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Let $\mathfrak{g}(A)$ be the Kac-Moody algebra with respect to a symmetrizable generalized Cartan matrix $A$. We give an explicit presentation of the fix-point Lie subalgebra $\mathfrak{k}(A)$ of $\mathfrak{g}(A)$ with respect to the Chevalley involution. It is a presentation of $\mathfrak{k}(A)$ involving inhomogeneous versions of the Serre relations, or, from a different perspective, a presentation generalizing the Dolan-Grady presentation of the Onsager algebra. In the finite and untwisted affine case we explicitly compute the structure constants of $\mathfrak{k}(A)$ in terms of a Chevalley type basis of $\mathfrak{k}(A)$. For the symplectic Lie algebra and its untwisted affine extension we explicitly describe the one-dimensional representations of $\mathfrak{k}(A)$., Comment: 20 pages. v2: typos corrected and references added

Nonsymmetric interpolation Laurent polynomials in $n$ variables are introduced, with the interpolation points depending on $q$ and on a $n$-tuple of parameters $\tau=(\tau_1,\ldots,\tau_n)$. When $\tau_i=st^{n-i}$ Okounkov's $3$-parameter $BC_n$-type interpolation Macdonald polynomials are recovered from the nonsymmetric interpolation Laurent polynomials through Hecke algebra symmetrisation with respect to a type $C_n$ Hecke algebra action. In the appendix we give some conjectures about extra vanishing, based on Mathematica computations in rank two., Comment: 32 pages, 9 figures; v5: thoroughly revised version, including a new introduction. To appear in Transformation Groups

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 the remarkable fact that their formulas continue to define a group action for general (unspecialized) parameters. In the first part of paper we give a conceptual explanation of this fact, by giving a uniform and elementary construction of the "metaplectic" representation for generic Hecke algebras as a suitable quotient of a parabolically induced affine Hecke algebra module, from which the associated Chinta-Gunnells Weyl group action follows through localization. In the second part of the paper we extend the metaplectic representation to the double affine Hecke algebra, which provides a generalization of Cherednik's basic representation. This allows us to introduce a new family of "metaplectic" polynomials, which generalize nonsymmetric Macdonald polynomials. In this paper, we provide the details of the construction of metaplectic polynomials in type A; the general case will be handled in the sequel to this paper., Comment: 39 pages. Version 2 is a significant revision. Added second part introducing a new family of "metaplectic" polynomials, which generalize nonsymmetric Macdonald polynomials and metaplectic Iwahori-Whittaker functions. Title has been changed and the introduction has been expanded

Cherednik attached to an affine Hecke algebra module a compatible system of difference equations, called quantum affine Knizhnik-Zamolodchikov (KZ) equations. In case of a principal series module we construct a basis of power series solutions of the quantum affine KZ equations. Relating the bases for different asymptotic sectors gives rise to a Weyl group cocycle, which we compute explicitly in terms of theta functions. For the spin representation of the affine Hecke algebra of type C the quantum affine KZ equations become the boundary qKZ equations associated to the Heisenberg spin-1/2 XXZ chain. We show that in this special case the results lead to an explicit 4-parameter family of elliptic solutions of the dynamical reflection equation associated to Baxter's 8-vertex face dynamical R-matrix. We use these solutions to define an explicit 9-parameter elliptic family of boundary quantum Knizhnik-Zamolodchikov-Bernard (KZB) equations., Comment: 47 pages. v2: small corrections; v.3: small corrections in the proof of Thm. 3.15

Basic Harish-Chandra series are asymptotically free meromorphic solutions of the system of basic hypergeometric difference equations associated to root systems. The associated connection coefficients are explicitly computed in terms of Jacobi theta functions. We interpret the connection coefficients as the transition functions for asymptotically free meromorphic solutions of Cherednik's root system analogs of the quantum Knizhnik-Zamolodchikov equations. They thus give rise to explicit elliptic solutions of root system analogs of dynamical Yang-Baxter and reflection equations. Applications to quantum c-functions, basic hypergeometric functions, reflectionless difference operators and multivariable Baker-Akhiezer functions are discussed., Comment: 34 pages. In the second version some additional references are included. In third version a typo in formula (1.10) is corrected, and references are updated

Nonpolynomial basic hypergeometric eigenfunctions of the Askey-Wilson second order difference operator are known to be expressible as very-well-poised 8phi7 series. In this paper we use this fact to derive various basic hypergeometric and theta function identities. We relate most of them to identities from the existing literature on basic hypergeometric series. This leads for example to a new derivation of a known quadratic transformation formula for very-well-poised 8phi7 series. We also provide a link to Chalykh's theory on (rank one, BC type) Baker-Akhiezer functions.

