Rational Cherednik Algebras Research Articles

Dirac cohomology of the Dunkl-Opdam subalgebra via inherited Drinfeld properties

In this paper, we define a new presentation for the Dunkl-Opdam subalgebra of the rational Cherednik algebra. This presentation uncovers the Dunkl-Opdam subalgebra as a Drinfeld algebra. We use this fact to define Dirac cohomology for the DO subalgebra. We also formalize generalized graded Hecke algebras and extend a Langlands classification to generalized graded Hecke algebras.

Signatures of representations of Hecke algebras and rational Cherednik algebras

Determining whether an irreducible representation of a group (or ⁎-algebra) admits a non-degenerate invariant, positive-definite Hermitian form is an important problem in representation theory. In this paper, we study the related notion of signatures. Let q∈C with |q|=1, κ=−1/c with c∈Q<0 with certain additional (mild) restrictions, and n∈Z≥2. We study representations Sλ(q) of Hn(q), the Hecke algebra of type An−1, and representations Mκ(λ) of Hκ, the rational Cherednik algebra of type An−1, which have unique (up to scaling) invariant Hermitian forms (here λ is a partition of n). The signature is the number of elements with positive norm minus the number of elements with negative norm, and we analogously define the signature character in the case that there is a natural grading on the module. We provide formulas for (1) signatures of modules over Hn(q) and (2) signature characters of modules over Hκ. We study the limit c→−∞, in which case the signature character has a simpler form in terms of inversions and descents of permutations in Sn. We provide examples corresponding to some special shapes, and small values of n. Finally, when q=e2πic, we show that the asymptotic signature character of the Hκ-module Mκ(τ) is the signature of the Hn(q)-module Sτ(q).

Characterization of quasi-Yetter–Drinfeld modules

In this paper, we characterize quasi-Yetter–Drinfeld modules over a Hopf algebra [Formula: see text], which was introduced in [Y. Bazlov and A. Berenstein, Braided doubles and rational Cherednik algebras, Adv. Math. 220 (2009), 1466–1530]. We first show that the quasi-Drinfeld center of the category of [Formula: see text]-modules is equivalent to the category [Formula: see text] of quasi-Yetter–Drinfeld modules. Next, we prove that [Formula: see text] is equivalent to the category of generalized Hopf bimodules. Finally, we show that [Formula: see text] is also equivalent to the category of quasi-coactions over some Majid’s braided group if [Formula: see text] is quasi-triangular.

Towards a classification of finite-dimensional representations of rational Cherednik algebras of type D

Using a combinatorial description due to Jacon and Lecouvey of the wall crossing bijections for cyclotomic rational Cherednik algebras, we show that the irreducible representations [Formula: see text] of the rational Cherednik algebra [Formula: see text] of type [Formula: see text] for symmetric bipartitions [Formula: see text] are infinite dimensional for all parameters [Formula: see text]. In particular, all finite dimensional irreducible representations of rational Cherednik algebras of type [Formula: see text] arise as restrictions of finite-dimensional irreducible representations of rational Cherednik algebras of type [Formula: see text].

Rational Cherednik Algebras and Schubert Cells

The representation theory of rational Cherednik algebras of type A at t = 0 gives rise, by considering supports, to a natural family of smooth Lagrangian subvarieties of the Calogero-Moser space. The goal of this article is to make precise the relationship between these Lagrangian families and Schubert cells in the adelic Grassmannian. In order to do this we show that the isomorphism, as constructed by Etingof and Ginzburg, from the spectrum of the centre of the rational Cherednik algebra to the Calogero-Moser space is compatible with the factorization property of both of these spaces. As a consequence, the space of homomorphisms between certain representations of the rational Cherednik algebra can be identified with functions on the intersection of some Schubert cells.

The Smallest Singular Values and Vector-Valued Jack Polynomials

There is a space of vector-valued nonsymmetric Jack polynomials associated with any irreducible representation of a symmetric group. Singular polynomials for the smallest singular values are constructed in terms of the Jack polynomials. The smallest singular values bound the region of positivity of the bilinear symmetric form for which the Jack polynomials are mutually orthogonal. As background there are some results about general finite reflection groups and singular values in the context of standard modules of the rational Cherednik algebra.

