Cherednik Algebras Research Articles

Quantum toroidal algebras and their representations

Quantum toroidal algebras (or double affine quantum algebras) are defined from quantum affine Kac–Moody algebras by using the Drinfeld quantum affinization process. They are quantum group analogs of elliptic Cherednik algebras (elliptic double affine Hecke algebras) to which they are related via Schur–Weyl duality. In this review paper, we give a glimpse of some aspects of their very rich representation theory in the context of general quantum affinizations. We illustrate the theory with several examples. We also announce new results and explain possible further developments, in particular on finite-dimensional representations at roots of unity.

Parking functions and vertex operators

We introduce several associative algebras and families of vector spaces associated to these algebras. Using lattice vertex operators, we obtain dimension and character formulae for these spaces. In particular, we define a family of representations of symmetric groups which turn out to be isomorphic to parking function modules. We also construct families of vector spaces whose dimensions are Catalan numbers and Fuss–Catalan numbers respectively. Conjecturally, these spaces are related to spaces of global sections of vector bundles on (zero fibres of) Hilbert schemes and representations of rational Cherednik algebras.

Radial Components, Prehomogeneous Vector Spaces, and Rational Cherednik Algebras

Let be a finite-dimensional representation of a connected reductive complex Lie group . Denote by G the derived subgroup of and assume that the categorical quotient is one dimensional, i.e. for a nonconstant polynomial f. In this situation there exists a homomorphism , the radial component map, where is the first Weyl algebra. We show that the image of is isomorphic to the spherical subalgebra of a rational Cherednik algebra whose multiplicity function is defined by the roots of the Bernstein-Sato polynomial of f. In the case where is also multiplicity free we describe the kernel of and prove a Howe duality result between representations of G occurring in and lowest-weight modules over the Lie algebra generated by f and the “dual” differential operator ; this extends results of H. Rubenthaler obtained when is a parabolic prehomogeneous vector space. If satisfies a Capelli-type condition, some applications are given to holonomic and equivariant D-modules on V. These applications are related to results proved by Muro [34, 37, 38] or Nang [40, 42, 44] in special cases of the representation .

Braided doubles and rational Cherednik algebras

We introduce and study a large class of algebras with triangular decomposition which we call braided doubles. Braided doubles provide a unifying framework for classical and quantum universal enveloping algebras and rational Cherednik algebras. We classify braided doubles in terms of quasi-Yetter–Drinfeld (QYD) modules over Hopf algebras which turn out to be a generalisation of the ordinary Yetter–Drinfeld modules. To each braiding (a solution to the braid equation) we associate a QYD-module and the corresponding braided Heisenberg double—this is a quantum deformation of the Weyl algebra where the role of polynomial algebras is played by Nichols–Woronowicz algebras. Our main result is that any rational Cherednik algebra canonically embeds in the braided Heisenberg double attached to the corresponding complex reflection group.

Hecke–Clifford algebras and spin Hecke algebras, II: The rational double affine type

The notion of rational spin double affine Hecke algebras (sDaHa) and rational double affine Hecke-Clifford algebras (DaHCa) associated to classical Weyl groups are introduced. The basic properties of these algebras such as the PBW basis and Dunkl operator representations are established. An algebra isomorphism relating the rational DaHCa to the rational sDaHa is obtained. We further develop a link between the usual rational Cherednik algebra and the rational sDaHa by introducing a notion of rational covering double affine Hecke algebras.

Factorization in generalized Calogero–Moser spaces

Using a recent construction of Bezrukavnikov and Etingof, [R. Bezrukavnikov, P. Etingof, Induction and restriction functors for rational Cherednik algebras, arXiv: 0803.3639], we prove that there is a factorization of the Etingof–Ginzburg sheaf on the generalized Calogero–Moser space associated to a complex reflection group. In the case W = S n , this confirms a conjecture of Etingof and Ginzburg, [P. Etingof, V. Ginzburg, Symplectic reflection algebras, Calogero–Moser space, and deformed Harish-Chandra homomorphisms, Invent. Math 147 (2002) 243–348].

Microlocalization of rational Cherednik algebras

We construct a microlocalization of the rational Cherednik algebras H of type Sn. This is achieved by a quantization of the Hilbert scheme HilbnC2 of n points in C2. We then prove the equivalence of the category of H-modules and that of modules over its microlocalization under certain conditions on the parameter

On elliptic Dunkl operators

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.

