Cherednik algebras and differential operators on quasi-invariants
Yuri Berest, Pavel Etingof, Victor Ginzburg
备注：
A few mistakes concerning certain KZ twists, which were overlooked in the published version, are corrected in the present version. The corrections involve Proposition 6.6 and Theorems 8.15(i) and 8.16. Also the statement of Conjecture 8.12 is corrected to include possible changes of sign of values of c
We develop representation theory of the rational Cherednik algebra H associated to a finite Coxeter group W in a vector space h. It is applied to show that, for integral values of parameter `c', the algebra H is simple and Morita equivalent to D(h)#W, the cross product of W with the algebra of polynomial differential operators on h.
We further study an algebra Q of quasi-invariant polynomials on h introduced by Chalykh, Feigin, and Veselov [CV], [FV], such that C[h]^W \subset Q \subset C[h]. We prove that the algebra D(Q) of differential operators on quasi-invariants is a simple algebra, Morita equivalent to D(h). The subalgebra D(Q)^W of W-invariant operators turns out to be isomorphic to the spherical subalgebra eHe \subset H. We also show that D(Q) is generated, as an algebra, by Q and its `Fourier dual Q*, and that D(Q) is a rank one projective (Q-Q*)-module (via multiplication-action on D(Q) on opposite sides).
We establish a link between two geometric approaches to the representation theory of rational Cherednik algebras of type A: one based on a noncommutative Proj construction, used in [GS]; the other involving quantum hamiltonian reduction of an algebra of differential operators, used in [GG]. In the present paper, we combine these two points of view by showing that the process of hamiltonian reduction intertwines a naturally defined geometric twist functor on D-modules with the shift functor for the Cherednik algebra. That enables us to give a direct and relatively short proof of the key result,…
We introduce and study deformations of finite-dimensional modules over rational Cherednik algebras. Our main tool is a generalization of usual harmonic polynomials for Coxeter groups -- the so-called quasiharmonic polynomials. A surprising application of this approach is the construction of canonical elementary symmetric polynomials and their deformations for all Coxeter groups.
We obtain several results about representations of rational Cherednik algebras, and discuss their applications. Our first result is the Cohen-Macaulayness property (as modules over the polynomial ring) of Cherednik algebra modules with minimal support. Our second result is an explicit formula for the character of an irreducible minimal support module in type A_{n-1} for c=m/n, and an expression of its quasispherical part (i.e., the isotypic part of "hooks") in terms of the HOMFLY polynomial of a torus knot colored by a Young diagram. We use this formula and the work of Calaque, Enriquez and Et…
The paper an elementary introduction for non-specialists to the theory of quasi-invariants of Coxeter groups. The main object of study is the variety X_m of quasi-invariants for a finite Coxeter group, which arose in a work of O.Chalykh and A.Veselov about 10 years ago, as the spectral variety of the quantum Calogero-Moser system. Despite being singular, this variety has very nice properties (Cohen-Macaulay, Gorenstein, simplicity of the ring of differential operators, explicitly given Hilbert series). It is interesting that although the definition of X_m is completely elementary, to understan…
For any smooth algebraic curve C, Pavel Etingof introduced a `global' Cherednik algebra as a natural deformation of the cross product of the algebra of differential operators on C^n and the symmetric group. We provide a construction of the global Cherednik algebra in terms of quantumn Hamiltonian reduction. We study a category of character D-modules on a representation scheme associated to C and define a Hamiltonian reduction functor from that category to category O for the global Cherednik algebra. In the special case where the curve C is the multiplicative group, the global Cherednik algebra…
This paper is an expanded and updated version of the preprint arXiv:math/0406499. It includes a more detailed description of the basics of the theory of Cherednik and Hecke algebras of varieties started in arXiv:math/0406499, as well as a new Section 4, which reviews the developments in this theory since 2004 with references to the relevant literature. Let $G$ be a finite group of linear transformations of a finite dimensional complex vector space $V$. To this data one can attach a family of algebras $H_{t,c}(V,G)$, parametrized by complex numbers $t$ and conjugation invariant functions $c$ on…
Braided doubles provide a unifying framework for classical and quantum universal enveloping algebras and rational Cherednik algebras. They are a class of algebras with triangular decomposition, arising from a deformation problem, the solutions to which are called quasi-Yetter-Drinfeld modules. A basic family of quasi-YD modules is provided by braidings (matrices satisfying the quantum Yang-Baxter equation); these give rise to quantum versions of the Weyl algebra, where the role of polynomial rings is played by Nichols-Woronowicz algebras. Rational Cherednik algebras for t = 0 emerge as subalge…
We show how a novel construction of the sheaf of Cherednik algebras on a quotient orbifold Y=X/G by virtue of formal geometry in author's prior work leads to results for the sheaf of Cherednik algebra which until recently were viewed as intractable. First, for every orbit type stratum in $X$, we define a trace density map for the Hochschild chain complex of the sheaf of Cherednik algebras, which generalises the standard Engeli-Felder's trace density construction for the sheaf of differential operators. Second, by means of the newly obtained trace density maps, we prove an isomorphism in the de…
In this paper we produce seven new algebras as confluences of the Cherednik algebra of type \check{C_1}C_1 and we characterise their spherical-sub-algebras. The limit of the spherical sub-algebra of the Cherednik algebra of type \check{C_1}C_1 is the monodromy manifold of the Painlevé VI equation. Here we prove that by considering the limits of the spherical sub-algebras of our new confluent algebras, one obtains the monodromy manifolds of all other Painlevé differential equations. Moreover, we introduce confluent versions of the Zhedanov algebra and prove that each of them (quotiented by thei…
