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We construct the basic representation of the double affine Hecke algebra at critical level $q=1$ associated to an irreducible reduced affine root system $R$ with a reduced gradient root system. For $R$ of untwisted type such a representation was studied by Oblomkov [O04] and further detailed by Gehles [G06] in the presence of minuscule weights., Comment: 7 pages

We construct a generalization of the $C^\vee C_1$-type double affine Hecke algebra for the skein algebra on the twice-punctured torus $\Sigma_{1,2}$ using the Heegaard dual of the Iwahori--Hecke operator recently introduced in our previous article. We show that the automorphisms of our algebra correspond to the Dehn twists about the curves on $\Sigma_{1,2}$. We also give the cluster algebraic construction of the the classical limit of the skein algebra, where the Dehn twists are given in terms of the cluster mutations., Comment: 15 pages, 4 figures

We prove that for generic parameters, the quantum radial parts map of Varagnolo and Vasserot gives an isomorphism between the spherical double affine Hecke algebra of $GL_n$ and a quantized multiplicative quiver variety, as defined by Jordan., Comment: v3, 40 pages. Fixed some errors and added more detail for the case of generic t (Section 5.3). Title changed. Submitted. Comments welcome!

We construct an isomorphism between the wrapped higher-dimensional Heegaard Floer homology of $\kappa$-tuples of cotangent fibers and $\kappa$-tuples of conormal bundles of homotopically nontrivial simple closed curves in $T^*\Sigma$ with a certain braid skein group, where $\Sigma$ is a closed oriented surface of genus $> 0$ and $\kappa$ is a positive integer. Moreover, we show this produces a (right) module over the surface Hecke algebra associated to $\Sigma$. This module structure is shown to be equivalent to the polynomial representation of DAHA in the case where $\Sigma=T^2$ and the cotangent fibers and conormal bundles of curves are both parallel copies., Comment: 30 pages. Comments are welcome! arXiv admin note: text overlap with arXiv:2202.05593 by other authors

Given a module $M$ for the algebra $\mathcal{D}_{\mathtt{q}}(G)$ of quantum differential operators on $G$, and a positive integer $n$, we may equip the space $F_n^G(M)$ of invariant tensors in $V^{\otimes n}\otimes M$, with an action of the double affine Hecke algebra of type $A_{n-1}$. Here $G= SL_N$ or $GL_N$, and $V$ is the $N$-dimensional defining representation of $G$. In this paper we take $M$ to be the basic $\mathcal{D}_{\mathtt{q}}(G)$-module, i.e. the quantized coordinate algebra $M= \mathcal{O}_{\mathtt{q}}(G)$. We describe a weight basis for $F_n^G(\mathcal{O}_{\mathtt{q}}(G))$ combinatorially in terms of walks in the type $A$ weight lattice, and standard periodic tableaux, and subsequently identify $F_n^G(\mathcal{O}_{\mathtt{q}}(G))$ with the irreducible "rectangular representation" of height $N$ of the double affine Hecke algebra., Comment: 37 pages, 14 figures; Several missing references added with discussion that includes proper citation to them

An inifinite-dimensional representation of the double affine Hecke algebra of rank 1 and type $(C_1^{\vee},C_1)$ in which all generators are tridiagonal is presented. This representation naturally leads to two systems of polynomials that are orthogonal on the unit circle. These polynomials can be considered as circle analogs of the Askey-Wilson polynomials. The corresponding polynomials orthogonal on an interval are constructed and discussed.

In this paper we construct representations of certain graded double affine Hecke algebras (DAHA) with possibly unequal parameters from geometry. More precisely, starting with a simple Lie algebra $\mathfrak{g}$ together with a $\mathbb{Z}/m\mathbb{Z}$-grading $\oplus_{i}\mathfrak{g}_{i}$ and a block of $G_{\underline{0}}$-equivariant complexes on the nilpotent cone of $\mathfrak{g}_{\underline{1}}$ as introduced in \cite{LY1}, we attach a graded DAHA and construct its action on the direct sum of spiral inductions in that block. This generalizes results of Vasserot \cite{V} and Oblomkov-Yun \cite{OY} which correspond to the case of the principal block., Comment: 33 pages

We give a full classification, in terms of periodic skew diagrams, of irreducible semisimple modules in category O for the degenerate double affine Hecke algebra of type A which can be realized as submodules of Verma modules.

We study a relationship between $Q$-polynomial distance-regular graphs and the double affine Hecke algebra of type $(C^{\vee}_1,C_1)$. Let $\Gamma$ denote a $Q$-polynomial distance-regular graph with vertex set $X$. We assume that $\Gamma$ has $q$-Racah type and contains a Delsarte clique $C$. Fix a vertex $x \in C$. We partition $X$ according to the path-length distance to both $x$ and $C$. This is an equitable partition. For each cell in this partition, consider the corresponding characteristic vector. These characteristic vectors form a basis for a $\mathbb{C}$-vector space ${\bf W}$. The universal double affine Hecke algebra of type $(C^{\vee}_1,C_1)$ is the $\mathbb{C}$-algebra $\hat{H}_q$ defined by generators $\{t^{\pm1}_n\}^3_{n=0}$ and relations (i) $t_nt_n^{-1}=t_n^{-1}t_n=1$; (ii) $t_n+t_n^{-1}$ is central; (iii) $t_0t_1t_2t_3 = q^{-1/2}$. In this paper, we display an $\hat{H}_q$-module structure for ${\bf W}$. For this module and up to affine transformation, (i) $t_0t_1+(t_0t_1)^{-1}$ acts as the adjacency matrix of $\Gamma$; (ii) $t_3t_0+(t_3t_0)^{-1}$ acts as the dual adjacency matrix of $\Gamma$ with respect to $C$; (iii) $t_1t_2+(t_1t_2)^{-1}$ acts as the dual adjacency matrix of $\Gamma$ with respect to $x$. To obtain our results we use the theory of Leonard systems., Comment: 79 pages, 1 fugure. arXiv admin note: text overlap with arXiv:1205.2144, arXiv:math/0608694, arXiv:1108.0458 by other authors

