Cherednik Algebras Research Articles

Representations of the rational Cherednik algebra [formula omitted] in positive characteristic

We study the rational Cherednik algebra Ht,c(S3,h) of type A2 in positive characteristic p, and its irreducible category O representations Lt,c(τ). For every possible value of p,t,c, and τ we calculate the Hilbert polynomial and the character of Lt,c(τ), and give explicit generators of the maximal proper graded submodule of the Verma module.

The centre of the Dunkl total angular momentum algebra

For a finite dimensional representation V of a finite reflection group W, we consider the rational Cherednik algebra Ht,c(V,W) associated with (V,W) at the parameters t≠0 and c. The Dunkl total angular momentum algebra Ot,c(V,W) arises as the centraliser algebra of the Lie superalgebra osp(1|2) containing a Dunkl deformation of the Dirac operator, inside the tensor product of Ht,c(V,W) and the Clifford algebra generated by V.We show that when dim⁡V≥3 and for every value of the parameter c, the centre of Ot,c(V,W) is isomorphic to a univariate polynomial ring. Notably, the generator of the centre changes depending on whether or not (−1)V is an element of the group W. Using this description of the centre, and using the projection of the pseudo scalar from the Clifford algebra into Ot,c(V,W), we establish results analogous to “Vogan's conjecture” for a family of operators depending on suitable elements of the double cover W˜.

Lie algebra actions on module categories for truncated shifted yangians

Abstract We develop a theory of parabolic induction and restriction functors relating modules over Coulomb branch algebras, in the sense of Braverman-Finkelberg-Nakajima. Our functors generalize Bezrukavnikov-Etingof’s induction and restriction functors for Cherednik algebras, but their definition uses different tools. After this general definition, we focus on quiver gauge theories attached to a quiver $\Gamma $ . The induction and restriction functors allow us to define a categorical action of the corresponding symmetric Kac-Moody algebra $\mathfrak {g}_{\Gamma }$ on category $ \mathcal {O}$ for these Coulomb branch algebras. When $ \Gamma $ is of Dynkin type, the Coulomb branch algebras are truncated shifted Yangians and quantize generalized affine Grassmannian slices. Thus, we regard our action as a categorification of the geometric Satake correspondence. To establish this categorical action, we define a new class of ‘flavoured’ KLRW algebras, which are similar to the diagrammatic algebras originally constructed by the second author for the purpose of tensor product categorification. We prove an equivalence between the category of Gelfand-Tsetlin modules over a Coulomb branch algebra and the modules over a flavoured KLRW algebra. This equivalence relates the categorical action by induction and restriction functors to the usual categorical action on modules over a KLRW algebra.

Harish-Chandra bimodules for type A rational Cherednik algebras

We study Harish-Chandra bimodules for the rational Cherednik algebra associated to the symmetric group Sn. In particular, we show that for any parameter c∈C, the category of Harish-Chandra Hc-bimodules admits a fully faithful embedding into the category Oc, and describe the irreducibles in the image. We also construct a duality on the category of Harish-Chandra bimodules, and in fact we do this in a greater generality of quantizations of Nakajima quiver varieties. We use this duality, along with induction and restriction functors, to describe, as an abelian category, the smallest Serre subcategory of the category of Harish-Chandra bimodules containing the regular bimodule, as well as to explicitly describe the tensor products of its irreducible objects.

Computational aspects of Calogero–Moser spaces

We present a series of algorithms for computing geometric and representation-theoretic invariants of Calogero–Moser spaces and rational Cherednik algebras associated with complex reflection groups. In particular, we are concerned with Calogero–Moser families (which correspond to the mathbb {C}^times -fixed points of the Calogero–Moser space) and cellular characters (a proposed generalization by Rouquier and the first author of Lusztig’s constructible characters based on a Galois covering of the Calogero–Moser space). To compute the former, we devised an algorithm for determining generators of the center of the rational Cherednik algebra (this algorithm has several further applications), and to compute the latter we developed an algorithmic approach to the construction of cellular characters via Gaudin operators. We have implemented all our algorithms in the Cherednik Algebra Magma Package by the second author and used this to confirm open conjectures in several new cases. As an interesting application in birational geometry we are able to determine for many exceptional complex reflection groups the chamber decomposition of the movable cone of a mathbb {Q}-factorial terminalization (and thus the number of non-isomorphic relative minimal models) of the associated symplectic singularity. Making possible these computations was also a source of inspiration for the first author to propose several conjectures about the geometry of Calogero–Moser spaces (cohomology, fixed points, symplectic leaves), often in relation with the representation theory of finite reductive groups.