Unitary representations of cyclotomic rational Cherednik algebras

We classify the irreducible unitary modules in category Oc for the rational Cherednik algebras of type G(r,1,n) and give explicit combinatorial formulas for their graded characters. More precisely, we produce a combinatorial algorithm determining, for each r-partition λ• of n, the closed semi-linear set of parameters c for which the contravariant form on the irreducible representation Lc(λ•) is positive definite. We use this algorithm to give a closed form answer for the Cherednik algebra of the symmetric group (recovering a result of Etingof–Stoica and the author) and the Weyl groups of classical type.

Dirac Induction for Rational Cherednik Algebras

AbstractWe introduce the local and global indices of Dirac operators for the rational Cherednik algebra $\mathsf{H}_{t,c}(G,\mathfrak{h})$, where $G$ is a complex reflection group acting on a finite-dimensional vector space $\mathfrak{h}$. We investigate precise relations between the (local) Dirac index of a simple module in the category $\mathcal{O}$ of $\mathsf{H}_{t,c}(G,\mathfrak{h})$, the graded $G$-character of the module, the Euler–Poincaré pairing, and the composition series polynomials for standard modules. In the global theory, we introduce integral-reflection modules for $\mathsf{H}_{t,c}(G,\mathfrak{h})$ constructed from finite-dimensional $G$-modules. We define and compute the index of a Dirac operator on the integral-reflection module and show that the index is, in a sense, independent of the parameter function $c$. The study of the kernel of these global Dirac operators leads naturally to a notion of dualised generalised Dunkl–Opdam operators.

The $\mathfrak{sl}_\infty$-crystal combinatorics of higher level Fock spaces

For integers $e,\ell\geq 2$, the level $\ell$ Fock space has an $\mathfrak{sl}_\infty$-crystal structure arising from the action of a Heisenberg algebra, intertwining the $\widehat{\mathfrak{sl}_e}$-crystal. The vertices of these crystals are charged $\ell$-partitions. We give the combinatorial rule for computing the arrows anywhere in the $\mathfrak{sl}_\infty$-crystal. This allows us to pinpoint the location of any charged $\ell$-partition. As an application, we compute the support of the spherical representation of a cyclotomic rational Cherednik algebra, and in particular, the set of parameters such that it is finite-dimensional. We also give an easy abacus characterization of all finite-dimensional representations of type $B$ Cherednik algebras.

Highest weight theory for finite-dimensional graded algebras with triangular decomposition

We show that the category of graded modules over a finite-dimensional graded algebra admitting a triangular decomposition can be endowed with the structure of a highest weight category. When the algebra is self-injective, we show furthermore that this highest weight category has tilting modules in the sense of Ringel. This provides a new perspective on the representation theory of such algebras, and leads to several new structures attached to them. There are a wide variety of examples in algebraic Lie theory to which this applies: restricted enveloping algebras, Lusztig's small quantum groups, hyperalgebras, finite quantum groups, and restricted rational Cherednik algebras.

Triple crystal action in Fock spaces

We make explicit a triple crystal structure on higher level Fock spaces, by investigating at the combinatorial level the actions of two affine quantum groups and of a Heisenberg algebra. To this end, we first determine a new indexation of the basis elements that makes the two quantum group crystals commute. Then, we define a so-called Heisenberg crystal, commuting with the other two. This gives new information about the representation theory of cyclotomic rational Cherednik algebras, relying on some recent results of Shan and Vasserot and of Losev. In particular, we give an explicit labelling of their finite-dimensional simple modules.

One-W-type Modules for Rational Cherednik Algebra and Cuspidal Two-sided Cells

We classify the simple modules for the rational Cherednik algebra that are irreducible when restricted to W, in the case when W is a finite Weyl group. The classification turns out to be closely related to the cuspidal two-sided cells in the sense of Lusztig. We compute the Dirac cohomology of these modules and use the tools of Dirac theory to find nontrivial relations between the cuspidal Calogero-Moser cells and the cuspidal two-sided cells.