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 cases, we give a characterization of the subrepresentations by vanishing conditions for Laurent polynomials., Comment: 45 pages

We introduce and study an affine analogue of skew Young diagrams and tableaux on them. It turns out that the double affine Hecke algebra of type $A$ acts on the space spanned by standard tableaux on each diagram. It is shown that the modules obtained this way are irreducible, and they exhaust all irreducible modules of a certain class over the double affine Hecke algebra. In particular, the classification of irreducible modules of this class, announced by Cherednik, is recovered., Comment: 34 pages. In ver.2, the title is changed and some typos are corrected

This paper builds on the previous paper arXiv:math/0612730 by the author, where a relationship between Zhedanov's algebra AW(3) and the double affine Hecke algebra (DAHA) corresponding to the Askey-Wilson polynomials was established. It is shown here that the spherical subalgebra of this DAHA is isomorphic to AW(3) with an additional relation that the Casimir operator equals an explicit constant. A similar result with q-shifted parameters holds for the antispherical subalgebra. Some theorems on centralizers and centers for the algebras under consideration will finally be proved as corollaries of the characterization of the spherical and antispherical subalgebra., Comment: This is a contribution to the Proc. of the Seventh International Conference ''Symmetry in Nonlinear Mathematical Physics'' (June 24-30, 2007, Kyiv, Ukraine), published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/ In v2 remarks about duality anti-algebra isomorphism and about shift operators added

We construct the representation of Double Affine Hecke Algebra whose symmetrization gives the center of the quantum group U_q(sl(2)) and by Kazhdan--Lusztig duality the Verlinde algebra of (1,p) models of logarithmic conformal field theory.

In this note we determine the values of parameters c for which the polynomial representation of the degenerate double affine Hecke algebra (DAHA), i.e. the trigonometric Cherednik algebra, is reducible. Namely, we show that c is a reducibility point for the polynomial representation of the trigonometric Cherednik algebra for a root system R if and only if it is a reducibility point for the rational Cherednik algebra for the Weyl group of some root subsystem R' of R of the same rank; such subsystems for any R are given by the well known Borel-de Siebenthal algorithm. This result has been proved by Cherednik using a case-by-case method., Comment: 7 pages, latex

Zhedanov's algebra AW(3) is considered with explicit structure constants such that, in the basic representation, the first generator becomes the second order q-difference operator for the Askey-Wilson polynomials. It is proved that this representation is faithful for a certain quotient of AW(3) such that the Casimir operator is equal to a special constant. Some explicit aspects of the double affine Hecke algebra (DAHA) related to symmetric and non-symmetric Askey-Wilson polynomials are presented and proved without requiring knowledge of general DAHA theory. Finally a central extension of this quotient of AW(3) is introduced which can be embedded in the DAHA by means of the faithful basic representations of both algebras., Comment: This is a contribution to the Vadim Kuznetsov Memorial Issue on Integrable Systems and Related Topics, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/ In v4 a slight error in formula (2.8) for Q_0 is corrected

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 ${\mathcal O}$ consisting of those modules which are semisimple with respect to the commutative subalgebra generated by Cherednik-Dunkl operators. A classification of all irreducible modules in ${\mathcal O}^{ss}$ for the rational Cherednik algebra is obtained from the corresponding result for the degenerate DAHA., Comment: 13 pages. In ver.2, some notation and wording were changed, and several typos and minor errors were corrected

We study a Laurent polynomial representation $V$ of the double affine Hecke algebra of type $GL_n$ for specialized parameters $t^{k+1}q^{r-1}=1$. We define a series of subrepresentations of $V$ by using a vanishing condition. For some cases, we give an explicit basis of the subrepresentation in terms of nonsymmetric Macdonald polynomials. These results are nonsymmetric versions of \cite{FJMM} and \cite{KMSV}., Comment: 21 pages

We classify the finite dimensional representations of the double affine Hecke algebra of type $C^{\vee}C_1$ in the case when $q$ is not a root of 1., Comment: 8 pages, no figures; exposition is improved

We introduce a class of induced representations of the degenerate double affine Hecke algebra of gl_N and analyze their structure mainly by means of intertwiners. We also construct them from modules of the affine Lie algebra using Knizhnik-Zamolodchikov connections in the conformal field theory. This construction provides a natural quotient of induced modules, which turns out to be the unique irreducible one under a certain condition. Some cunjectual formulas are presented for the symmetric part of these quotients., Comment: 33 pages, Latex, changed contents

Operators that intertwine representations of a degenerate version of the double affine Hecke algebra are introduced. Each of the representations is related to multi-variable orthogonal polynomials associated with Calogero-Sutherland type models. As applications, raising operators and shift operators for such polynomials are constructed., Comment: 26 pages, LaTeX using epsf.sty, one figure