V. -B. K. Rogov
The modified q-Bessel functions and the q-Bessel-Macdonald functions of the first and second kind are introduced. Their definition is based on representations as power series. Recurrence relations, the q-Wronskians, asymptotic decompositions and q-integral representations are received. In addition, the q-Bessel-Macdonald function of kind 3 is determined by its q-integral representation.
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The q-Bessel-Macdonald functions of kinds 1, 2 and 3 are considered. Their representations by classical integral are constructed.
The $q$ analog of Modified Bessel functions and Bessel-Macdonald functions, were defined in our previous work (q-alg/950913) as general solutions of a second order difference equations. Here we present a collection of their representations by the Jackson q-integral.
We define a q-analog of the modified Bessel and Bessel-Macdonald functions. As for the q-Bessel functions of Jackson there is a couple of functions of the both kind. They are arisen in the Harmonic analysis on quantum symmetric spaces similarly to their classical counterpart. Their definition is based on the power expansions. We derive the recurrence relations, difference equations, q-Wronskians, and an analog of asymptotic expansions which turns out is exact in some domain if $q\neq 1$. Some relations for the basic hypergeometric function which follow from this fact are discussed.
In this paper we study the variation diminishing kernel as a part of the $q$-calculus. We introduce the $q$-Macdonald function a newborne in the family of the $q$-special functions which play a central role in this study.
The aim of this paper is to give a group theoretical interpretation of the three types of Bessel-Jackson functions. We consider a family of quantum Lorentz groups and a family of quantum Lobachevsky spaces. For three members of quantum Lobachevsky spaces the Casimir operators give rise to the two-body relativistic open Toda lattice Hamiltonians. Their eigen-functions are the modified Bessel-Jackson functions of three types. We construct the principal series of unitary irreducible representations of the quantum Lorentz groups. Special matrix elements in the irreducible spaces are the Bessel-Mac…
This paper is based on the lectures given by the first author at Harvard in February and March, 2009. It begins with an introduction to the classical p-adic theory of the Macdonald, Matsumoto and Whittaker functions. Its major directions are as follows: 1) extending the theory of DAHA to arbitrary levels; 2) the affine Satake isomorphism and Hall functions via DAHA; 3) the spinor Dunkl operators for the Q-Toda equation; 4) applications to the nil-DAHA and q-Whittaker functions; 5) the technique of spinors in the differential theory and its applications to the AKZ-QMBP isomorphism theorem.
Elliptic Macdonald polynomials of sl(2)-type and level 2 are introduced. Suitable limits of elliptic Macdonald polynomials are the standard Macdonald polynomials and conformal blocks. Identities for elliptic Macdonald polynomials, in particular their modular properties, are studied.
In this paper, a link between $q$-difference equations, Jacobi operators and orthogonal polynomials is given. Replacing the variable $x$ by $ q^{-n}$ in a Sturm-Liouville $q$-difference equation we discovered the Jacobi operator. With appropriate initial conditions, the eigenfunctions of such operators are either $q$-orthogonal polynomials or the modified $q$-Bessel function and a newborn the $q$-Macdonald ones. The new Polynomial sequence we found is related to the $q$-Lommel polynomials introduced by Koelink and other. Adapting E. C. Titchmarsh's theory, we showed the existence of a solution…
In symmetric Macdonald polynomial theory the Pieri formula gives the branching coefficients for the product of the rth elementary symmetric function and the Macdonald polynomial. In this paper we give the nonsymmetric analogues for the cases r=1 and r=n-1. We do this by first deducing the the decomposition for the product of any nonsymmetric Macdonald polynomial with a linear function in terms of nonsymmetric Macdonald polynomials. As a corollary of finding the branching coefficients of the product of the first elementary function with a nonsymmetric Macdonald polynomial we evaluate the genera…
We consider the space of elliptic hypergeometric functions of the sl_2 type associated with elliptic curves with one marked point. This space represents conformal blocks in the sl_2 WZW model of CFT. The modular group acts on this space. We give formulas for the matrices of the action in terms of values at roots of unity of Macdonald polynomials of the sl_2 type.
