From representations of the rational Cherednik algebra to parabolic Hilbert schemes via the Dunkl-Opdam subalgebra
Eugene Gorsky, José Simental, Monica Vazirani
备注：
63 pages; This version fixes a couple of mistakes in Propositions 7.19 and 8.9 from the published version, concerning the use of dual lattices. The article title in v3 has also changed
In this note we explicitly construct an action of the rational Cherednik algebra $H_{1,m/n}(S_n,\mathbb{C}^n)$ corresponding to the permutation representation of $S_n$ on the $\mathbb{C}^{*}$-equivariant homology of parabolic Hilbert schemes of points on the plane curve singularity $\{x^{m} = y^{n}\}$ for coprime $m$ and $n$. We use this to construct actions of quantized Gieseker algebras on parabolic Hilbert schemes on the same plane curve singularity, and actions of the Cherednik algebra at $t = 0$ on the equivariant homology of parabolic Hilbert schemes on the non-reduced curve $\{y^{n} = 0\}.$ Our main tool is the study of the combinatorial representation theory of the rational Cherednik algebra via the subalgebra generated by Dunkl-Opdam elements.
In this paper we define a new presentation for the Dunkl-Opdam subalgebra of the rational Cherednik algebra. This shows that the Dunkl-Opdam subalgebra is a Drinfeld algebra. We use this fact to define Dirac cohomology for the DO subalgebra. We also formalise generalised graded Hecke algebras and define a Langlands classification to generalised graded Hecke algebras.
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…
In this note, a new proof for the positivity of Dunkl's intertwining operator in the crystallographic case is given. It is based on an asymptotic relationship between the Opdam-Cherednik kernel and the Dunkl kernel as recently observed by M. de Jeu, and on positivity results of S. Sahi for the Heckman-Opdam polynomials and their non-symmetric counterparts.
We introduce braided Dunkl operators that are acting on a q-polynomial algebra and q-commute. Generalizing the approach of Etingof and Ginzburg, we explain the q-commutation phenomenon by constructing braided Cherednik algebras for which the above operators form a representation. We classify all braided Cherednik algebras using the theory of braided doubles developed in our previous paper. Besides ordinary rational Cherednik algebras, our classification gives new algebras attached to an infinite family of subgroups of even elements in complex reflection groups, so that the corresponding braide…
We study the Dunkl-Opdam subalgebra of the rational Cherednik algebra for wreath products at $t=0$, and use this to describe the block decomposition of restricted rational Cherednik algebras for $G(m,d,n)$.
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.
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…
We prove a number of results on the structure and representation theory of the rational Cherednik algebra of the imprimitive reflection group $G(\ell,p,n)$. In particular, we: (1) show a relationship to the Coulomb branch construction of Braverman, Finkelberg and Nakajima, and 3-dimensional quantum field theory; (2) show that the spherical Cherednik algebra carries the structure of a principal Galois order; (3) construct a graded lift of category $\mathcal{O}$ and the larger category of Dunkl-Opdam modules, whose simple modules have the properties of a dual canonical basis and (4) give the fir…
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 consider the deformed versions of the classical Howe dual pairs $(O(r),\mathfrak{s}\mathfrak{l}(2))$ and $(O(r),\mathfrak{s}\mathfrak{p}\mathfrak{o}(2|2))$ in the context of a rational Cherednik algebra $H_c=H_c(W,\mathfrak{h})$ associated to a finite Coxeter group $W$ at the parameters $c$ and $t=1$. For the first pair, we compute the centraliser of the well-known copy of $\mathfrak{s}\cong\mathfrak{s}\mathfrak{l}(2)$ inside $H_c$. For the second pair, we show that the classical copy of $\mathfrak{g}\cong\mathfrak{s}\mathfrak{p}\mathfrak{o}(2|2)$ inside the Weyl-Clifford algebra $\mathcal{…
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.
