Related papers: Harmonic analysis on certain spherical varieties
Given a connected reductive algebraic group G over a finite field together with a representation of the dual group of G in GL(n), Braverman and Kazhdan defined an exotic Fourier operator on the space of complex valued functions on the…
We study quantizations of transverse slices to Schubert varieties in the affine Grassmannian. The quantization is constructed using quantum groups called shifted Yangians --- these are subalgebras of the Yangian we introduce which…
We construct a version of Fourier transform for a class of non-commutative algebras over abelian varieties which include algebras of twisted differential operators generalizing the previous construction of Laumon (alg-geom/9603004) and of…
The study of spherical harmonics in superspace, introduced in [J. Phys. A: Math. Theor. 40 (2007) 7193-7212], is further elaborated. A detailed description of spherical harmonics of degree k is given in terms of bosonic and fermionic…
We obtain a characterisation of the Fourier transform on the space of Schwartz class functions on $\mathbb{R}^n.$ The result states that any appropriately additive bijection of the Schwartz space onto itself, which interchanges convolution…
This paper conducts a geometric analysis of the Joint-Eigenspace Fourier transform of the symmetric space of the non-compact type. Our study shows how the Poisson transform builds up the well-known Helgason Fourier transform for an analysis…
Using his formulation of the potential theoretic notion of balayage and his deep results about this idea, Beurling gave sufficient conditions for Fourier frames in terms of balayage. The analysis makes use of spectral synthesis, due to…
In this article we prove a conjecture of Braverman-Kazhdan in \cite{BK} on the acyclicity of gamma sheaves in the de Rham setting. The proof relies on the techniques developed in \cite{BFO} on Drinfeld center of Harish-Chandra bimodules and…
The aim of this article is to give an overview of several types of Paley-Wiener theorems occuring in harmonic analysis related to symmetric spaces. This will serve as a motivation for the introduction of the $\Theta$-spherical functions,…
In this paper we make an attempt to extend L. Schwartz's classical result on spectral synthesis to several dimensions. Due to counterexamples of D. I. Gurevich this is impossible for translation invariant varieties. Our idea is to replace…
We consider the horospherical transform and its inversion in 3 examples of hyperboloids. We want to illustrate via these examples the fact that the horospherical inversion formulas can be directly extracted from the classical Radon…
This paper contains a non-trivial generalization of the Harish-Chandra transforms on a connected semisimple Lie group $G,$ with finite center, into what we term spherical convolutions. Among other results we show that its integral over the…
In this paper extensions of the classical Fourier, fractional Fourier and Radon transforms to superspace are studied. Previously, a Fourier transform in superspace was already studied, but with a different kernel. In this work, the…
Building upon Dyson's fundamental 1962 article known in random-matrix theory as 'the threefold way', we classify disordered fermion systems with quadratic Hamiltonians by their unitary and antiunitary symmetries. Important examples are…
We show that the Halphen transform of a Lam\'e equation can be written as the symmetric square of the Lam\'e equation followed by an Euler transform. We use this to compute a list of Lam\'e equations with arithmetic Fuchsian monodromy…
We present an approach to sums of random Hermitian matrices via the theory of spherical functions for the Gelfand pair $(\mathrm{U}(n) \ltimes \mathrm{Herm}(n), \mathrm{U}(n))$. It is inspired by a similar approach of Kieburg and K\"osters…
J.C.Lagarias (2000) conjectured that if $\mu$ is a complex measure on p-dimensional Euclidean space with a uniformly discrete support and its spectrum (Fourier transform) is also a measure with a uniformly discrete support, then the support…
We describe two new combinatorial algorithms (using the language of "triangular arrays") for computing the Fourier transforms of simple perverse sheaves on the moduli space of representations of an equioriented quiver of type A. (A rather…
Let $G$ be a connected reductive group over $\overline{\mathbb{F}}_q$ and let $\rho^\vee:G^\vee\rightarrow GL_n$ be an algebraic representation of the dual group $G^\vee$. Assuming that $G$ and $\rho^\vee$ are defined over $\mathbb{F}_q$,…
The unified transform method introduced by Fokas can be used to analyze initial-boundary value problems for integrable evolution equations. The method involves several steps, including the definition of spectral functions via nonlinear…