We interpret the five parameter family of Macdonald-Koornwinder polynomials as vector valued spherical functions on quantum Grassmannians.
The connections between q-Bessel functions of three types and q-exponential of three types are established. The q-exponentials and the q-Bessel functions are represented as the Laurent series. The asymptotic behaviour of the q-exponentials and the q-Bessel functions is investigated.
The $6j$-symbols for representations of the $\mathrm{SU}(2)$ quantum group are given by Hahn-Exton $q$-Bessel functions. This interpretation leads to several summation identities for the $q$-Bessel functions. Multivariate $q$-Bessel functions are defined, which are shown to be limit cases of multivariate Askey-Wilson polynomials. The multivariate $q$-Bessel functions occur as $3nj$-symbols.
A generalization of Newton's identity on symmetric functions is given. Using the generalized Newton identity we give a unified method to show the existence of Hall-Littlewood, Jack and Macdonald polynomials. We also give a simple proof of the Jing-Jözefiak formula for two-row Macdonald functions.
Based on a generalized Newton's identity, we construct a family of symmetric functions which deform the modular Hall-Littlewood functions. We also give a determinant formula for the Macdonald functions.
Let W be the complex reflection group G(e,1,n). In the author's previous paper, Hall-Littlewood functions associated to W were introduced. In the special case where W is a Weyl group of type B_n, they are closely related to Green polynomials of finite classical groups. In this paper, we introduce a two variables version of the above Hall-Littlewood functions, as a generalization of Macdonald functions associated to symmetric groups. A generalization of Macdonald operators is also constructed, and we characterize such functions by making use of Macdonald operators, assuming a certain conjecture…
Quantum analogues of the homogeneous spaces $\GL(n)/\SO(n)$ and $\GL(2n)/\Sp(2n)$ are introduced. The zonal spherical functions on these quantum homogeneous spaces are represented by Macdonald's symmetric polynomials $P_{\ld}=P_{\ld}(x_1,\cdots,x_n;q,t)$ with $t=q^{1 \over 2}$ or $t=q^2$.
A unified theory of quantum symmetric pairs is applied to q-special functions. Previous work characterized certain left coideal subalgebras in the quantized enveloping algebra and established an appropriate framework for quantum zonal spherical functions. Here a distinguished family of such functions, invariant under the Weyl group associated to the restricted roots, is shown to be a family of Macdonald polynomials, as conjectured by Koornwinder and Macdonald. Our results place earlier work for Lie algebras of classical type in a general context and extend to the exceptional cases.
The very well--poised elliptic Macdonald functions W_lambda in n independent variables are defined and their properties are investigated. The W_lambda are generalized by introducing an extra parameter to the elliptic Jackson coefficients omega_lambda and their properties are studied. BC_n multivariable Jackson sums in terms of both W_lambda and omega_lambda functions are proved.
A weight function which $q$-generalizes the ground state wave function of the multi-component Calogero-Sutherland quantum many body system is introduced. Conjectures, and some proofs in special cases, are given for a constant term identity involving this function. A Gram-Schmidt procedure with respect to the inner product associated with the weight function is used to define orthogonal polynomials in one of the components, which are conjectured to be the Macdonald polynomials $P_κ(w_1,\dots,w_{N_0};qt^p,t)$, and a proof is given in a special case. Conjectures are also given for an adjoint prop…
A new type of sl_3 basic hypergeometric series based on Macdonald polynomials is introduced. Besides a pair of Macdonald polynomials attached to two different sets of variables, a key-ingredient in the sl_3 basic hypergeometric series is a bisymmetric function related to Macdonald's commuting family of q-difference operators, to the sl_3 Selberg integrals of Tarasov and Varchenko, and to alternating sign matrices. Our main result for sl_3 series is a multivariable generalization of the celebrated q-binomial theorem. In the limit this q-binomial sum yields a new sl_3 Selberg integral for Jack p…
On the Nazarov-Sklyanin Lax operator for Macdonald functions
该研究扩展了杰克情形下的谱理论，将NS算子推广为带辅助变量多项式系数对称函数的Lax算子，得到麦克唐纳小伍德-理查森系数的双参数盒和恒等式及相关线性关系。
Multiple-integral representations of the (skew-)Macdonald symmetric functions are obtained. Some bosonization schemes for the integral representations are also constructed.
