One-dimensional double Hecke algebras and Gaussians
We use one-dimensional double affine Hecke algebras to introduce q-counterparts of the Gauss integrals and new types of Gauss-Selberg sums at roots of unity.
math.QA↗
arXiv subjects
Ivan Cherednik
Publications and source records attributed to Ivan Cherednik.
On q-analogues of Riemann's zeta
In the paper, we introduce $q$-deformations of the Riemann zeta function, extend them to the whole complex plane, and establish certain estimates of the number of roots. The construction is based on the recent difference generalization of the Harish-Chandra theory of zonal spherical functions. We also discuss numerical results, which indicate that the location of the zeros of the $q$-zeta functions is far from random.
math.QA↗
From Double Hecke algebra to analysis
We discuss q-counterparts of the Gauss integrals, a new type of Gauss-Selberg sums at roots of unity, and q-deformations of Riemann's zeta. The paper contains general results, one-dimensional formulas, and remarks about the current projects involving the double affine Hecke algebras.
math.QA↗
Calculating zeros of a q-zeta function numerically
The note is a continuation of the previous paper ``On q-analogues of Riemann's zeta'' (math.QA/980499). It contains an output of the computer program calculating the zeros of the ``sharp'' q-zeta function.
math.QA↗
Inverse Harish-Chandra Transform and Difference Operators
We apply a new technique based on double affine Hecke algebras to the Harish-Chandra theory of spherical zonal functions. The formulas for the Fourier transforms of the multiplications by the coordinates are obtained as well as a simple proof of the Harish-Chandra inversion theorem using the Opdam transform.
q-alg↗
Difference Macdonald-Mehta Conjecture
In the paper we formulate and verify a difference counterpart of the Macdonald-Mehta conjecture and its generalization for the Macdonald polynomials. Namely, we determine the Fourier transforms of the polynomials multiplied by the Gaussian, which is closely connected with the new difference Harish-Chandra theory.
q-alg↗
Intertwining Operators of Double Affine Hecke Algebras
A systematic study of the representation theory of double affine Hecke algebras and related harmonic analysis is started in this paper. Continuing the previous papers we use the technique of intertwining operators to create Macdonald polynomials, estimate their denominators, generalize the classical representations of p-adic affine Hecke algebras in the spaces of functions on affine Weyl groups, and to find out when induced representations are irreducible and co-spherical. The connection with recent results by Sahi and Knop on the integrality of the Macdonald polynomials is established.
q-alg↗
Kadell's two conjectures for Macdonald polynomials
This paper contains the proof of difference counterparts of the conjectures due to Keven Kadell on symmetric and anti-symmetric Macdonald polynomials.
q-alg↗
Macdonald's Evaluation Conjectures, Difference Fourier Transform, and Applications
This paper contains the proof of Macdonald's duality and evaluation conjectures, the definition of the difference Fourier transform, the recurrence theorem generalizing Pieri rules, and the action of GL(2,Z) on the Macdonald polynomials at roots of unity.
q-alg↗
Non-Symmetric Macdonald's Polynomials
We extend the previous paper "Macdonald's evaluation ... and applications" to the non-symmetric polynomilas recently introduced by Macdonald (as difference counterparts of Opdam's non-symmetric ones).
q-alg↗
Macdonald's Evaluation Conjectures and Difference Fourier Transform
In the previous author's paper the Macdonald norm conjecture (including the famous constant term conjecture) was proved. This paper contains the proof of the remaining two (the duality and evaluation conjectures). The evaluation theorem is in fact a $q,t$-generalization of the classic Weyl dimension formula. As to the duality theorem, it states that the generalized trigonometric-difference zonal Fourier transform is self-dual (at least formally). We define this transform in terms of double affine Hecke algebras related to elliptic braid groups. The duality appeared to be directly connected with the transposition of the periods of an elliptic curve.
q-alg↗
Difference-elliptic operators and root systems
Recently a new technique in the harmonic analysis on symmetric spaces was suggested based on certain remarkable representations of affine and double affine Hecke algebras in terms of Dunkl and Demazure operators instead of Lie groups and Lie algebras. In the classic case it resulted (among other applications) in a new theory of radial part of Laplace operators and their deformations including a related concept of the Fourier transform. In the present paper we demonstrate that the new technique works well even in the most general difference-elliptic case conjecturally corresponding to the $q$-Kac-Moody algebras. We discuss here only the construction of the generalized radial (zonal) Laplace operators and connect them with the difference-elliptic Ruijsenaars operators generalizing in its turn the Olshanetsky-Perelomov differential elliptic operators.
hep-th↗
Elliptic quantum many-body problem and double affine Knizhnik-Zamolodchikov equation
The elliptic-matrix quantum Olshanetsky-Perelomov problem is introduced for arbitrary root systems by means of an elliptic generalization of the Dunkl operators. Its equivalence with the double affine generalization of the Knizhnik-Zamolodchikov equation (in the induced representations) is established.
hep-th↗