An overview of the basic results on Macdonald(-Koornwinder) polynomials and double affine Hecke algebras is given. We develop the theory in such a way that it naturally encompasses all known cases. Among the basic properties of the Macdonald polynomials we treat are the quadratic norm formulas, duality and the evaluation formulas. This text is a provisional version of a chapter on Macdonald polynomials for volume 5 of the Askey-Bateman project, entitled "Multivariable special functions"., Comment: 61 pages. This text is a provisional version of a chapter on Macdonald polynomials for volume 5 of the Askey-Bateman project. Comments are welcome. In the revision the title is changed, some minor typos are corrected, the paragraph on affine root systems is moved to an appendix, and we explicitly list the affine Dynkin diagrams

We derive an explicit c-function expansion of a basic hypergeometric function associated to root systems. The basic hypergeometric function in question was constructed as explicit series expansion in symmetric Macdonald polynomials by Cherednik in case the associated twisted affine root system is reduced. Its construction was extended to the nonreduced case by the author. It is a meromorphic Weyl group invariant solution of the spectral problem of the Macdonald q-difference operators. The c-function expansion is its explicit expansion in terms of the basis of the space of meromorphic solutions of the spectral problem consisting of q-analogs of the Harish-Chandra series. We express the expansion coefficients in terms of a q-analog of the Harish-Chandra c-function, which is explicitly given as product of q-Gamma functions. The c-function expansion shows that the basic hypergeometric function formally is a q-analog of the Heckman-Opdam hypergeometric function, which in turn specializes to elementary spherical functions on noncompact Riemannian symmetric spaces for special values of the parameters., Comment: 42 pages; In version 2 we removed a technical condition on the lattice and corrected some small misprints; in version 3 we have made some minor revisions, and we have revised Subsections 5.2 and 5.3 establishing the link between the rank one case of the theory and the theory on basic hypergeometric series; in version 4 small typos corrected, references updated

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 Schr\"odinger equation with delta-potential. We use coordinate Bethe ansatz methods to construct solutions of the nonstationary Schr\"odinger 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 the vector valued root system analogs of the quantum Bose gas on the circle with pairwise delta-function interactions is indicated., Comment: 23 pages; Version 2: expanded introduction and misprints corrected

Cherednik's quantum affine Knizhnik-Zamolodchikov equations associated to an affine Hecke algebra module M form a holonomic system of q-difference equations acting on M-valued functions on a complex torus T. In this paper the quantum affine Knizhnik-Zamolodchikov equations are related to the Cherednik-Macdonald theory when M is induced from a character of a standard parabolic subalgebra of the affine Hecke algebra. We set up correspondences between solutions of the quantum affine KZ equations and, on the one hand, solutions to the spectral problem of the Cherednik-Dunkl q-difference reflection operators (generalizing work of Kasatani and Takeyama) and, on the other hand, solutions to the spectral problem of the Cherednik-Macdonald q-difference operators (generalizing work of Cherednik). The correspondences are applicable to all relevant spaces of functions on T and for all parameter values, including the cases that q and/or the Hecke algebra parameters are roots of unity., Comment: 47 pages

Reflection equation algebras and related U_q(g)-comodule algebras appear in various constructions of quantum homogeneous spaces and can be obtained via transmutation or equivalently via twisting by a cocycle. In this paper we investigate algebraic and representation theoretic properties of such so called `covariantized' algebras, in particular concerning their centres, invariants, and characters. Generalising M. Noumi's construction of quantum symmetric pairs we define a coideal subalgebra B_f of U_q(g) for each character f of a covariantized algebra. The locally finite part F_l(U_q(g)) of U_q(g) with respect to the left adjoint action is a special example of a covariantized algebra. We show that for each character f of F_l(U_q(g)) the centre Z(B_f) canonically contains the representation ring Rep(g) of the semisimple Lie algebra g. We show moreover that for g=sl_n(C) such characters can be constructed from any invertible solution of the reflection equation and hence we obtain many new explicit realisations of Rep(sl_n(C)) inside U_q(sl_n(C)). As an example we discuss the solutions of the reflection equation corresponding to the Grassmannian manifold Gr(m,2m) of m-dimensional subspaces in C^{2m}., Comment: 37 pages