The n-point correlation functions introduced by Bloch and Okounkov have already found several geometric connections and algebraic generalizations. In this Note we formulate a q,t-deformation of this n-point function. The key operator used in our formulation arises from the theory of Macdonald polynomials and affords a vertex operator interpretation. We obtain closed formulas for the n-point functions when n =1,2 in terms of the basic hypergeometric functions. We further generalize the q,t-deformed n-point function to more general vertex operators.
Quantum spherical functions of type $χ$ as Macdonald-Koornwinder polynomials
将量子对称对理论应用于q特殊函数，证明埃尔米特量子对称对的χ-球函数族中，受外尔群不变的特殊族在特定条件下为Macdonald-Koornwinder多项式。
The Macdonald polynomials can be obtained by acting on the constant 1 with creation operators. Three different expressions for these operators are derived, one from the other, in a rather succint way. When the last of these expressions is used, the formalism is seen to imply straightforwardly the integrality of the (q,t)-Kostka coefficients, that is of the expansion coefficients for the Macdonald functions in terms of Schur functions.
We establish a previously conjectured connection between $p$-adics and quantum groups. We find in Sklyanin's two parameter elliptic quantum algebra and its generalizations, the conceptual basis for the Macdonald polynomials, which ``interpolate'' between the zonal spherical functions of related real and $p$\--adic symmetric spaces. The elliptic quantum algebras underlie the $Z_n$\--Baxter models. We show that in the $n \air \infty$ limit, the Jost function for the scattering of {\em first} level excitations in the $Z_n$\--Baxter model coincides with the Harish\--Chandra\--like $c$\--function c…
In this paper we present a formula for Macdonald's polynomials for the root system A(n-1) which arises from the representation theory of quantum sl(n). This formula expresses Macdonald's polynomials via (weighted) traces of intertwining operators between certain modules over quantum sl(n). We also describe the commutative system of Macdonald's difference operators using the generators of the center of the quantum universal enveloping algebra, and use this description to prove a trace formula for generic eigenfunctions of these operators. These functions are generalized q-hypergeometric functio…
We extend some results about shifted Schur functions to the general context of shifted Macdonald polynomials. We obtain two explicit formulas for these polynomials: a $q$-integral representation and a combinatorial formula. Our main tool is a $q$-integral representation for ordinary Macdonald polynomials. We also discuss duality for shifted Macdonald polynomials and Jack degeneration of these polynomials.
In this paper we construct examples of commutative rings of difference operators with matrix coefficients from representation theory of quantum groups, generalizing the results of our previous paper to the $q$-deformed case. A generalized Baker-Akhiezer function $Ψ$ is realized as a matrix character of a Verma module and is a common eigenfunction for a commutative ring of difference operators. In particular, we obtain the following result in Macdonald theory: at integer values of the Macdonald parameter $k$, there exist difference operators commuting with Macdonald operators which are not poly…
We show how a certain limit of the nonsymmetric Macdonald polynomials appears in the representation theory of semisimple groups over p--adic fields as matrix coefficients for the unramified principal series representations. The result is the nonsymmetric counterpart of a classical result relating the same limit of the symmetric Macdonald polynomials to zonal spherical functions on groups of p--adic type.
This paper is a continuation of our papers [EK1, EK2]. In [EK2] we showed that for the root system A_n-1 one can obtain Macdonald's polynomials - a new interesting class of symmetric functions recently defined by I. Macdonald {M1] - as weighted traces of intertwining operators between certain finite-dimensional representations of U_q sl_n. The main goal of the present paper is to use this construction to give a representation-theoretic proof of Macdonald's inner product and symmetry identities for the root system A_n-1. Macdonald's inner product identities (see [M2]) have been proved by combin…
A quantum deformation of the Virasoro algebra is defined. The Kac determinants at arbitrary levels are conjectured. We construct a bosonic realization of the quantum deformed Virasoro algebra. Singular vectors are expressed by the Macdonald symmetric functions. This is proved by constructing screening currents acting on the bosonic Fock space.
