Cyclotomic Rational Cherednik Research Articles

Unitary representations of the Cherednik algebra: $$V^*$$-homology

We give a non-negative combinatorial formula, in terms of Littlewood-Richardson numbers, for the $$V^*$$ -homology of the unitary representations of the cyclotomic rational Cherednik algebra, and as a consequence, for the graded Betti numbers for the ideals of a class of subspace arrangements arising from the reflection arrangements of complex reflection groups.

Subspace Arrangements and Cherednik Algebras

Abstract The purpose of this article is to study the relationship between numerical invariants of certain subspace arrangements coming from reflection groups and numerical invariants arising in the representation theory of Cherednik algebras. For instance, we observe that knowledge of the equivariant graded Betti numbers (in the sense of commutative algebra) of any irreducible representation in category ${\mathscr{O}}$ is equivalent to knowledge of the Kazhdan–Lusztig character of the irreducible object (we use this observation in joint work with Fishel–Manosalva). We then explore the extent to which Cherednik algebra techniques may be applied to ideals of linear subspace arrangements: we determine when the radical of the polynomial representation of the Cherednik algebra is a radical ideal and, for the cyclotomic rational Cherednik algebra, determine the socle of the polynomial representation and characterize when it is a radical ideal. The subspace arrangements that arise include various generalizations of the $k$-equals arrangement. In the case of the radical, we apply our results with Juteau together with an idea of Etingof–Gorsky–Losev to observe that the quotient is Cohen–Macaulay for positive choices of parameters. In the case of the socle (in cyclotomic type), we give an explicit vector space basis in terms of certain specializations of nonsymmetric Jack polynomials, which in particular determines its minimal generators and Hilbert series and answers a question posed by Feigin and Shramov.

Cyclotomic double affine Hecke algebras

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. Namely, we give a $q$-deformation of each of the above four descriptions of the partially spherical rational Cherednik algebra, replacing differential operators with difference operators, degenerate DAHA with DAHA, and homology with K-theory, and show that they give the same algebra. In addition, we show that spherical cyclotomic DAHA are quantizations of certain multiplicative quiver and bow varieties, which may be interpreted as K-theoretic Coulomb branches of a framed quiver gauge theory. Finally, we apply cyclotomic DAHA to prove new flatness results for various kinds of spaces of $q$-deformed quasiinvariants. In the appendix by H. Nakajima and D. Yamakawa (added in version 2), the authors explain the relations between multiplicative bow varieties and (various versions of) multiplicative quiver varieties for a cyclic quiver.

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].

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.

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.

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.

ROUQUIER’S CONJECTURE AND DIAGRAMMATIC ALGEBRA

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 representations of the Hecke algebra that are hit by projectives under the $\mathsf{KZ}$-functor from the Cherednik category ${\mathcal{O}}$ in this case, with an explicit basis. This algebra has a number of beautiful structures including categorifications of many aspects of Fock space. It can be understood quite explicitly using a homogeneous cellular basis which generalizes such a basis given by Hu and Mathas for cyclotomic KLR algebras. Thus, we can transfer results proven in this diagrammatic formalism to category ${\mathcal{O}}$ for a cyclotomic rational Cherednik algebra, including the connection of decomposition numbers to canonical bases mentioned above, and an action of the affine braid group by derived equivalences between different blocks.

Proof of Varagnolo–Vasserot conjecture on cyclotomic categories $${\mathcal {O}}$$ O

We prove an asymptotic version of a conjecture by Varagnolo and Vasserot on an equivalence between the category \({\mathcal {O}}\) for a cyclotomic rational Cherednik algebra and a suitable truncation of an affine parabolic category \({\mathcal {O}}\) that, in particular, implies Rouquier’s conjecture on the decomposition numbers in the former. Our proof uses two ingredients: an extension of Rouquier’s deformation approach as well as categorical actions on highest weight categories and related combinatorics.

Highest weight 𝔰𝔩₂-categorifications II: Structure theory

This paper continues the study of highest weight categorical s l 2 \mathfrak {sl}_2 -actions initiated in part I. We start by refining the definition given there and showing that all examples considered in part I are also highest weight categorifications in the refined sense. Then we prove that any highest weight s l 2 \mathfrak {sl}_2 -categorification can be filtered in such a way that the successive quotients are so-called basic highest weight s l 2 \mathfrak {sl}_2 -categorifications. For a basic highest weight categorification we determine minimal projective resolutions of standard objects. We use this, in particular, to examine the structure of tilting objects in basic categorifications and to show that the Ringel duality is given by the Rickard complex. We apply some of these structural results to categories O \mathcal {O} for cyclotomic Rational Cherednik algebras.

Crystal Isomorphisms in Fock Spaces and Schensted Correspondence in Affine Type A

We are interested in the structure of the crystal graph of level l Fock spaces representations of $\mathcal{U}_{q}^{\prime} (\widehat{\mathfrak{s}\mathfrak{l}_{e}}) $ . Since the work of Shan (Ann. Sci. Éc Norm. Supér. 44:147–182, 2011), we know that this graph encodes the branching rule for a corresponding cyclotomic rational Cherednik algebra. Besides, it appears to be closely related to the Harish-Chandra branching graph for the appropriate finite unitary group, according to [8]. In this paper, we make explicit a particular isomorphism between connected components of the crystal graphs of Fock spaces. This so-called “canonical” crystal isomorphism turns out to be expressible only in terms of: We explain how this can be seen as an analogue of the bumping algorithm for affine type A. Moreover, it yields a combinatorial characterisation of the vertices of any connected component of the crystal of the Fock space.

Abelian Localization for Cyclotomic Cherednik Algebras

In this paper we prove the abelian localization theorem for modules over cyclotomic Rational Cherednik algebras.

Derived equivalence for quantum symplectic resolutions

Using techniques from the homotopy theory of derived categories and noncommutative algebraic geometry, we establish a general theory of derived microlocalization for quantum symplectic resolutions. In particular, our results yield a new proof of derived Beilinson–Bernstein localization and a derived version of the more recent microlocalization theorems of Gordon–Stafford (Gordon and Stafford in Adv Math 198(1):222–274, 2005; Duke Math J 132(1):73–135, 2006) and Kashiwara–Rouquier (Kashiwara and Rouquier in Duke Math J 144(3):525–573, 2008) as special cases. We also deduce a new derived microlocalization result linking cyclotomic rational Cherednik algebras with quantized Hilbert schemes of points on minimal resolutions of cyclic quotient singularities.

Highest weight $$\mathfrak{sl }_2$$ -categorifications I: crystals

We define highest weight categorical actions of \(\mathfrak{sl }_2\) on highest weight categories and show that basically all known examples of categorical \(\mathfrak{sl }_2\)-actions on highest weight categories (including rational and polynomial representations of general linear groups, parabolic categories \(\mathcal O \) of type \(A\), categories \(\mathcal O \) for cyclotomic Rational Cherednik algebras) are highest weight in our sense. Our main result is an explicit combinatorial description of (the labels of) the crystal on the set of simple objects. A new application of this is to determining the supports of simple modules over the cyclotomic Rational Cherednik algebras starting from their labels.