In the case of rational Cherednik algebras associated with cyclic groups, we give an alternative proof that the projective object $P_{\text{KZ}}$ representing the KZ-functor is isomorphic to the $Δ$-module associated with the coinvariant algebra for a subset of parameter values from which all parameter values can be obtained by integral translations. We also specify the exact parameter values for which this isomorphism occurs. Furthermore, we determine the action of the cyclotomic Hecke algebra on $P_{\text{KZ}}$ for these parameter values, thereby giving a complete algebraic description of the KZ-functor in this case.
Bezrukavnikov and Etingof introduced some functors between the categories O for rational Cherednik algebras. Namely, they defined two induction functors Ind_b, ind_λand two restriction functors Res_b,res_λ. They conjectured that one has functor isomorphisms $Ind_b\cong ind_λ, Res_b\cong res_λ$. The goal of this paper is to prove this conjecture.
Ginzburg, Guay, Opdam and Rouquier established an equivalence of categories between a quotient category of the category $\mathcal{O}$ for the rational Cherednik algebra and the category of finite dimension modules of the Hecke algebra of a complex reflection group $W$. We establish two generalizations of this result. On the one hand to the extension of the Hecke algebra associated to the normaliser of a reflection subgroup and on the other hand to the extension of the Hecke algebra by a lattice.
We prove that the KZ functor from a certain category of modules for the Cherednik algebra to finite dimensional modules over the Hecke algebra is essentially surjective. Then we begin to use this result to study the analog of category O for Cherednik algebras on Riemann surfaces and on products of elliptic curves. In particular we give conditions on the parameters under which these categories are nonzero.
Following the work of Kashiwara-Rouquier and Gan-Ginzburg, we define a family of exact functors from category $\mathcal O$ for the rational Cherednik algebra in type $A$ to representations of certain "coloured braid groups" and calculate the dimensions of the representations thus obtained from standard modules. To show that our constructions also make sense in a more general context, we also briefly study the case of the rational Cherednik algebra corresponding to complex reflection group $\mathbb Z/l\mathbb Z$.
We introduce parabolic induction and restriction functors for rational Cherednik algebras, and study their basic properties. Then we discuss applications of these functors to representation theory of rational Cherednik algebras. In particular, we prove the Gordon-Stafford theorem about Morita equivalence of the rational Cherednik algebra for type A and its spherical subalgebra, without the assumption that c is not a half-integer, which was required up to now. Also, we classify representations from category O over the rational Cherednik algebras of type A which do not contain an S_n-invariant v…
In this paper, we shall establish the Mackey formulas in the following two set ups: (i) on the tensor induction and restriction functors on the modules over cyclotomic Hecke algebras (Ariki-Koike algebras) and their standard subalgebras of parabolic subgroups. (ii) on the Bezrukavnikov-Etingof induction and restriction functors among categories O of rational Cherednik algebras for the complex reflection group of type G(r, 1, n) and their parabolic subgroups.
In this paper we define and study a critical-level generalization of the Suzuki functor, relating the affine general linear Lie algebra to the rational Cherednik algebra of type A. Our main result states that this functor induces a surjective algebra homomorphism from the centre of the completed universal enveloping algebra at the critical level to the centre of the rational Cherednik algebra at t=0. We use this homomorphism to obtain several results about the functor. We compute it on Verma modules, Weyl modules, and their restricted versions. We describe the maps between endomorphism rings i…
We introduce the notion of a totally aspherical parameter for a Rational Cherednik algebra. We get an explicit construction of the projective object defining the KZ functor for such parameters. We establish the existence of sufficiently many totally aspherical parameters for the groups G(l,1,n).
We show that the spherical subalgebra of the rational Cherednik algebra associated to the wreath product of a symmetric group and a cyclic group is isomorphic to a quotient of the ring of invariant differential operators on a space of representations of the cyclic quiver. This confirms a version of a conjecture of Etingof and Ginzburg in the case of cyclic groups. The proof is a straightforward application of work of Oblomkov on the deformed Harish-Chandra homomorphism, and of Crawley-Boevey and of Gan and Ginzburg on preprojective algebras.