We motivate and prove a series of identities which form a generalization of the Euler's pentagonal number theorem, and are closely related to specialized Macdonald's identities for powers of the Dedekind $η$--function. More precisely, we show that what we call ``denominator formula'' for the Virasoro algebra has ``higher analogue'' for all $c_{s,t}$-minimal models. We obtain one identity per series which is in agreement with features of conformal field theory such as {\em fusion} and {\em modular invariance} that require all the irreducible modules of the series. In particular, in the case of …
We study the interpolation Macdonald functions, remarkable inhomogeneous generalizations of Macdonald functions, and a sequence $A^1, A^2, \ldots$ of commuting operators that are diagonalized by them. Such a sequence of operators arises in the projective limit of finite families of commuting q-difference operators studied by Okounkov, Knop and Sahi. The main theorem is an explicit formula for the operators $A^k$. Our formula involves the family of Hall-Littlewood functions and a new family of inhomogeneous Hall-Littlewood functions, for which we give an explicit construction and identify as a …
In the basic representation of $U_q(\hat{sl}(2))$ realized via the algebra of symmetric functions we compare the canonical basis with the basis of Macdonald polynomials with $q=t^2$. We show that the Macdonald polynomials are invariant with respect to the bar involution defined abstractly on the representations of quantum groups. We also prove that the Macdonald scalar product coincides with the abstract Kashiwara form. This implies, in particular, that the Macdonald polynomials form an intermediate basis between the canonical basis and the dual canonical basis, and the coefficients of the tra…
Macdonald defined two-parameter Kostka functions K_{λμ}(q,t) where λ, μare partitions. The main purpose of this paper is to extend his definition to include all compositions as indices. Following Macdonald, we conjecture that also these more general Kostka functions are polynomials in q and t^{1/2} with non-negative integers as coefficients. If q=0 then our Kostka functions are Kazhdan-Lusztig polynomials of a special type. Therefore, our positivity conjecture combines Macdonald positivity and Kazhdan-Lusztig positivity and hints towards a connection between Macdonald and Kazhdan-Lusztig theor…
The Macdonald process is a stochastic process on the collection of partitions that is a $(q,t)$-deformed generalization of the Schur process. In this paper, we approach the Macdonald process identifying the space of symmetric functions with a Fock representation of a Heisenberg algebra. By using the free field realization of operators diagonalized by the Macdonald symmetric functions, we propose a method of computing several correlation functions with respect to the Macdonald process. It is well-known that expectation value of several observables for the Macdonald process admit determinantal e…
Our main aim with these notes is to introduce the combinatorial and symmetric function tools that relate to the description of the Poincare polynomial of the triply graded Khovanov-Rozansky homology of torus links, a.k.a. the (reduced) superpolynomial of these links.
The Macdonald symmetric functions are used to define measures on the set of all partitions of all integers. Probabilistic algorithms are given for growing partitions according to these measures. The case of Hall-Littlewood polynomials is related to the finite classical groups, and the corresponding algorithms simplify. The case of Schur functions leads to a $q$-analog of Plancharel measure, and a conditioned version of the corresponding algorithms yields generalizations of the hook walk of combinatorics.
We demonstrate in some detail how Macdonald polynomials emerge from the recently introduced 3-Schur functions when the plane-partition vector time-variables are projected onto the ordinary scalar times under non-vanishing angles, which depend on $q$ and $t$. We also explain how the cut-and-join operators smoothly interpolate between different cases. Most of consideration is restricted to level two.
Using the Lax operator formalism, we construct a family of pairwise commuting operators such that the Macdonald symmetric functions of infinitely many variables and of two parameters $q,t$ are their eigenfunctions. We express our operators in terms of the Hall-Littlewood symmetric functions of the same variables and of the parameter $t$ corresponding to the partitions with one part only. Our expression is based on the notion of Baker-Akhiezer function.