We consider irreducible lowest-weight representations of Cherednik algebras associated to certain classes of complex reflection groups in characteristic p. In particular, we study maximal graded submodules of Verma modules associated to these algebras. Various results and conjectures are presented concerning generators of these maximal submodules, which are found by computing singular polynomials of Dunkl operators.
We attach elliptic Dunkl operators to an abelian variety with a finite group action. This generalizes elliptic Dunkl operators for Weyl groups, defined by Buchstaber, Felder, and Veselov in 1994. We show that these operators commute, and use them to define representations from category O of elliptic Cherednik algebras. We also consider the monodromy representations of differential equations defined by elliptic Dunkl operators, and show that they yield finite dimensional rrepresentations of generalized double affine Hecke algebras.
The Heckman-Opdam hypergeometric functions of type BC extend classical Jacobi functions in one variable and include the spherical functions of non-compact Grassmann manifolds over the real, complex or quaternionic numbers. There are various limit transitions known for such hypergeometric functions. In the present paper, we use an explicit form of the Harish-Chandra integral representation as well as an interpolated variant, in order to obtain limit results for three continuous classes of hypergeometric functions of type BC which are distinguished by explicit, sharp and uniform error bounds. Th…
The Dunkl--Dirac symmetry algebra is an associative subalgebra of the tensor product of a Clifford algebra and the faithful polynomial representation of a rational Cherednik algebra. In previous work, the finite-dimensional representations of the Dunkl--Dirac symmetry algebra in three dimensions linked with a dihedral group were given. We give here the necessary results to proceed to the same construction when the Clifford algebra in the tensor product has negative signature.
We 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…
We define a family of Dirac operators for the Dunkl angular momentum algebra depending on certain central elements of the group algebra of the Pin cover of the Weyl group inherent to the rational Cherednik algebra. We prove an analogue of Vogan's conjecture for this family of operators and use this to show that the Dirac cohomology, when non-zero, determines the central character of representations of the angular momentum algebra. Furthermore, interpreting this algebra in the framework of (deformed) Howe dualities, we show that the natural Dirac element we define yields, up to scalars, a squar…
For any module over the affine Weyl group we construct a representation of the associated trigonometric Cherednik algebra $A(k)$ at critical level in terms of Dunkl type operators. Under this representation the center of $A(k)$ produces quantum conserved integrals for root system generalizations of quantum spin-particle systems on the circle with delta function interactions. This enables us to translate the spectral problem of such a quantum spin-particle system to questions in the representation theory of $A(k)$. We use this approach to derive the associated Bethe ansatz equations. They are e…
The Dunkl--Dirac operator is a deformation of the Dirac operator by means of Dunkl derivatives. We investigate the symmetry algebra generated by the elements supercommuting with the Dunkl--Dirac operator and its dual symbol. This symmetry algebra is realised inside the tensor product of a Clifford algebra and a rational Cherednik algebra associated with a reflection group or root system. For reducible root systems of rank three, we determine all the irreducible finite-dimensional representations and conditions for unitarity. Polynomial solutions of the Dunkl--Dirac equation are given as a real…
We realize an extended version of the trigonometric Cherednik algebra as affine Dunkl operators involving Heaviside functions. We use the quadratic Casimir element of the extended trigonometric Cherednik algebra to define an explicit nonstationary Schrödinger equation with delta-potential. We use coordinate Bethe ansatz methods to construct solutions of the nonstationary Schrödinger equation in terms of generalized Bethe wave functions. It is shown that the generalized Bethe wave functions satisfy affine difference Knizhnik-Zamolodchikov equations in their spectral parameter. The relation to t…
In [M. Rösler and M. Voit. Integral Representation and Uniform Limits for Some Heckman-Opdam Hypergeometric Functions of type BC, Transactions of the American Mathematical Society, Vol. 368, No. 8, 6005-6032, 2016.], Rösler and Voit give a formula for generalized spherical functions of type $BC$ in terms of the spherical functions of type A. We use this formula to describe precisely the support of the associated generalized Abel transform. Furthermore, we derive a similar formula for the generalized spherical functions in the rational Dunkl setting. The support of the intertwining operator V i…
In this article, we conjecture exact estimates for the Weyl-invariant Opdam-Cherednik hypergeometric functions. We prove the conjecture for the root system $A_n$ and for all rank 1 cases. We provide other evidence that the conjecture might be true in general.