In this article, we describe all two sided ideals of a cyclotomic rational Cherednik algebra $H_\mathbf{c}$ and its spherical subalgebra $eH_\mathbf{c} e$ with a Weil generic aspherical parameter $\mathbf{c}$, and further describe the simple modules in the category $\mathcal{O}^{sph}_\mathbf{c}$ . The main tools we use are categorical Kac-Moody actions on catogories $\mathcal{O}_\mathbf{c}$ and restriction functors for Harish-Chandra bimodules.
In this paper we prove the existence of the Dunkl weight function $K_{c, λ}$ for any irreducible representation $λ$ of any finite Coxeter group $W$, generalizing previous results of Dunkl. In particular, $K_{c, λ}$ is a family of tempered distributions on the real reflection representation of $W$ taking values in $\text{End}_\mathbb{C}(λ)$, with holomorphic dependence on the complex multi-parameter $c$. When the parameter $c$ is real, the distribution $K_{c, λ}$ provides an integral formula for Cherednik's Gaussian inner product $γ_{c, λ}$ on the Verma module $Δ_c(λ)$ for the rational Cheredni…
In this note we are interested in labelling the irreducible representations of non-semisimple specialisations of Hecke algebras of complex reflection groups. We will use category O for the rational Cherednik algebra and the KZ functor together with elementary algebraic and combinatorial arguments to construct "canonical basic sets" in many cases. We will also show that the images of the standard modules through the KZ functor agree with the appropriate cell modules, whenever the Hecke algebra has a cellular structure.
In this paper, we classify the irreducible representations of the rational Cherednik algebras of rank 1 in characteristic p > 0. There are two cases. One is the "quantum" case, where "Planck's constant" is nonzero and generic irreducible representations have dimension pr, where r is the order of the cyclic group contained in the algebra. The other is the "classical" case, where "Planck's constant" is zero and generic irreducible representations have dimension r.
We study the representation theory of the rational Cherednik algebra $H_κ= H_κ({\mathbb Z}_l)$ for the cyclic group ${\mathbb Z}_l = {\mathbb Z} / l {\mathbb Z}$ and its connection with the geometry of the quiver variety $M_θ(δ)$ of type $A_{l-1}^{(1)}$. We consider a functor between the categories of $H_κ$-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 $H_κ$-modules with good filtration to the category of coherent sheaves on $M_θ(δ)$. We prove that the image of the reg…
We give an overview of the representation theory of restricted rational Cherednik algebras. These are certain finite-dimensional quotients of rational Cherednik algebras at t=0. Their representation theory is connected to the geometry of the Calogero-Moser space, and there is a lot of evidence that they contain certain information about Hecke algebras even though the precise connection is so far unclear. We outline the basic theory along with some open problems and conjectures, and give explicit results in the cyclic and dihedral cases.
We study canonical intertwining operators between modules of the trigonometric Cherednik algebra, induced from the standard modules of the degenerate affine Hecke algebra. We show that these operators correspond to the Zhelobenko operators for the affine Lie algebra $\widehat{\mathfrak{sl}}_m$. To establish the correspondence, we use the functor of Arakawa, Suzuki and Tsuchiya which maps certain $\widehat{\mathfrak{sl}}_m$-modules to modules of the Cherednik algebra.
We study holonomic D-modules on SL_n(C)xC^n, called mirabolic modules, analogous to Lusztig's character sheaves. We describe the supports of simple mirabolic modules. We show that a mirabolic module is killed by the functor of Hamiltonian reduction from the category of mirabolic modules to the category of representations of the trigonometric Cherednik algebra if and only if the characteristic variety of the module is contained in the unstable locus. We introduce an analogue of the Verdier specialization functor for representations of Cherednik algebras which agrees, on category O, with the res…
Let $W$ be a complex reflection group of the form $G(l,1,n)$. Following [BK12, BPW12, Gor06, GS05, GS06, KR08, MN11], the theory of deform quantising conical symplectic resolutions allows one to study the category of modules for the spherical Cherednik algebra, $U_\textbf{c}(W)$, via a functor, $\mathbb T_{\textbf{c},θ}$, which takes invariant global sections of certain twisted sheaves on some Nakajima quiver variety $Y_θ$. A parameter for the Cherednik algebra, $\textbf{c}$, is considered `good' if there exists a choice of GIT parameter $θ$, such that $\mathbb T_{\textbf{c},θ}$ is exact and `…
We determine the structure of category $\cO$ for the rational Cherednik algebra of $G(m,1,n)$ in the case where the $\KZ$ functor satisfies a condition called \emph{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.