We give a proof of Awata and Yamada's conjecture for the explicit formula of Whittaker vector of the deformed Virasoro algebra realized in the Fock space. The formula is expressed as a summation over Macdonald symmetric functions with factored coefficients. In the proof we fully use currents appearing in the Fock representation of Ding-Iohara-Miki quantum algebra. We also mention an interpretation of Whittaker vector in terms of the geometry of the Hilbert schemes of points on the affine plane.
We construct a family of pairwise commuting operators such that the Macdonald symmetric functions of infinitely many variables $x_1,x_2,...$ and of two parameters $q,t$ are their eigenfunctions. These operators are defined as limits at $N\to\infty$ of renormalised Macdonald operators acting on symmetric polynomials in the variables $x_1,...,x_N$. They are differential operators in terms of the power sum variables $p_n=x_1^n+x_2^n+...$ and we compute their symbols by using the Macdonald reproducing kernel. We express these symbols in terms of the Hall-Littlewood symmetric functions of the varia…
We propose periodic Macdonald processes as a $(q,t)$-deformation of periodic Schur processes and a periodic analogue of Macdonald processes. It is known that, in the theory of stochastic processes related to a family of symmetric functions, the Cauchy-like identity gives an explicit expression of a partition function. We compute the partition functions of periodic Macdonald processes relying on the free field realization of the Macdonald theory. We also study several families of observables for periodic Macdonald processes and give formulas of their moments. The technical tool is the free fiel…
Generalized Hall-Littlewood polynomials (Macdonald spherical functions) and generalized Kostka-Foulkes polynomials ($q$-weight multiplicities) arise in many places in combinatorics, representation theory, geometry, and mathematical physics. This paper attempts to organize the different definitions of these objects and prove the fundamental combinatorial results from ``scratch'', in a presentation which, hopefully, will be accessible and useful for both the nonexpert and researchers currently working in this very active field. The combinatorics of the affine Hecke algebra plays a central role. …
An explicit formula is obtained for the generalized Macdonald functions on the $N$-fold Fock tensor spaces, calculating a certain matrix element of a composition of several screened vertex operators. As an application, we prove the factorization property of the arbitrary matrix elements of the multi-valent intertwining operator (or refined topological vertex operator) associated with the Ding--Iohara--Miki algebra (DIM algebra) with respect to the generalized Macdonald functions, which was conjectured by Awata, Feigin, Hoshino, Kanai, Yanagida and one of the authors. Our proof is based on the …
Let $Λ$ be the space of symmetric functions and $V_k$ be the subspace spanned by the modified Schur functions $\{S_λ[X/(1-t)]\}_{λ_1\leq k}$. We introduce a new family of symmetric polynomials, $\{A_λ^{(k)}[X;t]\}_{λ_1\leq k}$, constructed from sums of tableaux using the charge statistic. We conjecture that the polynomials $A_λ^{(k)}[X;t]$ form a basis for $V_k$ and that the Macdonald polynomials indexed by partitions whose first part is not larger than $k$ expand positively in terms of our polynomials. A proof of this conjecture would not only imply the Macdonald positivity conjecture, but wo…
We provide Harish-Chandra type formulas for the multivariate Bessel functions and Heckman-Opdam hypergeometric functions as representation-valued integrals over dressing orbits. Our expression is the quasi-classical limit of the realization of Macdonald polynomials as traces of intertwiners of quantum groups given by Etingof-Kirillov Jr. Integration over the Liouville tori of the Gelfand-Tsetlin integrable system and adjunction for higher Calogero-Moser Hamiltonians recovers and gives a new proof of the integral realization over Gelfand-Tsetlin polytopes which appeared in the recent work of Bo…