Rational Cherednik algebras of G(ℓ,p,n) from the Coulomb perspective

We prove a number of results on the structure and representation theory of the rational Cherednik algebra of the imprimitive reflection group G(ℓ,p,n). In particular, we: (1) show a relationship to the Coulomb branch construction of Braverman, Finkelberg, and Nakajima, and 3-dimensional quantum field theory; (2) show that the spherical Cherednik algebra carries the structure of a principal Galois order; (3) construct a graded lift of category O and the larger category of Dunkl-Opdam modules, whose simple modules have the properties of a dual canonical basis and (4) give the first classification of simple Dunkl-Opdam modules for the rational Cherednik algebra of the imprimitive reflection group G(ℓ,p,n).

Skew cellularity of the Hecke algebras of type 𝐺(ℓ,𝑝,𝑛)

This paper introduces (graded) skew cellular algebras, which generalise Graham and Lehrer’s cellular algebras. We show that all of the main results from the theory of cellular algebras extend to skew cellular algebras and we develop a “cellular algebra Clifford theory” for the skew cellular algebras that arise as fixed point subalgebras of cellular algebras. As an application of this general theory, the main result of this paper proves that the Hecke algebras of type G ( ℓ , p , n ) G(\ell ,p,n) are graded skew cellular algebras. In the special case when p = 2 p = 2 this implies that the Hecke algebras of type G ( ℓ , 2 , n ) G(\ell ,2,n) are graded cellular algebras. The proofs of all of these results rely, in a crucial way, on the diagrammatic Cherednik algebras of Webster and Bowman. Our main theorem extends Geck’s result that the one parameter Iwahori-Hecke algebras are cellular algebras in two ways. First, our result applies to all cyclotomic Hecke algebras in the infinite series in the Shephard-Todd classification of complex reflection groups. Secondly, we lift cellularity to the graded setting. As applications of our main theorem, we show that the graded decomposition matrices of the Hecke algebras of type G ( ℓ , p , n ) G(\ell ,p,n) are unitriangular, we construct and classify their graded simple modules and we prove the existence of “adjustment matrices” in positive characteristic.

Expression of Concern: Trace Densities and Algebraic Index Theorems for Sheaves of Formal Cherednik Algebras

Expression of Concern: Trace Densities and Algebraic Index Theorems for Sheaves of Formal Cherednik Algebras

Gaudin algebras, RSK and Calogero–Moser cells in Type A

AbstractWe study the spectrum of a family of algebras, the inhomogeneous Gaudin algebras, acting on the ‐fold tensor representation of the Lie algebra . We use the work of Halacheva–Kamnitzer–Rybnikov–Weekes to demonstrate that the Robinson–Schensted–Knuth correspondence describes the behaviour of the spectrum as we move along special paths in the family. We apply the work of Mukhin–Tarasov–Varchenko, which proves that the rational Calogero–Moser phase space can be realised as a part of this spectrum, to relate this to behaviour at of rational Cherednik algebras of . As a result, we confirm for symmetric groups a conjecture of Bonnafé–Rouquier which proposes an equality between the Calogero–Moser cells they defined and the well‐known Kazhdan–Lusztig cells.