We study the category O of representations of the rational Cherednik algebra A attached to a complex reflection group W. We construct an exact functor, called Knizhnik-Zamolodchikov functor, from O to the category of H-modules, where H is the (finite) Iwahori-Hecke algebra associated to W. We prove that the Knizhnik-Zamolodchikov functor induces an equivalence between O/O_tor, the quotient of O by the subcategory of A-modules supported on the discriminant and the category of finite-dimensional H-modules. The standard A-modules go, under this equivalence, to certain modules arising in Kazhdan-L…
We study a connection between the representation theory of the rational Cherednik algebra of type $GL_n$ and the representation theory of the degenerate double affine Hecke algebra (the degenerate DAHA). We focus on an algebra embedding from the rational Cherednik algebra to the degenerate DAHA and investigate the induction functor through this embedding. We prove that this functor embeds the category ${\mathcal O}$ for the rational Cherednik algebra fully faithfully into the category ${\mathcal O}$ for the degenerate DAHA. We also study the full subcategory ${\mathcal O}^{ss}$ of ${\mathcal O…
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…
Arakawa, Suzuki and Tsuchiya constructed a correspondence between certain modules of the trigonometric Cherednik algebra $\mathfrak{C}_N$ depending on a parameter $κ\in\mathbb{C}$, and certain modules of the affine Lie algebra $\widehat{\mathfrak{sl}}_m$ of level $κ-m$. We give a detailed proof of this correspondence by working with the affine Lie algebra $\widehat{\mathfrak{gl}}_m$ alongside of $\widehat{\mathfrak{sl}}_m$. We also relate this construction to a correspondence between certain modules of the degenerate affine Hecke algebra $\mathfrak{H}_N$ and all modules of $\mathfrak{sl}_m$ or…
We give an alternate presentation of the cyclotomic rational Cherednik algebra, which has the useful feature of compatibility with the Opdam-Dunkl subalgebra. This presentation has a diagrammatic flavor, and it provides a simple explanation of several surprising facts about this algebra. It allows direct proof of the connection of category $\mathcal{O}$ to weighted KLR algebras, allows us to classify the simple Dunkl-Opdam modules over the Cherednik algebra and provides an algebraic construction of the KZ functor. Furthermore, one of prime motivations for considering this approach is to provid…
For a complex reflection group $W$ with reflection representation $\mathfrak{h}$, we define and study a natural filtration by Serre subcategories of the category $\mathcal{O}_c(W, \mathfrak{h})$ of representations of the rational Cherednik algebra $H_c(W, \mathfrak{h})$. This filtration refines the filtration by supports and is analogous to the Harish-Chandra series appearing in the representation theory of finite groups of Lie type. Using the monodromy of the Bezrukavnikov-Etingof parabolic restriction functors, we show that the subquotients of this filtration are equivalent to categories of …
We study a scheme M closely related to the set of pairs of n by n-matrices with rank 1 commutator. We show that M is a reduced complete intersection with n+1 irreducible components, which we describe. There is a distinguished Lagrangian subvariety Nil in M. We introduce a category, C, of D-modules whose characteristic variety is contained in Nil. Simple objects of that category are analogous to Lusztig's character sheaves. We construct a functor of Quantum Hamiltonian reduction from category C to the category O for type A rational Cherednik algebra.
We compare crystal combinatorics of the level 2 Fock space with the classification of unitary irreducible representations of type B rational Cherednik algebras to study how unitarity behaves under parabolic restriction. First, we show that any finite-dimensional unitary irreducible representation of such an algebra is labeled by a bipartition consisting of a rectangular partition in one component and the empty partition in the other component. This is a new proof of a result that can be deduced from theorems of Montarani and Etingof-Stoica. Second, we show that the crystal operators that remov…
In this article, we define an algebraic version of the Knizhnik--Zamolodchikov 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.
