arXiv 0807.21552008-07math.QAmath.RT

Whittaker limits of difference spherical functions

Ivan Cherednik

备注：: V2: a discussion of the one-dimensional case was added. V3: Jackson integration and growth estimates were added. V4: a q-variant of the Harish-Chandra asymptotic formula for spherical functions was added. V5: editing, some improvements, adding references. V6: General editing

英文摘要：

We introduce the (global) q-Whittaker function as the limit at t=0 of the q,t-spherical function extending the symmetric Macdonald polynomials to arbitrary eigenvalues. The construction heavily depends on the technique of the q-Gaussians developed by the author (and Stokman in the non-reduced case). In this approach, the q-Whittaker function is given by a series convergent everywhere, a kind of generating function for multi-dimensional q-Hermite polynomials (closely related to the level 1 Demazure characters). One of the applications is a q-version of the Shintani- Casselman- Shalika formula, which appeared directly connected with q-Mehta- Macdonald identities in terms of the Jackson integrals. This formula generalizes that of type A due to Gerasimov et al. to arbitrary reduced root systems. At the end of the paper, we obtain a q,t-counterpart of the Harish-Chandra asymptotic formula for the spherical functions, including the Whittaker limit.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

A new take on spherical, Whittaker and Bessel functions (Spherical and Whittaker functions via DAHA I,II)

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.

2012-10-29math.QA

Nonsymmetric difference Whittaker functions

Starting with nonsymmetric global difference spherical functions, we define and calculate spinor (nonsymmetric) global q-Whittaker functions for arbitrary reduced root systems, which are reproducing kernels of the DAHA-Fourier transforms of Nil-DAHA and solutions of the q-Toda-Dunkl eigenvalue problem. We introduce the spinor q-Toda-Dunkl operators as limits of the difference Dunkl operators in DAHA theory under the spinor variant of the Ruijsenaars procedure. Their general algebraic theory (any reduced root systems) is the key part of this paper, based on the new technique of W-spinors and co…

2013-04-22math.QAmath.RT

Iwahori-metaplectic duality

We construct a family of solvable lattice models whose partition functions include $p$-adic Whittaker functions for general linear groups from two very different sources: from Iwahori-fixed vectors and from metaplectic covers. Interpolating between them by Drinfeld twisting, we uncover unexpected relationships between Iwahori and metaplectic Whittaker functions. This leads to new Demazure operator recurrence relations for spherical metaplectic Whittaker functions. In prior work of the authors it was shown that the row transfer matrices of certain lattice models for spherical metaplectic Whitta…

2022-09-08math.NTmath.QAmath.RT

A path model for Whittaker vectors

In this paper we construct weighted path models to compute Whittaker vectors in the completion of Verma modules, as well as Whittaker functions of fundamental type, for all finite-dimensional simple Lie algebras, affine Lie algebras, and the quantum algebra $U_q(\mathfrak{sl}_{r+1})$. This leads to series expressions for the Whittaker functions. We show how this construction leads directly to the quantum Toda equations satisfied by these functions, and to the $q$-difference equations in the quantum case. We investigate the critical limit of affine Whittaker functions computed in this way.

2014-07-31math-phmath.MPmath.QA

Vertex operators, solvable lattice models and metaplectic Whittaker functions

We show that spherical Whittaker functions on an $n$-fold cover of the general linear group arise naturally from the quantum Fock space representation of $U_q(\widehat{\mathfrak{sl}}(n))$ introduced by Kashiwara, Miwa and Stern (KMS). We arrive at this connection by reconsidering solvable lattice models known as `metaplectic ice' whose partition functions are metaplectic Whittaker functions. First, we show that a certain Hecke action on metaplectic Whittaker coinvariants agrees (up to twisting) with a Hecke action of Ginzburg, Reshetikhin, and Vasserot. This allows us to expand the framework o…

2020-06-12math.NTmath.QAmath.RT

A Yang-Baxter equation for metaplectic ice

We will give new applications of quantum groups to the study of spherical Whittaker functions on the metaplectic $n$-fold cover of $GL(r,F)$, where $F$ is a nonarchimedean local field. Earlier Brubaker, Bump, Friedberg, Chinta and Gunnells had shown that these Whittaker functions can be identified with the partition functions of statistical mechanical systems. They postulated that a Yang-Baxter equation underlies the properties of these Whittaker functions. We confirm this, and identify the corresponding Yang-Baxter equation with that of the quantum affine Lie superalgebra $U_{\sqrt{v}}(\wideh…

2018-08-31math.NTmath.QAmath.RT

更多相关论文收起更多论文

Yang-Baxter Equations for General Metaplectic Ice

In this paper, we extend results connecting quantum groups to spherical Whittaker functions on metaplectic covers of $GL_r(F)$, for $F$ a nonarchimedean local field. Brubaker, Buciumas, and Bump showed that for a certain metaplectic $n$-fold cover of $GL_r(F)$ a set of Yang-Baxter equations model the action of standard intertwiners on principal series Whittaker functions. These equations arise from a Drinfeld twist of the quantum affine Lie superalgebra $U_{\sqrt{v}}(\widehat{\frak{gl}}(n)),$ where $v = q^{-1}$ for $q$ the cardinality of the residue field. We extend their results to all metapl…