These are lecture notes of a minicourse given by the first author at the Summer School on Quantization at the University of Notre Dame in June 2011. The notes were written up and expanded by the second author who took the liberty of adding a few interesting results and proofs from the literature. In a broad sense, our goal is to give an introduction to representation theory of rational Cherednik algebras and some of its recent applications. More specifically, we focus on the two concepts featuring in the title (Dunkl operators and quasi-invariants) and explain the relation between them. The co…
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 a finite dimensional representation $V$ of a finite reflection group $W$, we consider the rational Cherednik algebra $\mathsf{H}_{t,c}(V,W)$ associated with $(V,W)$ at the parameters $t\neq 0$ and $c$. The Dunkl total angular momentum algebra $O_{t,c}(V,W)$ arises as the centraliser algebra of the Lie superalgebra $\mathfrak{osp}(1|2)$ containing a Dunkl deformation of the Dirac operator, inside the tensor product of $\mathsf{H}_{t,c}(V,W)$ and the Clifford algebra generated by $V$. We show that, for every value of the parameter $c$, the centre of $O_{t,c}(V,W)$ is isomorphic to a univaria…
We prove that the quotient of the polynomial representation of the double affine Hecke algebra (DAHA) by the radical of the duality pairing is always irreducible assuming that it is finite dimensional (apart from the roots of unity). We also find necessary and sufficient conditions for the radical to be zero, which is a q-generalization of Opdam's formula for the singular k-parameters with the multiple zero-eigenvalues of the corresponding Dunkl operators.
The notion of rational spin double affine Hecke algebras (sDaHa) and rational double affine Hecke-Clifford algebras (DaHCa) associated to classical Weyl groups are introduced. The basic properties of these algebras such as the PBW basis and Dunkl operator representations are established. An algebra isomorphism relating the rational DaHCa to the rational sDaHa is obtained. We further develop a link between the usual rational Cherednik algebra and the rational sDaHa by introducing a notion of rational covering double affine Hecke algebras.
In this paper, we study the polynomial representation of the double affine Hecke algebra of type $(C^\vee_n, C_n)$ for specialized parameters. Inductively and combinatorially, we give a linear basis of the representation in terms of linear combinations of non-symmetric Koornwinder polynomials. The basis consists of generalized eigenfunctions with respect to $q$-Dunkl-Cherednik operators $\hat{Y}_i$, and it gives a way to cancel out poles of non-symmetric Koornwinder polynomials. We examine irreducibility and $Y$-semisimplicity of the representation for the specialized parameters. For some case…
Commutative sets of Jucys-Murphyelements for affine braid groups of $A^{(1)},B^{(1)},C^{(1)},D^{(1)}$ types were defined. Construction of $R$-matrix representations of the affine braid group of type $C^{(1)}$ and its distinguish commutative subgroup generated by the $C^{(1)}$-type Jucys--Murphy elements are given. We describe a general method to produce flat connections for the two-boundary quantum Knizhnik-Zamolodchikov equations as necessary conditions for Sklyanin's type transfer matrix associated with the two-boundary multicomponent Zamolodchikov algebra to be invariant under the action of…
We consider the polynomial representation of Double Affine Hecke Algebras (DAHAs) and construct its submodules as ideals of functions vanishing on the special collections of affine planes. This generalizes certain results of Kasatani in types A_n, (C_n^\vee,C_n). We obtain commutative algebras of difference operators given by the action of invariant combinations of Cherednik-Dunkl operators in the corresponding quotient modules of the polynomial representation. This gives known and new generalized Macdonald-Ruijsenaars systems. Thus in the cases of DAHAs of types A_n and (C_n^\vee,C_n) we deri…