We give a detailed account of a combinatorial construction, due to Cherednik, of cyclic generators for irreducible modules of the affine Hecke algebra of the general linear group with generic parameter q.
We introduce an analogue of the composition of the Cherednik and Drinfeld functor for twisted Yangians. Our definition is based on the Howe duality, and originates from the centralizer construction of twisted Yangians due to Olshanski. Using our functor, we establish a correspondence between intertwining operators on the tensor products of certain modules over twisted Yangians, and the extremal cocycle on the hyperoctahedral group.
We introduce a skew analogue of the composition of the Cherednik and Drinfeld functors for twisted Yangians. Our definition is based on the skew Howe duality, and originates from the centralizer construction of twisted Yangians due to Olshanski. Using our functor, we establish a correspondence between intertwining operators on the tensor products of certain modules over twisted Yangians, and the extremal cocycle on the hyperoctahedral group.
We determine the composition factors of the $A$-fibered Burnside functor $kB^A$ for $p$-groups over a field $k$ of characteristic $q$ with $q\neq p$ and cyclic fiber group $A$. We also show that, in this case, $kB^A$ is uniserial.
We prove a conjecture of Rouquier relating the decomposition numbers in category $\mathcal{O}$ for a cyclotomic rational Cherednik algebra to Uglov's canonical basis of a higher level Fock space. Independent proofs of this conjecture have also recently been given by Rouquier, Shan, Varagnolo and Vasserot and by Losev, using different methods. Our approach is to develop two diagrammatic models for this category $\mathcal{O}$; while inspired by geometry, these are purely diagrammatic algebras, which we believe are of some intrinsic interest. In particular, we can quite explicitly describe the re…
Let $W$ be a complex reflection group. We formulate a conjecture relating blocks of the corresponding restricted rational Cherednik algebras and Rouquier families for cyclotomic Hecke algebras. We verify the conjecture in the case that $W$ is a wreath product of a symmetric group with a cyclic group of order $l$.
We prove that the simple root functors E and F appearing in the categorification of irreducible highest weight modules of quantum groups via cyclotomic Khovanov-Lauda-Rouquier algebras is a biadjoint pair.
We study a composition of two functors. The first one, from the category of modules over the Lie algebra $\gl_m$ to the category of modules over the degenerate affine Hecke algebra of $GL_N$, was introduced by I. Cherednik. The second functor is a skew version of the functor from the latter category to the category of modules over the Yangian $\Y(\gl_n)$ due to V. Drinfeld. We give a representation theoretic explanation of a link between intertwining operators on tensor products of $\Y(\gl_n)$-modules, and the `extremal cocycle' on the Weyl group of $\gl_m$ introduced by D. Zhelobenko. We esta…
For a finite cyclic p-group G and a discrete valuation domain R of characteristic 0 with maximal ideal pR the R[G]-permutation modules are characterized in terms of the vanishing of first degree cohomology on all sub- groups (cf. Thm. A). As a consequence any R[G]-lattice can be presented by R[G]-permutation modules (cf. Thm. C). The proof of these results is based on a detailed analysis of the category of cohomological G-Mackey functors with values in the category of R-modules. It is shown that this category has global dimension 3 (cf. Thm. E). A crucial step in the proof of Theorem E is the …
We study the composition of the functor from the category of modules over the Lie algebra gl_m to the category of modules over the degenerate affine Hecke algebra of GL_N introduced by I. Cherednik, with the functor from the latter category to the category of modules over the Yangian Y(gl_n) due to V. Drinfeld. We propose a representation theoretic explanation of a link between the intertwining operators on the tensor products of Y(gl_n)-modules, and the `extremal cocycle' on the Weyl group of gl_m defined by D. Zhelobenko. We also establish a connection between the composition of two functors…
We show that the partially spherical cyclotomic rational Cherednik algebra (obtained from the full rational Cherednik algebra by averaging out the cyclotomic part of the underlying reflection group) has four other descriptions: (1) as a subalgebra of the degenerate DAHA of type A given by generators; (2) as an algebra given by generators and relations; (3) as an algebra of differential-reflection operators preserving some spaces of functions; (4) as equivariant Borel-Moore homology of a certain variety. Also, we define a new $q$-deformation of this algebra, which we call cyclotomic DAHA. Namel…
We begin the study of unitary representations of Hecke algebras of complex reflections groups. We obtain a complete classification for the Hecke algebra of the symmetric group $\mathfrak{S}_n$ over the complex numbers. Interestingly, the unitary representations in the category $\OO$ for the rational Cherednik algebra of type A studied in \cite{ESG} correspond to the unitary representations of the corresponding Hecke algebra via the KZ functor defined in \cite{GGOR}.