According to a statement by Pavel Etingof, in the special case of an affine variety $X$ with a faithful action by a finite group $G$, the sheaf of (twisted) Cherednik algebras $\mathcal{H}_{1, c, ψ, X, G}$ with formal parameters $c, ψ$ is a universal formal deformation of $\mathcal{D}_X\rtimes G$ where $\mathcal{D}_X$ is the sheaf of differential operators on $X$. In the current note, we generalize Etingof's result to the non-affine case. We prove that for a generic smooth analytic or algebraic variety $X$, the sheaf $\mathcal{H}_{1, c, ψ, X, G}$ with formal $c$ and $ψ$ is a universal filtered…
Generalized power sums are linear combinations of i-th powers of coordinates. We consider subalgebras of the polynomial algebra generated by generalized power sums, and study when such algebras are Cohen-Macaulay. It turns out that the Cohen-Macaulay property of such algebras is rare, and tends to be related to quantum integrability and representation theory of Cherednik algebras. Using representation-theoretic results and deformation theory, we establish Cohen-Macaulayness of the algebra of $q,t$-deformed power sums defined by Sergeev and Veselov, and of some generalizations of this algebra, …
For every pair of positive coprime integers, m and n, with m<n, there is an associated generalized Cremmer-Gervais r-matrix r_{m,n} for the Lie algebra sl_n which provides a nonstandard quasitriangular solution to the classical Yang-Baxter equation. We give an interpretation of r_{n-2,n} (for n odd) in terms of differential-Dunkl operators related to the polynomial representation of dihedral-type rational Cherednik algebras. Finally, we use this interpretation to partially answer a conjecture of Gerstenhaber and Giaquinto concerning boundary solutions to the classical Yang-Baxter equation.
These are lecture notes of a minicourse given by the first author at the Summer School on Quantization at the University of Notre Dame in June 2011. The notes were written up and expanded by the second author who took the liberty of adding a few interesting results and proofs from the literature. In a broad sense, our goal is to give an introduction to representation theory of rational Cherednik algebras and some of its recent applications. More specifically, we focus on the two concepts featuring in the title (Dunkl operators and quasi-invariants) and explain the relation between them. The co…
We attach elliptic Dunkl operators to an abelian variety with a finite group action. This generalizes elliptic Dunkl operators for Weyl groups, defined by Buchstaber, Felder, and Veselov in 1994. We show that these operators commute, and use them to define representations from category O of elliptic Cherednik algebras. We also consider the monodromy representations of differential equations defined by elliptic Dunkl operators, and show that they yield finite dimensional rrepresentations of generalized double affine Hecke algebras.
We introduce quasi-invariant polynomials for an arbitrary finite complex reflection group W. Unlike in the Coxeter case, the space Q_k of quasi-invariants of a given multiplicity is not, in general, an algebra but a module over the coordinate ring of some (singular) affine variety X_k. We extend the main results of Etingof, Ginzburg and the first author (see [BEG]) to this setting: in particular, we show that the variety X_k and the module Q_k are Cohen-Macaulay, and the rings of differential operators on X_k and Q_k are simple rings, Morita equivalent to the Weyl algebra A_n(C), where n = dim…
We describe the ring structure of the cohomology of the Hilbert scheme of points for a smooth surface X. When the canonical class K_X = 0, this was done by Lehn and Sorger, extending earlier work when X = C^2. Their approach does not generalize. Instead we recast this problem as a question of finding integrals of motion for a Hamiltonian which describes ``intersection with the boundary''. To do so, we use the identification of the cohomology of the Hilbert schemes with a Fock space modelled on the lattice H(X). With this identification, Lehn computed the operator of intersection with the bound…
We prove that an irreducible quasifinite module over the central extension of the Lie algebra of $N\times N$-matrix differential operators on the circle is either a highest or lowest weight module or else a module of the intermediate series. Furthermore, we give a complete classification of indecomposable uniformly bounded modules.
We introduce a notion of quasi-lisse vertex algebras, which generalizes admissible affine vertex algebras. We show that the normalized character of an ordinary module over a quasi-lisse vertex operator algebra has a modular invariance property, in the sense that it satisfies a modular linear differential equation. As an application we obtain the explicit character formulas of simple affine vertex algebras associated with the Deligne exceptional series at level $-h^{\vee}/6-1$, which express the homogeneous Schur indices of 4d SCFTs studied by Beem, Lemos, Liendo, Peelaers, Rastelli and van Ree…
We integrate nonabelian Toda field equations for root systems of types A, B, C, for functions with values in any associative algebra. The solution is expressed via quasideterminants. In the appendix we review some results concerning nonabelian soliton equations.
We classify positive energy representations with finite degeneracies of the Lie algebra $W_{1+\infty}\/$ and construct them in terms of representation theory of the Lie algebra $\hatgl ( \infty R_m )\/$ of infinite matrices with finite number of non-zero diagonals over the algebra $R_m = \C [ t ] / ( t^{m + 1} )\/$. The unitary ones are classified as well. Similar results are obtained for the sin-algebras.
Let X=GM be a finite group factorisation. It is shown that the quantum double D(H) of the associated bicrossproduct Hopf algebra $H=kM\cobicross k(G)$ is itself a bicrossproduct $kX\cobicross k(Y)$ associated to a group YX, where $Y=G\times M^{op}$. This provides a class of bicrossproduct Hopf algebras which are quasitriangular. We also construct a subgroup $Y^θX^θ$ associated to every order-reversing automorphism $θ$ of X. The corresponding Hopf algebra $kX^θ\cobicross k(Y^θ)$ has the same coalgebra as H. Using related results, we classify the first order bicovariant differential calculi on H…
