Macdonald operators at infinity

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→∞ of renormalized Macdonald operators acting on symmetric polynomials in the variables x ₁,…,x _N . They are differential operators in terms of the power sum variables \(p_{n}=x_{1}^{n}+x_{2}^{n}+\cdots\) 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 variables x ₁,x ₂,…. Our result also yields elementary step operators for the Macdonald symmetric functions.     

Matrix elements of vertex operators of the deformed W A n -algebra and the Harish-Chandra solutions to Macdonald's difference equations

In this paper we prove that certain matrix elements of vertex operators of the deformed W A _n -algebra satisfy Macdonald's difference equations and form a natural (n + 1)!-dimensional space of solutions. These solutions are the analogues of the Harish-Chandra solutions of the radial parts of the Laplace-Casimir operators on noncompact Riemannian symmetric spaces G/K with prescribed asymptotic behavior. We obtain formulas for analytic continuation of our Harish-Chandra type solutions as a consequence of braiding properties (obtained earlier by Y. Asai, M. Jimbo, T. Miwa, and Y. Pugay) of certain vertex operators of the deformed W A _n -algebra.     

The correlation functions of vertex operators and Macdonald polynomials

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.     

Nonsymmetric Macdonald polynomials and matrix coefficients for unramified principal series

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.     

Gaussian series and daubechies operators

The subspace [Mtilde] of L²(Cⁿ) which is composed of Gaussian series and contains the subspace M spanned by Gaussian functions given in the paper [6] by Du and Wong has the proporety that the product of two Daubechies operators with symbols in [Mtilde] is a Daubechies operator with symbol H in [Mtilde]. Furthermore, an explicit expression for the symbol H is given     

On multidimensional difference operators and equations

We study the solvability of multidimensional difference equations in Sobolev–Slobodetskii spaces. In the simplest model case, we describe the solvability picture for such equations. In the general case, we present conditions for the Fredholm property and a theorem on the zero index.     

Integrable discrete-time systems and difference operators

M. V. Lomonosov Moscow State University. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 22, No. 2, pp. 1–13, April–June, 1988.     

Macdonald positivity via the Harish-Chandra <Emphasis Type="Italic">D</Emphasis>-module

Using the Harish-Chandra D-module, we give a proof of Haiman’s theorem on the positivity of Macdonald polynomials. Ginzburg’s work on the connection between this D-module and the isospectral commuting variety is fundamental to this approach.     

Stabilization of optimal difference operators

Zusammenfassung Dahlquist hat in [2] gezeigt, dass stabile Differenzenoperatoren mit dem Gradk höchstens die Ordnungp=k+2 erreichen können. Solche Operatoren zeigen die Erscheinung der schwachen Stabilität (marginal stability) [1,2,3].In der vorliegenden Arbeit werden stabile. Differenzenoperatoren gezeigt, die trotz maximaler Ordnung nicht mehr schwach stabil sind. Das Auftreten von schwacher Stabilität kann vermieden werden, wenn man die Koeffizienten des Operators von der Schrittweiteh des entsprechenden Verfahrens und von einem geeignet zu wählenden Parameter abhängen lässt.