Harish-Chandra Modules over Hopf Galois Orders

Abstract The theory of Galois orders was introduced by Futorny and Ovsienko [9]. We introduce the notion of $\mathcal {H}$-Galois $\Lambda $-orders. These are certain noncommutative orders $F$ in a smash product of the fraction field of a noetherian integral domain $\Lambda $ by a Hopf algebra ${\mathcal {H}}$ (or, more generally, by a coideal subalgebra of a Hopf algebra). They are generalizations of Webster’s [25] principal flag orders. Examples include Cherednik algebras, as well as examples from Hopf Galois theory. We also define spherical Galois orders, which are the corresponding generalizations of principal Galois orders introduced by the author [12]. The main results are (1) for every maximal ideal $\mathfrak {m}$ of $\Lambda $ of finite codimension, there exists a simple Harish-Chandra $F$-module in the fiber of $\mathfrak {m}$; (2) for every character of $\Lambda $, we construct a canonical simple Harish-Chandra module as a subquotient of the module of local distributions; (3) if a certain stabilizer coalgebra is finite-dimensional, then the corresponding fiber of simple Harish-Chandra modules is finite; and (4) centralizers of symmetrizing idempotents are spherical Galois orders and every spherical Galois order appears that way.

Deformed Calogero–Moser Operators and Ideals of Rational Cherednik Algebras

We introduce a class of hyperplane arrangements $$\mathcal {A}$$ in $${\mathbb {C}}^n$$ that generalise the locus configurations of Chalykh, Feigin and Veselov. To such an arrangement we associate a second order partial differential operator of Calogero–Moser type and prove that this operator is completely integrable (in the sense that its centraliser in $$\mathcal {D}({\mathbb {C}}^n\!\setminus \!\mathcal {A})$$ contains a maximal commutative subalgebra of Krull dimension n). Our approach is based on the study of shift operators and associated ideals in spherical Cherednik algebras that may be of independent interest. Examples include all known completely integrable deformations of Calogero–Moser operators with rational potentials. In addition, we construct new families of examples, including a BC-type generalisation of the deformed Calogero-Moser operators recently found by Gaiotto and Rapčák. We describe these examples in a unified representation-theoretic framework of rational Cherednik algebras.

On the basic representation of the double affine Hecke algebra at critical level

We construct the basic representation of the double affine Hecke algebra at critical level [Formula: see text] associated to an irreducible reduced affine root system [Formula: see text] with a reduced gradient root system. For [Formula: see text] of untwisted type such a representation was studied by Oblomkov [A. Oblomkov, Double affine Hecke algebras and Calogero–Moser spaces, Represent. Theory 8(2004) 243–266] and further detailed by Gehles [K. E. Gehles, Properties of Cherednik algebras and graded Hecke algebras, PhD thesis, University of Glasgow (2006)] in the presence of minuscule weights.

Twists of rational Cherednik algebras

Abstract We show that braided Cherednik algebras introduced by Bazlov and Berenstein are cocycle twists of rational Cherednik algebras of the imprimitive complex reflection groups $G(m,p,n)$, when m is even. This gives a new construction of mystic reflection groups which have Artin–Schelter regular rings of quantum polynomial invariants. As an application of this result, we show that a braided Cherednik algebra has a finite-dimensional representation if and only if its rational counterpart has one.

Wreath Macdonald polynomials at 𝑞=𝑡 as characters of rational Cherednik algebras

Using the theory of Macdonald [Symmetric functions and Hall polynomials, The Clarendon Press, Oxford University Press, New York, 1995], Gordon [Bull. London Math. Soc. 35 (2003), pp. 321–336] showed that the graded characters of the simple modules for the restricted rational Cherednik algebra by Etingof and Ginzburg [Invent. Math. 147 (2002), pp. 243–348] associated to the symmetric group S n \mathfrak {S}_n are given by plethystically transformed Macdonald polynomials specialized at q = t q=t . We generalize this to restricted rational Cherednik algebras of wreath product groups C ℓ ≀ S n C_\ell \wr \mathfrak {S}_n and prove that the corresponding characters are given by a specialization of the wreath Macdonald polynomials defined by Haiman in [Combinatorics, symmetric functions, and Hilbert schemes, Int. Press, Somerville, MA, 2003, pp. 39–111].