Based on a construction by Kashiwara and Rouquier, we present an analogue of the Beilinson- Bernstein localization theorem for hypertoric varieties. In this case, sheaves of differential operators are replaced by sheaves of W-algebras. As a special case, our result gives a localization theorem for rational Cherednik algebras associated to cyclic groups.
In this article we prove that there exists an explicit bijection between nice $d$-pre-Calabi-Yau algebras and $d$-double Poisson differential graded algebras, where $d \in \mathbb{Z}$, extending a result proved by N. Iyudu and M. Kontsevich. We also show that this correspondence is functorial in a quite satisfactory way, giving rise to a (partial) functor from the category of $d$-double Poisson dg algebras to the partial category of $d$-pre-Calabi-Yau algebras. Finally, we further generalize it to include double $P_{\infty}$-algebras, as introduced by T. Schedler.
We equip the categorified quantum group attached to a KLR algebra and an arbitrary choice of scalars with duality functor which is cyclic, that is, such that f=f^** for all 2-morphisms f. This is accomplished via a modified diagrammatic formalism.
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(\C)$; $ (b) $ simple modules over deformed preprojective algebras $ Π_λ(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 …
We describe the derived functor DRep_V(A) of the affine representation scheme Rep_V(A), parametrizing the representations of an associative k-algebra A on a finite-dimensional vector space V. We construct the characteristic maps Tr_V(A)_n: HC_n(A) \to H_n[DRep_V(A)], extending the canonical trace Tr_V(A): HC_0(A) \to k[Rep_V(A)] to the higher cyclic homology of the algebra A, and describe a related derived version of the representation functor introduced recently by M. Van den Bergh. We study various operations on the homology of DRep_V(A) induced by known operations on cyclic and Hochschild h…
We introduce and begin to study Lie theoretical analogs of symplectic reflection algebras for a finite cyclic group, which we call "cyclic double affine Lie algebra". We focus on type A : in the finite (resp. affine, double affine) case, we prove that these structures are finite (resp. affine, toroidal) type Lie algebras, but the gradings differ. The case which is essentially new involves $\mathbb{C}[u,v]$. We describe its universal central extensions and start the study of its representation theory, in particular of its highest weight integrable modules and Weyl modules. We also consider the …
The universal enveloping algebra $U(\mathfrak{tr}_n)$ of a Lie algebra associated to the classical Yang-Baxter equation was introduced in [BEER06] where it was shown to be Koszul. This algebra appears as the $A_{n-1}$ case in a general class of braided Hopf algebras in [BB09] for any complex reflection group. In this paper, we show that the algebras corresponding to the series $B_n$ and $D_n$, which are again universal enveloping algebras, are Koszul. We further show how results of [BB09] can be used to produce pairs of adjoint functors between categories of rational Cherednik algebra represen…
Restriction of functions from a reductive p-adic group G to its compact subgroups defines an operator on the Hochschild and cyclic homology of the Hecke algebra of G. We study the commutation relations between this operator and others coming from representation theory: Jacquet functors, idempotents in the Bernstein centre, and characters of admissible representations.
Motivated by Lusztig's $G$-stable pieces, we consider the combinatorial pieces: the pairs $(w, K)$ for elements $w$ in the Weyl group and subsets $K$ of simple reflections that are normalized by $w$. We generalize the notion of cyclic shift classes on the Weyl groups to the set of combinatorial pieces. We show that the partial cyclic shift classes of combinatorial pieces associated with minimal-length elements have nice representatives. As applications, we prove the left-right symmetry and the compatibility of the induction functors of the parabolic character sheaves.
