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.
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…
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.
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…
We study those finite dimensional quotients of the rational Cherednik algebra at t=0 that are supported at a point of the centre. It is shown that each such quotient is Morita equivalent to a certain cuspidal quotient of a rational Cherednik algebra associated to a parabolic subgroup of W.
We study a family of abelian categories O_{c, t} depending on complex parameters c, t which are interpolations of the O-category for the rational Cherednik algebra H_c(t) of type A, where t is a positive integer. We define the notion of a Verma object in such a category (a natural analogue of the notion of Verma module). We give some necessary conditions and some sufficient conditions for the existence of a non-trivial morphism between two such Verma objects. We also compute the character of the irreducible quotient of a Verma object for sufficiently generic values of parameters c, t, and prov…
We study equivalences for category O_p of the rational Cherednik algebras H_p of type G_l(n) = μ_l^n\rtimes S_n: a highest weight equivalence between O_p and O_{σ(p)} for σ\in S_l and an action of S_l on a non-empty Zariski open set of parameters p; a derived equivalence between O_p and O_{p'} whenever p and p' have integral difference; a highest weight equivalence between O_p and a parabolic category O for the general linear group, under a non-rationality assumption on the parameter p. As a consequence, we confirm special cases of conjectures of Etingof and of Rouquier.
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…
Varagnolo and Vasserot conjectured an equivalence between the category O for a cyclotomic Rational Cherednik algebra and a truncation of an affine parabolic category O of type A. In this paper we reduce their conjecture to some purely combinatorial conjecture. Our principal tool for the reduction is the theory of categorical sl_2-actions on highest weight categories. Then we prove our combinatorial conjecture in some cases.
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 derive a Givental-type stationary phase integral representation for the specified $\Gr_{m,N}$-Whittaker function introduced in \cite{GLO2}, which presumably describes the $S^1\times U_N$-equivariant Gromov-Witten invariants of Grassmann variety $\Gr_{m,N}$. Our main tool is a generalization of Whittaker model for principal series $\CU(\frak{gl}_N)$-modules. In particular, our construction includes a representation theory interpretation of the Batyrev--Ciocan-Fontanine--Kim--van Straten toric degeneration of Grassmannian, providing a direct connection between this toric degeneration of $\Gr_…
We define highest weight categorical actions of sl_2 on highest weight categories and show that basically all known examples of categorical sl_2-actions on highest weight categories (including rational and polynomial representations of general linear groups, parabolic categories O of type $A$, categories 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 Ra…
We define and study the Drinfeld-Lafforgue-Vinberg compactification of the moduli stack of G-bundles Bun_G for an arbitrary reductive group G; its definition is given in terms of the Vinberg semigroup of G, and is due to Drinfeld (unpublished). Throughout the article we prefer to view the compactification as a canonical multi-parameter degeneration of Bun_G which we call the Drinfeld-Lafforgue-Vinberg degeneration VinBun_G. We construct local models for the degeneration VinBun_G which "factorize in families" and use them to study its singularities, generalizing results of the article [Sch1] wh…
We show that Hilbert schemes of planar curve singularities and their parabolic variants can be interpreted as certain generalized affine Springer fibers for $GL_n$, as defined by Goresky-Kottwitz-MacPherson. Using a generalization of affine Springer theory for Braverman-Finkelberg-Nakajima's Coulomb branch algebras, we construct a rational Cherednik algebra action on the homology of the Hilbert schemes, and compute it in examples. Along the way, we generalize to the parahoric setting the recent construction of Hilburn-Kamnitzer-Weekes, which may be of independent interest. In the spherical cas…
Chinta and Gunnells introduced a rather intricate multi-parameter Weyl group action on rational functions on a torus, which, when the parameters are specialized to certain Gauss sums, describes the functional equations of Weyl group multiple Dirichlet series associated to metaplectic (n-fold) covers of algebraic groups. In subsequent joint work with Puskas, they extended this action to a "metaplectic" representation of the equal parameter affine Hecke algebra, which allowed them to obtain explicit formulas for the p-parts of these Dirichlet series. They have also verified by a computer check t…
To any finite group G of automorphisms of a symplectic vector space V we associate a new multi-parameter deformation, H_k, of the smash product of G with the polynomial algebra on V. The algebra H_k, called a symplectic reflection algebra, is related to the coordinate ring of a universal Poisson deformation of the quotient singularity V/G. If G is the Weyl group of a root system in a vector space h and V=h\oplus h^*, then the algebras H_k are `rational' degenerations of Cherednik's double affine Hecke algebra. Let G=S_n, the Weyl group of g=gl_n. We construct a 1-parameter deformation of the H…
Let X be an affine spherical variety, possibly singular, and $L^+X$ its arc space. The intersection complex of $L^+X$, or rather of its finite-dimensional formal models, is conjectured to be related to special values of local unramified L-functions. Such relationships were previously established in Braverman-Finkelberg-Gaitsgory-Mirkovic for the affine closure of the quotient of a reductive group by the unipotent radical of a parabolic, and in Bouthier-Ngo-Sakellaridis for toric varieties and L-monoids. In this paper, we compute this intersection complex for the large class of those spherical …
