.-L. Brylinski, Differential operators on homogeneous spaces, III: Characteristic varieties of Harish-Chandra modules and of primitive ideals, Invent. Math. 80 (1985), 1--68.
• T. Bridgeland, A. King, and M. Reid, The McKay correspondence as an equivalence of derived categories, J. Amer. Math. Soc. 14 (2001), 535--554.
• J.-L. Brylinski and M. Kashiwara, Kazhdan-Lusztig conjecture and holonomic systems, Invent. Math. 64 (1981), 387--410.
• N. Chriss and V. Ginzburg, Representation Theory and Complex Geometry, Birkhäuser, Boston, 1997.
• N. Conze, Algèbres d'opérateurs différentiels et quotients des algèbres enveloppantes, Bull. Soc. Math. France 102 (1974), 379--415.
• C. DezéLéE, Représentations de dimension finie de l'algèbre de Cherednik rationnelle, Bull. Soc. Math. France 131 (2003), 465--482.
• C. F. Dunkl, Singular polynomials for the symmetric groups, Internat. Math. Res. Notices 2004, no. 67, 3607--3635.
• P. Etingof and V. Ginzburg, Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism, Invent. Math. 147 (2002), 243--348.
• O. Gabber, The integrability of the characteristic variety, Amer. J. Math. 103 (1981), 445--468.
• W. L. Gan and V. Ginzburg, Almost commuting variety, $\mathcalD$-modules, and Cherednik algebras, to appear in Int. Math. Res. Not., preprint.
• A. M. Garsia and M. Haiman, A remarkable $q,t$-Catalan sequence and $q$-Lagrange inversion, J. Algebraic Combin. 5 (1996), 191--244.
• —, Some natural bigraded $S\sb n$-modules and $q,t$-Kostka coefficients, Electron. J. Combin. 3 (1996), R24.
• V. Ginzburg, N. Guay, E. Opdam, and R. Rouquier, On the category $\mathcalO$ for rational Cherednik algebras, Invent. Math. 154 (2003), 617--651.
• I. Gordon, On the quotient by diagonal invariants, Invent. Math. 153 (2003), 503--518.
• I. Gordon and J. T. Stafford, Rational Cherednik algebras and Hilbert schemes, Adv. Math. 198 (2005), 222--274.
• I. Grojnowski, Instantons and affine algebras, I: The Hilbert scheme and vertex operators, Math. Res. Lett. 3 (1996), 275--291.
• A. Grothendieck, Éléments de géométrie algébrique, II: Étude globale élémentaire de quelques classes de morphismes, Inst. Hautes Études Sci. Publ. Math. 8 (1961).
• N. Guay, Projective modules in the category $\OO$ for the Cherednik algebra, J. Pure Appl. Algebra 182 (2003), 209--221.
• M. Haiman, $t,q$-Catalan numbers and the Hilbert scheme, Discrete Math. 198 (1998), 201--224.
• —, Hilbert schemes, polygraphs and the Macdonald positivity conjecture, J. Amer. Math. Soc. 14 (2001), 941--1006.
• —, Vanishing theorems and character formulas for the Hilbert scheme of points in the plane, Invent. Math. 149 (2002), 371--407.
• A. Joseph and J. T. Stafford, Modules of $\mathfrakk$-finite vectors over semisimple Lie algebras, Proc. London Math. Soc. (3) 49 (1984), 361--384.
• G. R. Krause and T. H. Lenagan, Growth of algebras and Gelfand-Kirillov dimension, rev. ed., Grad. Stud. Math. 22, Amer. Math. Soc., Providence, 2000.
• B. Leclerc and J.-Y. Thibon, Canonical bases of $q$-deformed Fock spaces, Internat. Math. Res. Notices 1996, no. 9, 447--456.
• T. Levasseur, Relèvements d'opérateurs différentiels sur les anneaux d'invariants'' in Operator Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory (Paris, 1989), Progr. Math. 92, Birkhäuser, Boston, 1990.
• J. C. Mcconnell and J. C. Robson, Noncommutative Noetherian Rings, rev. ed., Grad. Stud. Math. 30, Amer. Math. Soc., Providence, 2001.
• I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd ed., Oxford Math. Monogr., Oxford Univ. Press, Oxford, 1995.
• H. Matsumura, Commutative Algebra, 2nd ed., Math. Lecture Note Ser. 56, Benjamin/Cummings, Reading, Mass., 1980.
• H. Nakajima, Lectures on Hilbert Schemes of Points on Surfaces, Univ. Lecture Ser. 18, Amer. Math. Soc., Providence, 1999.
• R. Rouquier, $q$-Schur algebras and complex reflection groups, I, preprint.
• M. Varagnolo and E. Vasserot, On the decomposition matrices of the quantized Schur algebra, Duke Math. J. 100 (1999), 267--297.
• N. Wallach, Invariant differential operators on a reductive Lie algebra and Weyl group representations, J. Amer. Math. Soc. 6 (1993), 779--816.
• K. Watanabe, Certain invariant subrings are Gorenstein, I, Osaka J. Math. 11 (1974), 1--8.