2021-02-22math.NTmath.QAmath.RT

超几何积分、钩子公式与Whittaker向量

Hypergeometric integrals, hook formulas and Whittaker vectors

本文确定了两类与Young图相关的多维超几何积分之间的比例系数，即加权钩子乘积的倒数，并由此给出gl_n泛包络代数中心在Whittaker向量空间上作用的显式同时特征向量基。

2023-08-09math-phmath.AGmath.MP

A Relativistic Conical Function and its Whittaker Limits

In previous work we introduced and studied a function $R(a_{+},a_{-},{\bf c};v,\hat{v})$ that is a generalization of the hypergeometric function ${}_2F_1$ and the Askey-Wilson polynomials. When the coupling vector ${\bf c}\in{\mathbb C}^4$ is specialized to $(b,0,0,0)$, $b\in{\mathbb C}$, we obtain a function ${\mathcal R}(a_{+},a_{-},b;v,2\hat{v})$ that generalizes the conical function specialization of ${}_2F_1$ and the $q$-Gegenbauer polynomials. The function ${\mathcal R}$ is the joint eigenfunction of four analytic difference operators associated with the relativistic Calogero-Moser syste…

2011-11-01math-phmath.CAmath.MP

Fermionic formulas for eigenfunctions of the difference Toda Hamiltonian

We use the Whittaker vectors and the Drinfeld Casimir element to show that eigenfunctions of the difference Toda Hamiltonian can be expressed via fermionic formulas. Motivated by the combinatorics of the fermionic formulas we use the representation theory of the quantum groups to prove a number of identities for the coefficients of the eigenfunctions.

2018-01-24math-phmath.MPmath.QA

Yang-Baxter random fields and stochastic vertex models

Bijectivization refines the Yang-Baxter equation into a pair of local Markov moves which randomly update the configuration of the vertex model. Employing this approach, we introduce new Yang-Baxter random fields of Young diagrams based on spin $q$-Whittaker and spin Hall-Littlewood symmetric functions. We match certain scalar Markovian marginals of these fields with (1) the stochastic six vertex model; (2) the stochastic higher spin six vertex model; and (3) a new vertex model with pushing which generalizes the $q$-Hahn PushTASEP introduced recently by Corwin-Matveev-Petrov (arXiv:1811.06475).…

2019-07-17cond-mat.stat-mechmath-phmath.CO

量子矩阵球函数

Quantum Matrix Spherical Functions as Orthogonal Polynomials

本文提出了一种量子矩阵球函数理论，通过将经典三元组转化为量子三元组，得到具有差反射算子代数对角化的向量值正交多项式，并将其与中间Macdonald多项式联系起来。

2026-08-28math.QAmath.RT

Genus Two Generalization of $A_1$ spherical DAHA

We consider a system of three commuting difference operators in three variables $x_{12},x_{13},x_{23}$ with two generic complex parameters $q,t$. This system and its eigenfunctions generalize the trigonometric $A_1$ Ruijsenaars-Schneider model and $A_1$ Macdonald polynomials, respectively. The principal object of study in this paper is the algebra generated by these difference operators together with operators of multiplication by $x_{ij} + x_{ij}^{-1}$. We represent the Dehn twists by outer automorphisms of this algebra and prove that these automorphisms satisfy all relations of the mapping c…

2017-04-10hep-thmath.QAmath.RT

Q-deformed Painleve tau function and q-deformed conformal blocks

We propose $q$-deformation of the Gamayun-Iorgov-Lisovyy formula for Painlevé $τ$ function. Namely we propose formula for $τ$ function for $q$-difference Painlevé equation corresponding to $A_7^{(1)}{}'$ surface (and $A_1^{(1)}$ symmetry) in Sakai's classification. In this formula $τ$ function equals the series of $q$-Virasoro Whittaker conformal blocks (equivalently Nekrasov partition functions for pure $SU(2)$ 5d theory).

2019-01-01hep-thmath-phmath.MP

Quantum affine Knizhnik-Zamolodchikov equations and quantum spherical functions, I

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…

2010-01-15math.QA

Macdonald 对偶与 Quantum Q-system 猜想的证明

Macdonald Duality and the proof of the Quantum Q-system conjecture

本文利用 Macdonald 和 Koornwinder 多项式的对偶性质，证明了在 $q$-Whittaker 极限下时间演化的 Macdonald 算子构成离散可积量子 Q-systems 的表示，从而证实了该猜想。

2023-09-15math-phmath.COmath.MP

Difference Fourier transforms for nonreduced root systems

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.

2001-11-20math.CAmath.QA