We use the double affine Hecke algebra of type GL_N to construct an explicit consistent system of q-difference equations, which we call the bispectral quantum Knizhnik-Zamolodchikov (BqKZ) equations. BqKZ includes, besides Cherednik's quantum affine KZ equations associated to principal series representations of the underlying affine Hecke algebra, a compatible system of q-difference equations acting on the central character of the principal series representations. We construct a meromorphic self-dual solution \Phi of BqKZ which, upon suitable specializations of the central character, reduces to symmetric self-dual Laurent polynomial solutions of quantum KZ equations. We give an explicit correspondence between solutions of BqKZ and solutions of a particular bispectral problem for the Ruijsenaars' commuting trigonometric q-difference operators. Under this correspondence \Phi becomes a self-dual Harish-Chandra series solution \Phi^+ of the bispectral problem. Specializing the central character as above, we recover from \Phi^+ the symmetric self-dual Macdonald polynomials., Comment: 52 pages

We derive generalizations of the Cherednik-Macdonald constant term identities associated to root systems which depend, besides on the usual multiplicity function, symmetrically on two quasi-periods. They are natural analogues of the Cherednik-Macdonald constant term q-identities in which the deformation parameter q is allowed to have modulus one. They unite the Cherednik-Macdonald constant term q-identities with closely related Jackson p-integral identities due to Macdonald, where the deformation parameter p is related to q by modular inversion., Comment: 13 pages

We analyze the centralizer of the Macdonald difference operator in an appropriate algebra of Weyl group invariant difference operators. We show that it coincides with Cherednik's commuting algebra of difference operators via an analog of the Harish-Chandra isomorphism. Analogs of Harish-Chandra series are defined and realized as solutions to the system of basic hypergeometric difference equations associated to the centralizer algebra. These Harish-Chandra series are then related to both Macdonald polynomials and Chalykh's Baker-Akhiezer functions., Comment: 38 pages. Small typos corrected

Based on Spiridonov's analysis of elliptic generalizations of the Gauss hypergeometric function, we develop a common framework for 7-parameter families of generalized elliptic, hyperbolic and trigonometric univariate hypergeometric functions. In each case we derive the symmetries of the generalized hypergeometric function under the Weyl group of type E_7 (elliptic, hyperbolic) and of type E_6 (trigonometric) using the appropriate versions of the Nassrallah-Rahman beta integral, and we derive contiguous relations using fundamental addition formulas for theta and sine functions. The top level degenerations of the hyperbolic and trigonometric hypergeometric functions are identified with Ruijsenaars' relativistic hypergeometric function and the Askey-Wilson function, respectively. We show that the degeneration process yields various new and known identities for hyperbolic and trigonometric special functions. We also describe an intimate connection between the hyperbolic and trigonometric theory, which yields an expression of the hyperbolic hypergeometric function as an explicit bilinear sum in trigonometric hypergeometric functions., Comment: 46 pages

In this paper we study root system generalizations of the quantum Bose-gas on the circle with pair-wise delta function interactions. The underlying symmetry structures are shown to be governed by the associated graded of Cherednik's (suitably filtered) degenerate double affine Hecke algebra, acting by Dunkl-type differential-reflection operators. We use Gutkin's generalization of the equivalence between the impenetrable Bose-gas and the free Fermi-gas to derive the Bethe ansatz equations and the Bethe ansatz eigenfunctions., Comment: 36 pages. The analysis of the propagation operator in Sections 5 and 6 is corrected and simplified. To appear in Comm. Math. Phys

We interpret the five parameter family of Macdonald-Koornwinder polynomials as vector valued spherical functions on quantum Grassmannians., Comment: 47 pages, no figures

It is shown that a dynamical quantum group arising from a vertex-IRF transformation has a second realization with untwisted dynamical multiplication but nontrivial bigrading. Applied to the $\hbox{SL}(2;\mathbb{C})$ dynamical quantum group, the second realization is naturally described in terms of Koornwinder's twisted primitive elements. This leads to an intrinsic explanation why harmonic analysis on the ``classical'' $\hbox{SL}(2;\mathbb{C})$ quantum group with respect to twisted primitive elements, as initiated by Koornwinder, is the same as harmonic analysis on the $\hbox{SL}(2;\mathbb{C})$ dynamical quantum group., Comment: 24 pages