On refined filtration by supports for rational Cherednik categories $$\mathcal {O}$$O

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 finite-dimensional representations over generalized Hecke algebras. When W is a finite Coxeter group, we give a method for producing explicit presentations of these generalized Hecke algebras in terms of finite-type Iwahori–Hecke algebras. This yields a method for counting the number of irreducible objects in $$\mathcal {O}_c(W, \mathfrak {h})$$ of given support. We apply these techniques to count the number of irreducible representations in $$\mathcal {O}_c(W, \mathfrak {h})$$ of given support for all exceptional Coxeter groups W and all parameters c, including the unequal parameter case. This completes the classification of the finite-dimensional irreducible representations of $$\mathcal {O}_c(W, \mathfrak {h})$$ for exceptional Coxeter groups W in many new cases.

Hyperplane arrangements associated to symplectic quotient singularities

We study the hyperplane arrangements associated, via the minimal model programme, to symplectic quotient singularities. We show that this hyperplane arrangement equals the arrangement of CM-hyperplanes coming from the representation theory of restricted rational Cherednik algebras. We explain some of the interesting consequences of this identification for the representation theory of restricted rational Cherednik algebras. We also show that the Calogero–Moser space is smooth if and only if the Calogero–Moser families are trivial. We describe the arrangements of CM-hyperplanes associated to several exceptional complex reflection groups, some of which are free.

A CASSELMAN–OSBORNE THEOREM FOR RATIONAL CHEREDNIK ALGEBRAS

We define Lie algebra cohomology associated with the half-Dirac operators for representations of rational Cherednik algebras and show that it has property described in the Casselman–Osborne Theorem by establishing a version of the Vogan's conjecture for the half-Dirac operators. Moreover, we study the relationship between Lie algebra cohomology and Dirac cohomology in analogy of the representations for semisimple Lie algebras.

Holonomic modules over Cherednik algebras, I

The goal of this paper is to generalize several basic results from the theory of D-modules to the representation theory of rational Cherednik algebras. We relate characterizations of holonomic modules in terms of singular support and Gelfand–Kirillov dimension. We study pullback, pushforward, and dual on the derived category of (holonomic) Cherednik modules for certain classes of maps between varieties. We prove, in the case of generic parameters for the rational Cherednik algebra, that pushforward with respect to an open affine inclusion preserves holonomicity.

Harish-Chandra bimodules over rational Cherednik algebras

We study Harish-Chandra bimodules over the rational Cherednik algebra Hc(W) associated to a complex reflection group W with parameter c. Our results allow us to partially reduce the study of these bimodules to smaller algebras. We classify those pairs of parameters (c,c′) for which there exist fully supported Harish-Chandra bimodules, and give a description of the category of all Harish-Chandra bimodules modulo those without full support. When W is a symmetric group we are able to classify all irreducible Harish-Chandra bimodules. Our proofs are based on localization techniques, the action of the Namikawa–Weyl group on the set of parameters, and the study of partial KZ functors.

Parabolic degeneration of rational Cherednik algebras

We introduce parabolic degenerations of rational Cherednik algebras of complex reflection groups, and use them to give necessary conditions for finite-dimensionality of an irreducible lowest weight module for the rational Cherednik algebra of a complex reflection group, and for the existence of a non-zero map between two standard modules. The latter condition reproduces and enhances, in the case of the symmetric group, the combinatorics of cores and dominance order, and in general shows that the c-ordering on category $$\mathcal {O}_c$$ may be replaced by a much coarser ordering. The former gives a new proof of the classification of finite dimensional irreducible modules for the Cherednik algebra of the symmetric group.

Supports of representations of the rational Cherednik algebra of type A

We first consider the rational Cherednik algebra corresponding to the action of a finite group on a complex variety, as defined by Etingof. We define a category of representations of this algebra which is analogous to “category O” for the rational Cherednik algebra of a vector space. We generalise to this setting Bezrukavnikov and Etingof's results about the possible support sets of such representations. Then we focus on the case of Sn acting on Cn, determining which irreducible modules in this category have which support sets. We also show that the category of representations with a given support, modulo those with smaller support, is equivalent to the category of finite dimensional representations of a certain Hecke algebra.

Clifford theory on rational Cherednik algebras of imprimitive groups

Ram and Rammage have introduced an automorphism and Clifford theory on affine Hecke algebras. Here we will extend them to cyclotomic Hecke algebras and rational Cherednik algebras.