Morita Invariance of the Filter Dimension and of the Inequality of Bernstein

It is proved that the filter dimension is Morita invariant. A direct consequence of this fact is the Morita invariance of the inequality of Bernstein: if an algebra A is Morita equivalent to the ring \({\cal D} (X)\) of differential operators on a smooth irreducible affine algebraic variety X of dimension n ≥ 1 over a field of characteristic zero then the Gelfand–Kirillov dimension \( {\rm GK} (M)\geq n = \frac{{\rm GK} (A)}{2}\) for all nonzero finitely generated A-modules M. In fact, a stronger result is proved, namely, a Morita invariance of the holonomic number for finitely generated algebra. A direct consequence of this fact is that an analogue of the inequality of Bernstein holds for the (simple) rational Cherednik algebras H c for integral c: \({\rm GK} (M)\geq n =\frac{{\rm GK} (H_c)}{2}\) for all nonzero finitely generated H c -modules M. For these class of algebras, it gives an affirmative answer to a question of Ken Brown about symplectic reflection algebras.

$q,t$-Fuß-Catalan numbers for complex reflection groups

In type $A$, the $q,t$-Fuß-Catalan numbers $\mathrm{Cat}_n^{(m)}(q,t)$ can be defined as a bigraded Hilbert series of a module associated to the symmetric group $\mathcal{S}_n$. We generalize this construction to (finite) complex reflection groups and exhibit some nice conjectured algebraic and combinatorial properties of these polynomials in $q$ and $t$. Finally, we present an idea how these polynomials could be related to some graded Hilbert series of modules arising in the context of rational Cherednik algebras. This is work in progress. Dans le cas du type $A$, les $q,t$-nombres de Fuß-Catalan $\mathrm{Cat}_n^{(m)}(q,t)$ peuvent être définis comme la série de Hilbert bigraduée d'un certain module associé au groupe symétrique $\mathcal{S}_n$. Nous généralisons cette construction aux groupes de réflexion complexes (finis) et nous formulons de jolies propriétés (conjecturales) algébriques et combinatoires de ces polynômes en $q$ et $t$. Enfin, nous décrivons une idée sur la manière dont ces polynômes pourraient être liés à certaines séries de Hilbert de modules apparaissant dans le contexte des algèbres de Cherednik rationnelles. Ceci est un travail en cours.

On category $\mathcal{O}$ for the rational Cherednik algebra of $G(m,1,n)$: the almost semisimple case

We determine the structure of category $\mathcal{O}$ for the rational Cherednik algebra of the wreath product complex reflection group $G(m,1,n)$ in the case where the $\mathsf{KZ}$ functor satisfies a condition called separating simples. As a consequence, we show that the property of having exactly $N-1$ simple modules, where $N$ is the number of simple modules of $G(m,1,n)$, determines the Ariki-Koike algebra up to isomorphism.

Q-Schur Algebras and Complex Reflection Groups

We show that the category O for a rational Cherednik algebra of type A is equivalent to modules over a q-Schur algebra (parameter not a half integer), providing thus character formulas for simple modules. We give some generalization to B_n(d). We prove an ``abstract'' translation principle. These results follow from the unicity of certain highest categories covering Hecke algebras. We also provide a semi-simplicity criterion for Hecke algebras of complex reflection groups.

Characteristic cycles of standard modules for the rational Cherednik algebra of type $\mathbb{Z}/l\mathbb{Z}$

We study the representation theory of the rational Cherednik algebra $H_{\kappa}=H_{\kappa}(\mathbb{Z}_{l})$ for the cyclic group $\mathbb{Z}_{l}= \mathbb{Z}/l\mathbb{Z}$ and its connection with the geometry of the quiver variety $\mathfrak{M}_{\theta}(\delta )$ of type $A^{(1)}_{l-1}$. We consider a functor between the categories of $H_{\kappa}$-modules with different parameters, called the shift functor, and give the condition when it is an equivalence of categories. We also consider a functor from the category of $H_{\kappa}$-modules with good filtration to the category of coherent sheaves on $\mathfrak{M}_{\theta}(\delta )$. We prove that the image of the regular representation of $H_{\kappa}$ by this functor is the tautological bundle on $\mathfrak{M}_{\theta}(\delta )$. As a corollary, we determine the characteristic cycles of the standard modules. It gives an affirmative answer to a conjecture given in [Go] in the case of $\mathbb{Z}_{l}$.

Recollement of Deformed Preprojective Algebras and the Calogero–Moser Correspondence

The aim of this paper is to clarify the relation between the following objects: (a) rank 1 projective modules (ideals) over the first Weyl algebra A_1; (b) simple modules over deformed preprojective algebras $\Pi_{\lambda}(Q)$ introduced by Crawley-Boevey and Holland; and (c) simple modules over the rational Cherednik algebras $H_{0,c}(S_n)$ associated to symmetric groups. The isomorphism classes of each type of these objects can be parametrized geometrically by the same space (namely, the Calogero-Moser algebraic varieties); however, no natural functors between the corresponding module categories seem to be known. We construct such functors by translating our earlier results on A-infinity modules over A_1 to a more familiar setting of representation theory. In the last section we extend our construction to the case of Kleinian singularities $C^2/\Gamma$, where $\Gamma$ is a finite cyclic subgroup of $SL(2,C)$.