The resonance relations are identities between coordinates of functions with values in tensor products of representations of the quantum group Uq(sl2). We show that the space of hypergeometric solutions of the associated qKZB equations is characterized as the space of functions of Baker-Akhiezer type, satisfying the resonance relations. We give an alternative representation-theoretic construction of this space, using the traces of regularized intertwining operators for the quantum group, and thus establish the equivalence between hypergeometric and trace function solutions of the qKZB equation…
Integrality of two variable Kostka functions
本文证明双变量Kostka函数至少是具有整数系数的多项式，通过建立递推关系和 affine Hecke 代数来证明非对称Macdonald多项式的相关性质。
Wreath Macdonald polynomials, a survey
本文概述了Wen关于Wreath Macdonald多项式的研究成果，探讨了其几何基础和反射 functor 的作用，并提出了新的猜想。
We refine the statement of the denominator and evaluation conjectures for affine Macdonald polynomials proposed by Etingof-Kirillov Jr. and prove the first non-trivial cases of these conjectures. Our results provide a q-deformation of the computation of genus 1 conformal blocks via elliptic Selberg integrals by Felder-Stevens-Varchenko. They allow us to give precise formulations for the affine Macdonald conjectures in the general case which are consistent with computer computations. Our method applies recent work of the second named author to relate these conjectures in the case of $U_q(\wideh…
We introduce a new family of symmetric multivariate polynomials, whose coefficients are meromorphic functions of two parameters $(q,t)$ and polynomial in a further two parameters $(u,v)$. We evaluate these polynomials explicitly as a matrix product. At $u=v=0$ they reduce to Macdonald polynomials, while at $q=0$, $u=v=s$ they recover a family of inhomogeneous symmetric functions originally introduced by Borodin.
We establish orthogonality relations for the Baker-Akhiezer (BA) eigenfunctions of the Macdonald difference operators. We also obtain a version of Cherednik-Macdonald-Mehta integral for these functions. As a corollary, we give a simple derivation of the norm identity and Cherednik-Macdonald-Mehta integral for Macdonald polynomials. In the appendix written by the first author, we prove a summation formula for BA functions. We also consider more general identities of Cherednik type, which we use to introduce and construct more general, twisted BA functions. This leads to a construction of new qu…
Generalized Macdonald functions and quantum toroidal gl(1) algebra
本研究将普通Macdonald对称函数的$e_1$-Pieri规则、与Whittaker向量相关恒等式等公式推广到量子环面$\rm \frak{gl}(1)$代数对应的广义情形，提出五项关系与Fourier/Hopf配对的推广，并证明了广义Macdonald核的因子化猜想。
Let Sym denote the algebra of symmetric functions and $P_μ(\,\cdot\,;q,t)$ and $Q_μ(\,\cdot\,;q,t)$ be the Macdonald symmetric functions (recall that they differ by scalar factors only). The $(q,t)$-Cauchy identity $$ \sum_μP_μ(x_1,x_2,\dots;q,t)Q_μ(y_1,y_2,\dots;q,t)=\prod_{i,j=1}^\infty\frac{(x_iy_jt;q)_\infty}{(x_iy_j;q)_\infty} $$ expresses the fact that the $P_μ(\,\cdot\,;q,t)$'s form an orthogonal basis in Sym with respect to a special scalar product $\langle\,\cdot\,,\,\cdot\,\rangle_{q,t}$. The present paper deals with the inhomogeneous \emph{interpolation} Macdonald symmetric function…
A basic triad in Macdonald theory
本文证明Noumi-Shiraishi级数在$t=q^{-m}$时约化为多变量BA函数，从而揭示Macdonald多项式、BA函数与NS级数构成定义不同却直接相连的基本三元组。
Higher Rank Macdonald Polynomials
本文构造高阶非对称与对称 Macdonald 多项式，证明其 Knop–Sahi 关系、权组合描述、稳定性及椭圆 Hall 代数和双 Dyck 路径代数表示意义。
We define cylindric generalisations of skew Macdonald functions when one of their parameters is set to zero. We define these functions as weighted sums over cylindric skew tableaux: fixing two integers n>2 and k>0 we shift an ordinary skew diagram of two partitions, viewed as a subset of the two-dimensional integer lattice, by the period vector (n,-k). Imposing a periodicity condition one defines cylindric skew tableaux as a map from the periodically continued skew diagram into the integers. The resulting cylindric Macdonald functions appear in the coproduct of a commutative Frobenius algebra,…
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