On automorphicL-functions for the Jacobi group of degree one and a relation withL-functions for Jacobi forms

We try to attach anL-function to an automorphic representations of the Jacobi group by defining local factors via certain zeta-integrals. We come up with two kinds of factors which are compared to factors appearing in theL-functions associated to Jacobi forms (of index 1).     

Uniqueness and existence of Whittaker models for the metaplictic group

We introduce the notion of Whittaker models for representations of a metaplectic covering group of GL (2) and establish the uniqueness and existence of such models. Our results generalize corresponding results of Jacquet-Langlands, but the methods are new. Sloan Foundation Fellow     

On the flatness of local models for the symplectic group

We investigate the bad reduction of certain Shimura varieties (associated to the symplectic group). More precisely, we look at a model of the Shimura variety at a prime p, with parahoric level structure at p. We show that this model is flat, as conjectured by Rapoport and Zink (Ann. of Math. Stud. 141 (1996)), and that its special fibre is reduced.A crucial ingredient is Faltings’ theorem on the normality of Schubert varieties in the affine flag variety.     

On Whittaker models and the vanishing of Fourier coefficients of cusp forms

The purpose of this paper is to construct examples of automorphic cuspidal representations which possess a ψ-Whittaker model even though their ψ-Fourier coefficients vanish identically. This phenomenon was known to be impossible for the groupGL(n), but in general remained an open problem. Our examples concern the metaplectic group and rely heavily upon J L Waldspurger’s earlier analysis of cusp forms on this group. This research was partially supported by Grant No. 8400139 from the United States-Israel Bi National Science Foundation (BSF), Jerusalem, Israel.     

On the local Picard group

In his book SGA2, A. Grothendieck developed Lefschetz theorems for the Picard group, the aim being to compare the Picard group of a projective variety with the one of a hyperplane section. An intermediate object is the Picard group of the formal completion along the hyperplane section. Here we proceed similarly but in the local complex analytic context. The use of the exponential sequence leads to analytic as well as topological depth conditions.     

On the Jacobi series

Summary The Jacobi series of a functionf is an expansion in a series of ascending powers of a prescribed polynomialP of degreen in which the coefficients are polynomials of lesser degree. These coefficients are usually expressed as contour integrals or are determined by their interpolatory properties. We show how they may be expressed as generalized derivatives off with respect toP. In so doing we also show how the Jacobi series may be expressed (in yet another way) as a generalized Taylor series. In addition, we obtain a number of interesting relations among the generalized derivatives.Dedicated to Professor Otto Haupt with best wishes on his 100th birthday.     

On the Erdelyi-Magnus-Nevai Conjecture for Jacobi Polynomials

T. Erdelyi, A.P. Magnus and P. Nevai conjectured that for the orthonormal Jacobi polynomials satisfy the inequality

On the local fluctuations of last-passage percolation models

Using the fact that the Airy process describes the limiting fluctuations of the Hammersley last-passage percolation model, we prove that it behaves locally like a Brownian motion. Our method is quite straightforward, and it is based on a certain monotonicity and good control over the equilibrium measures of the Hammersley model (local comparison).     

On one class of dynamic transportation models

A model is studied that describes the process of good transportation occurring in some technologies. Transportation regimes satisfying a given management system are examined. Such regimes are described by traveling-wave solutions to a nonlinear finite-difference analogue of a parabolic equation. Possible transportation regimes are described, and the stability of stationary regimes is analyzed.     

Complete signal processing bases and the Jacobi group

The continuous windowed Fourier and wavelet transforms are created from the actions of the Heisenberg and affine groups, respectively. Both wavelet and windowed Fourier bases are known to be complete; that is, the only signal which is orthogonal to every element of each basis is the zero signal. The Jacobi group is a group which contains both the Heisenberg and affine groups, and it can also be used to produce bases for signal processing. This paper investigates completeness for bases of one and two real variables which are produced by the Jacobi group.