Knizhnik–Zamolodchikov functor for degenerate double affine Hecke algebras: algebraic theory

In this article, we define an algebraic version of the Knizhnik–Zamolodchikov (KZ) functor for the degenerate double affine Hecke algebras (a.k.a. trigonometric Cherednik algebras). We compare it with the KZ monodromy functor constructed by Varagnolo–Vasserot. We prove the double centraliser property for our functor and give a characterisation of its kernel. We establish these results for a family of algebras, called quiver double Hecke algebras, which includes the degenerate double affine Hecke algebras as special cases.

From Representations of the Rational Cherednik Algebra to Parabolic Hilbert Schemes via the Dunkl-Opdam Subalgebra

From Representations of the Rational Cherednik Algebra to Parabolic Hilbert Schemes via the Dunkl-Opdam Subalgebra

Algebra of Dunkl Laplace–Runge–Lenz vector

We introduce the Dunkl version of the Laplace–Runge–Lenz vector associated with a finite Coxeter group W acting geometrically in mathbb R^N and with a multiplicity function g. This vector generalizes the usual Laplace–Runge–Lenz vector and its components commute with the Dunkl–Coulomb Hamiltonian given as the Dunkl Laplacian with an additional Coulomb potential gamma /r. We study the resulting symmetry algebra R_{g, gamma }(W) and show that it has the Poincaré–Birkhoff–Witt property. In the absence of a Coulomb potential, this symmetry algebra R_{g,0}(W) is a subalgebra of the rational Cherednik algebra H_g(W). We show that a central quotient of the algebra R_{g, gamma }(W) is a quadratic algebra isomorphic to a central quotient of the corresponding Dunkl angular momenta algebra H_g^{so(N+1)}(W). This gives an interpretation of the algebra H_g^{so(N+1)}(W) as the hidden symmetry algebra of the Dunkl–Coulomb problem in mathbb {R}^N. By specialising R_{g, gamma }(W) to g=0, we recover a quotient of the universal enveloping algebra U(so(N+1)) as the hidden symmetry algebra of the Coulomb problem in {mathbb R}^N. We also apply the Dunkl Laplace–Runge–Lenz vector to establish the maximal superintegrability of the generalised Calogero–Moser systems.

Dirac Operators for the Dunkl Angular Momentum Algebra

We define a family of Dirac operators for the Dunkl angular momentum algebra depending on certain central elements of the group algebra of the Pin cover of the Weyl group inherent to the rational Cherednik algebra. We prove an analogue of Vogan's conjecture for this family of operators and use this to show that the Dirac cohomology, when non-zero, determines the central character of representations of the angular momentum algebra. Furthermore, interpreting this algebra in the framework of (deformed) Howe dualities, we show that the natural Dirac element we define yields, up to scalars, a square root of the angular part of the Calogero-Moser Hamiltonian.

Poisson Noether's Problem and Poisson rationality

In this paper we show that the Poisson analogue of the Noether's Problem has a positive solution for essentially all symplectic reflection groups — the analogue of complex reflection groups in the symplectic world. Our proofs are constructive, and generalize and refine previously known results. The results of this paper can be thought as analogues of the Noncommutative Noether Problem and the Gelfand-Kirillov Conjecture for rational Cherednik algebras in the quasi-classical limit. An abstract framework to understand these results is introduced. As a consequence for complex reflection groups, we obtain the Poisson rationality of the Calogero-Moser spaces associated to any of them, and we verify the Gelfand-Kirillov Conjecture for trigonometric Cherednik algebras and the Poisson rationality of their corresponding Calogero-Moser spaces.

Cellularity of endomorphism algebras of tilting objects

We show that, in a highest weight category with duality, the endomorphism algebra of a tilting object is naturally a cellular algebra. Our proof generalizes a recent construction of Andersen, Stroppel, and Tubbenhauer [4]. This result raises the question of whether all cellular algebras can be realized in this way. The construction also works without the presence of a duality and yields standard bases, in the sense of Du and Rui, which have similar combinatorial features to cellular bases. As an application, we obtain standard bases—and thus a general theory of “cell modules”—for Hecke algebras associated to finite complex reflection groups (as introduced by Broué, Malle, and Rouquier) via category O of the rational Cherednik algebra. For real reflection groups these bases are cellular.