Cherednik, Hecke and quantum algebras as free Frobenius and Calabi–Yau extensions

We show how the existence of a PBW-basis and a large enough central subalgebra can be used to deduce that an algebra is Frobenius. We apply this to rational Cherednik algebras, Hecke algebras, quantised universal enveloping algebras, quantum Borels and quantised function algebras. In particular, we give a positive answer to [R. Rouquier, Representations of rational Cherednik algebras, in: Infinite-Dimensional Aspects of Representation Theory and Applications, Amer. Math. Soc., 2005, pp. 103–131] stating that the restricted rational Cherednik algebra at the value t = 0 is symmetric.

Generalized double affine Hecke algebras of rank 1 and quantized del Pezzo surfaces

Let D be a simply laced Dynkin diagram of rank r whose affinization has the shape of a star (i.e., D 4 , E 6 , E 7 , E 8 ). To such a diagram one can attach a group G whose generators correspond to the legs of the affinization, have orders equal to the leg lengths plus 1, and the product of the generators is 1. The group G is then a 2-dimensional crystallographic group: G = Z ℓ ⋉ Z 2 , where ℓ is 2, 3, 4, and 6, respectively. In this paper, we define a flat deformation H ( t , q ) of the group algebra C [ G ] of this group, by replacing the relations saying that the generators have prescribed orders by their deformations, saying that the generators satisfy monic polynomial equations of these orders with arbitrary roots (which are deformation parameters). The algebra H ( t , q ) for D 4 is the Cherednik algebra of type C ∨ C 1 , which was studied by Noumi, Sahi, and Stokman, and controls Askey–Wilson polynomials. We prove that H ( t , q ) is the universal deformation of the twisted group algebra of G, and that this deformation is compatible with certain filtrations on C [ G ] . We also show that if q is a root of unity, then for generic t the algebra H ( t , q ) is an Azumaya algebra, and its center is the function algebra on an affine del Pezzo surface. For generic q, the spherical subalgebra e H ( t , q ) e provides a quantization of such surfaces. We also discuss connections of H ( t , q ) with preprojective algebras and Painlevé VI.

Rational Cherednik algebras and diagonal coinvariants of [formula omitted

We construct a quotient ring of the ring of diagonal coinvariants of the complex reflection group W = G ( m , p , n ) and determine its graded character. This generalises a result of Gordon for Coxeter groups. The proof uses a study of category O for the rational Cherednik algebra of W.

Affine Yangians and deformed double current algebras in type A

We study the structure of Yangians of affine type and deformed double current algebras, which are deformations of the enveloping algebras of matrix W 1 + ∞ -algebras. We prove that they admit a PBW-type basis, establish a connection (limit construction) between these two types of algebras and toroidal quantum algebras, and we give three equivalent definitions of deformed double current algebras. We construct a Schur–Weyl functor between these algebras and rational Cherednik algebras.

Generalized Graded Hecke Algebras of Types B and D

For a root system of type B we study an algebra similar to a graded Hecke algebra, isomorphic to a subalgebra of the rational Cherednik algebra. We introduce principal series modules over it and prove an irreducibility criterion for these modules. We deduce similar results for an algebra associated to a root system of type D.

The non-degeneracy of the bilinear form of m-Quasi-Invariants

We give here a new proof of the non-degeneracy of the fundamental bilinear form for S n - m-Quasi-Invariants and for m-Quasi-Invariants of classical Weyl groups. We also indicate how our approach can be extended to other Coxeter groups. This bilinear form plays a crucial role in the original proof [P. Etingof, V. Ginzburg, On m-quasi-invariants of a Coxeter group, arXiv: math.QA/0106175 v1, June 2001] that m-Quasi-Invariants are a free module over the invariants as well as in all subsequent proofs [Y. Berest, P. Etingof, V. Ginsburg, Cherednik algebras and differential operators on quasi-invariants, math.QA/0111005; A. Garsia, N. Wallach, Some new applications of orbit harmonics, Sém. Lothar. Combin. 50 (2005), Article B50j]. However, in previous literature this non-degeneracy was stated and used without proof with reference to some deep results of Opdam [E.M. Opdam, Some applications of shift operators, Invent. Math. 98 (1989) 1–18] on shift-differential operators. This result hinges on the validity of a deceptively simple identity on Dunkl operators which, at least in the S n case, begs for an elementary painless proof. An elementary but by all means not painless proof of this identity can be found in a paper of Dunkl and Hanlon [C. Dunkl, P. Hanlon, Integrals of polynomials associated with tableaux and the Garsia–Haiman conjecture, Math. Z. 228 (1998) 537–567. 71]. Our proof here is not elementary but hopefully it should be painless and informative.