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We give a non-negative combinatorial formula, in terms of Littlewood-Richardson numbers, for the 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., 41 pages

We classify the irreducible unitary modules in category O 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 of n, the closed semi-linear set of parameters for which the contravariant form on the irreducible representation with the given r-partition as lowest weight 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., Comment: 39 pages; version 2 contains major changes: a new title to more accurately reflect content, greatly expanded exposition, completely explicit results for r=2. Also: pictures!

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 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 arrangment. 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 by the radical 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 non-symmetric Jack polynomials, which in particular determines its minimal generators and Hilbert series and answers a question posed by Feigin and Shramov., Comment: 26 pages. 2nd version: added a Theorem on Cohen-Macaulayness of the top of the spherical representation for positive choices of parameters

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 O 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., 35 pages

Given a complex reflection group W we compute the support of the spherical irreducible module of the rational Cherednik algebra of W in terms of the simultaneous eigenfunction of the Dunkl operators and Schur elements for finite Hecke algebras., Comment: 21 pages

We construct (q,t)-Catalan polynomials and q-Fuss-Catalan polynomials for any irreducible complex reflection group W. The two main ingredients in this construction are Rouquier's formulation of shift functors for the rational Cherednik algebras of W, and Opdam's analysis of permutations of the irreducible representations of W arising from the Knizhnik-Zamolodchikov connection., Comment: 9 pages

We apply the Dunkl-Opdam operators and generalized Jack polynomials to study category O for the rational Cherednik algebra of type G(r,1,n). We determine the set of aspherical values, and answer a question of Iain Gordon on the ordering of category O., 20 pages; v2 updated based on reviewer feedback, now with somewhat expanded exposition and proofs

We study blocks of category O for the Cherednik algebra having the property that every irreducible module in the block admits a BGG resolution, and as a consequence prove a character formula conjectured by Oblomkov-Yun., Comment: 34 pages, color figures

We show that for Jack parameter \alpha = -(k+1)/(r-1), certain Jack polynomials studied by Feigin-Jimbo-Miwa-Mukhin vanish to order r when k+1 of the coordinates coincide. This result was conjectured by Bernevig and Haldane, who proposed that these Jack polynomials are model wavefunctions for fractional quantum Hall states. Special cases of these Jack polynomials include the wavefunctions of Laughlin and Read-Rezayi. In fact, along these lines we prove several vanishing theorems known as clustering properties for Jack polynomials in the mathematical physics literature, special cases of which had previously been conjectured by Bernevig and Haldane. Motivated by the method of proof, which in case r = 2 identifies the span of the relevant Jack polynomials with the S_n-invariant part of a unitary representation of the rational Cherednik algebra, we conjecture that unitary representations of the type A Cherednik algebra have graded minimal free resolutions of Bernstein-Gelfand-Gelfand type; we prove this for the ideal of the (k+1)-equals arrangement in the case when the number of coordinates n is at most 2k+1. In general, our conjecture predicts the graded S_n-equivariant Betti numbers of the ideal of the (k+1)-equals arrangement with no restriction on the number of ambient dimensions., Comment: 19 pages; v2: corrected typos and updated first author's name

We prove that Macdonald polynomials are characters of irreducible Cherednik algebra modules., Comment: 6 pages; v2 logical content essentially the same, but main theorem is now explicitly an equivalence between the n! theorem and the character formula for irreducible H-modules

We prove the conjectures of Graham-Kumar and Griffeth-Ram concerning the alternation of signs in the structure constants for torus-equivariant K-theory of generalized flag varieties G/P. These results are immediate consequences of an equivariant homological Kleiman transversality principle for the Borel mixing spaces of homogeneous spaces, and their subvarieties, under a natural group action with finitely many orbits. The computation of the coefficients in the expansion of the equivariant K-class of a subvariety in terms of Schubert classes is reduced to an Euler characteristic using the homological transversality theorem for non-transitive group actions due to S. Sierra. A vanishing theorem, when the subvariety has rational singularities, shows that the Euler characteristic is a sum of at most one term--the top one--with a well-defined sign. The vanishing is proved by suitably modifying a geometric argument due to M. Brion in ordinary K-theory that brings Kawamata-Viehweg vanishing to bear., Comment: 28 pages; v2 has slightly expanded exposition and fixes an error in v1 that treated dualizing sheaves of Schubert varieties as if they were line bundles; v3 is the published version, but includes corrections of the signs of weights in Section 2.3 and the definition of a torus action in Section 6; v4 corrects and simplifies the proofs of Proposition 8.1 and Lemma 10.2

Using a combinatorial approach that avoids geometry, this paper studies the structure of KT(G/B), the T-equivariant K-theory of the generalized flag variety G/B. This ring has a natural basis {[Oxw] | w ∈ W} (the double Grothendieck polynomials), where Oxw is the structure sheaf of the Schubert variety Xw. For rank two cases we compute the corresponding structure constants of the ring KT(G/B) and, based on this data, make a positivity conjecture for general G which generalizes the theorems of M. Brion (for K(G/B)) and W. Graham (for HT* (G/B)). Let [Xλ] ∈ KT (G/B) be the class of the homogeneous line bundle on G/B corresponding to the character of T indexed by λ. For general G we prove "Pieri-Chevalley formulas" for the products [Xλ][OXw], [X-λ][OXw], [Xw0λ] [OXw], and [OXw0si] [Oxw], where λ is dominant. By using the Chern character and comparing lowest degree terms the products which are computed in this paper also give results for the Grothendieck polynomials, double Schubert polynomials, and ordinary Schubert polynomials in, respectively K(G/B), HT* (G/B) and H*(G/B).