Hyperbolic beta integrals are analogues of Euler's beta integral in which the role of Euler's gamma function is taken over by Ruijsenaars' hyperbolic gamma function. They may be viewed as $(q,\widetilde{q})$-bibasic analogues of the beta integral in which the two bases $q$ and $\widetilde{q}$ are interrelated by modular inversion, and they entail $q$-analogues of the beta integral for $|q|=1$. The integrals under consideration are the hyperbolic analogues of the Ramanujan integral, the Askey-Wilson integral and the Nassrallah-Rahman integral. We show that the hyperbolic Nassrallah-Rahman integral is a formal limit case of Spiridonov's elliptic Nassrallah-Rahman integral., Comment: 35 pages. Remarks and references to recent new developments are added. To appear in Adv. Math

Eigenfunctions of the Askey-Wilson second order $q$-difference operator for $0

Generalized flag manifolds endowed with the Bruhat-Poisson bracket are compact Poisson homogeneous spaces, whose decompositions in symplectic leaves coincide with their stratifications in Schubert cells. In this note it is proved that the irreducible *-representations of the corresponding quantized flag manifolds are also parametrized by their Schubert cells. An important step is the determination of suitable algebraic generators of the quantized flag manifolds. These algebraic generators can be naturally expressed in terms of quantum Plucker coordinates. This note complements the paper of the author and Dijkhuizen in Commun. Math. Phys. 203 (1999), pp. 297-324, in which these results were established for a special subclass of generalized flag manifolds., Comment: 10 pages

In the first part of the paper kernels are constructed which meromorphically extend the Macdonald-Koornwinder polynomials in their degrees. In the second part of the paper the kernels associated with rank one root systems are used to define nonsymmetric variants of the spherical Fourier transform on the quantum SU(1,1) group. Related Plancherel and inversion formulas are derived using double affine Hecke algebra techniques., Comment: 68 pages, no figures

The Askey-Wilson function transform is a q-analogue of the Jacobi function transform with kernel given by an explicit non-polynomial eigenfunction of the Askey-Wilson second order q-difference operator. The kernel is called the Askey-Wilson function. In this paper an explicit expansion formula for the Askey-Wilson function in terms of Askey-Wilson polynomials is proven. With this expansion formula at hand, the image under the Askey-Wilson function transform of an Askey-Wilson polynomial multiplied by an analogue of the Gaussian is computed explicitly. As a special case of these formulas a q-analogue (in one variable) of the Macdonald-Mehta integral is obtained, for which also two alternative, direct proofs are presented., Comment: 24 pages. Some remarks added in section 6 on the connection with moment problems

In this paper we present an explicit (rank one) function transform which contains several Jacobi-type function transforms and Hankel-type transforms as degenerate cases. The kernel of the transform, which is given explicitly in terms of basic hypergeometric series, thus generalizes the Jacobi function as well as the Bessel function. The kernel is named the Askey-Wilson function, since it provides an analytical continuation of the Askey-Wilson polynomial in its degree. In this paper we establish the $L^2$-theory of the Askey-Wilson function transform, and we explicitely determine its inversion formula., Comment: LaTeX2e file. 19 pages

We study Askey-Wilson type polynomials using representation theory of the double affine Hecke algebra. In particular, we prove bi-orthogonality relations for non-symmetric and anti-symmetric Askey-Wilson polynomials with respect to a complex measure. We give duality properties of the non-symmetric Askey-Wilson polynomials, and we show how the non-symmetric Askey-Wilson polynomials can be created from Sahi's intertwiners. The diagonal terms associated to the bi-orthogonality relations (which replace the notion of quadratic norm evaluations for orthogonal polynomials) are expressed in terms of residues of the complex weight function using intertwining properties of the non-symmetric Askey-Wilson transform under the action of the double affine Hecke algebra. We evaluate the constant term, which is essentially the well-known Askey-Wilson integral, using shift operators. We furthermore show how these results reduce to well-known properties of the symmetric Askey-Wilson polynomials, as were originally derived by Askey and Wilson using basic hypergeometric series theory., Comment: 35 pages

In this paper we present an addition to Askey's scheme of q-hypergeometric orthogonal polynomials involving classes of q-special functions which do not consist of polynomials only. The special functions are q-analogues of the Jacobi and Bessel function, and are Askey-Wilson functions, big q-Jacobi functions and little q-Jacobi functions and the corrresponding q-Bessel functions. The generalised orthogonality relations and the second order q-difference equations for these families are given. Limit transitions between these families are discussed. The quantum group theoretic interpretations are discussed shortly., Comment: 17 pages, 2 figures, AMS-TeX Some formulas corrected, reference updated

We give a detailed description of the resolution of the identity of a second order $q$-difference operator considered as an unbounded self-adjoint operator on two different Hilbert spaces. The $q$-difference operator and the two choices of Hilbert spaces naturally arise from harmonic analysis on the quantum group $SU_q(1,1)$ and $SU_q(2)$. The spectral analysis associated to $SU_q(1,1)$ leads to the big $q$-Jacobi function transform together with its Plancherel measure and inversion formula. The dual orthogonality relations give a one-parameter family of non-extremal orthogonality measures for the continuous dual $q^{-1}$-Hahn polynomials with $q^{-1}>1$, and explicit sets of functions which complement these polynomials to orthogonal bases of the associated Hilbert spaces. The spectral analysis associated to $SU_q(2)$ leads to a functional analytic proof of the orthogonality relations and quadratic norm evaluations for the big $q$-Jacobi polynomials., Comment: 40 pages

The quantum complex Grassmannian U_q/K_q of rank l is the quotient of the quantum unitary group U_q=U_q(n) by the quantum subgroup K_q=U_q(n-l)xU_q(l). We show that (U_q,K_q) is a quantum Gelfand pair and we express the zonal spherical functions, i.e. K_q-biinvariant matrix coefficients of finite- dimensional irreducible representations of U_q, as multivariable little q-Jacobi polynomials depending on one discrete parameter. Another type of biinvariant matrix coefficients is identified as multivariable big q-Jacobi polynomials. The proof is based on earlier results by Noumi, Sugitani and the first author relating Koornwinder polynomials to a one-parameter family of quantum complex Grassmannians, and certain limit transitions from Koornwinder polynomials to multivariable big and little q-Jacobi polynomials studied by Koornwinder and the second author., Comment: 39 pages, no figures, Latex2e

We study irreducible *-representations of a certain quantization of the algebra of polynomial functions on a generalized flag manifold regarded as a real manifold. All irreducible *-representations are classified for a subclass of flag manifolds containing in particular the irreducible compact Hermitian symmetric spaces. For this subclass it is shown that the irreducible *-representations are parametrized by the symplectic leaves of the underlying Poisson bracket. We also discuss the relation between the quantized flag manifolds studied in this paper and the quantum flag manifolds studied by Soibelman, Lakshimibai and Reshetikhin, Jurco and Stovicek, and Korogodsky., Comment: AMS-LaTeX v1.2, 27 pages, no figures

The five parameter family of multivariable Askey-Wilson polynomials is studied with four parameters generically complex. The multivariable Askey-Wilson polynomials form an orthogonal system with respect to an explicit (in general complex) measure. A partially discrete orthogonality measure is obtained by shifting the contour to the torus while picking up residues. A parameter domain is given for which the partially discrete orthogonality measure is positive. The orthogonality relations and norm evaluations for multivariable q-Racah polynomials and multivariable big and little q-Jacobi polynomials are proved by taking suitable limits in the orthogonality relations for the multivariable Askey-Wilson polynomials. In particular new proofs of several well known q-analogues of the Selberg integral are obtained., Comment: AMS-LaTeX v1.2, 53 pages, no figures

The Koornwinder-Macdonald multivariable generalization of the Askey-Wilson polynomials is studied for parameters satisfying a truncation condition such that the orthogonality measure becomes discrete with support on a finite grid. For this parameter regime the polynomials may be seen as a multivariable counterpart of the (one-variable) $q$-Racah polynomials. We present the discrete orthogonality measure, expressions for the normalization constants converting the polynomials into an orthonormal system (in terms of the normalization constant for the unit polynomial), and we discuss the limit $q\to 1$ leading to multivariable Racah type polynomials. Of special interest is the situation that $q$ lies on the unit circle; in that case it is found that there exists a natural parameter domain for which the discrete orthogonality measure (which is complex in general) becomes real-valued and positive. We investigate the properties of a finite-dimensional discrete integral transform for functions over the grid, whose kernel is determined by the multivariable $q$-Racah polynomials with parameters in this positivity domain., Comment: AMS-LaTeX v1.2, 38 pages