arXiv subjects

Angela Pasquale

Publications and source records attributed to Angela Pasquale.

8 recordsLinked to original sources

arXiv2022

The resonances of the Capelli operators for small split orthosymplectic dual pairs

Let $(G,G')$ be a reductive dual pair in $Sp(W)$ with rank $G\leq$ rank $G'$ and $G'$ semisimple. The image of the Casimir element of the universal enveloping algebra of $G'$ under the Weil representation $ω$ is a Capelli operator. It is a hermitian operator acting on the smooth vectors of the representation space of $ω$. We compute the resonances of a natural multiple of a translation of this operator for small split orthosymplectic dual pairs. The corresponding resonance representations turn out to be $GG'$-modules in Howe's correspondence. We determine them explicitly.

math.RT↗

arXiv2012

Ramanujan's Master theorem for the hypergeometric Fourier transform on root systems

Ramanujan's Master theorem states that, under suitable conditions, the Mellin transform of an alternating power series provides an interpolation formula for the coefficients of this series. Ramanujan applied this theorem to compute several definite integrals and power series, which explains why it is referred to as the "Master Theorem". In this paper we prove an analogue of Ramanujan's Master theorem for the hypergeometric Fourier transform on root systems. This theorem generalizes to arbitrary positive multiplicity functions the results previously proven by the same authors for the spherical Fourier transform on semisimple Riemannian symmetric spaces.

math.CA↗

arXiv2012

Ramanujan's Master Theorem for Riemannian symmetric spaces

Ramanujan's Master theorem states that, under suitable conditions, the Mellin transform of a power series provides an interpolation formula for the coefficients of this series. Based on the duality of Riemannian symmetric spaces of compact and noncompact type inside a common complexification, we prove an analogue of Ramanujan's Master Theorem for the spherical Fourier transform of a spherical Fourier series. This extend the results proven by Bertram for Riemannian symmetric spaces of rank-one.

math.RT↗

arXiv2011

On the Uniqueness of Solutions of the Schrödinger Equation on Riemannian Symmetric Spaces of the Noncompact Type

Let X be a Riemannian symmetric space of the noncompact type. We prove that the solution of the time-dependent Schrödinger equation on X with square integrable initial condition f is identically zero at all times t whenever f and the solution at a time t0 > 0 are simultaneously very rapidly decreasing. The stated condition of rapid decrease is of Beurling type. Conditions respectively of Gelfand-Shilov, Cowling-Price and Hardy type are deduced.

math.AP↗

arXiv2011

The $Cos^λ$ and $Sin^λ$ Transforms as Intertwining Operators between generalized principal series Representations of SL (n+1,K)

In this article we connect topics from convex and integral geometry with well known topics in representation theory of semisimple Lie groups by showing that the $Cos^\lamda$ and $Sin^λ$-transforms on the Grassmann manifolds $Gr_p(K)=SU (n+1,K)/S (U (p,K)\times U (n+1-p,K))$ are standard intertwining operators between certain generalized principal series representations induced from a maximal parabolic subgroup $P_p$ of $SL (n+1,K)$. The index ${}_p$ indicates the dependence of the parabolic on p. The general results of Knapp and Stein and Vogan and Wallach then show that both transforms have meromorphic extension to C and are invertible for generic $λ\in C$. Furthermore, known methods from representation theory combined with a Selberg type integral allow us to determine the K-spectrum of those operators.

math.RT↗

arXiv2005

The Paley-Wiener Theorem for the Jacobi Transform and the Local Huygens' Principle for Root Systems with Even Multiplicities

This note is a continuation of the previous paper math.AP/0411383 by the same authors. Its purpose is to extend the results of math.AP/0411383 to the context of root systems with even multiplicities. Under the even multiplicity assumption, we prove a local Paley-Wiener theorem for the Jacobi transform and the strong Huygens' principle for the wave equation associated with the modified compact Laplace operator.

math.AP↗

arXiv2004

The Paley-Wiener Theorem and the Local Huygens' Principle for Compact Symmetric Spaces

We prove a Paley-Wiener Theorem for a class of symmetric spaces of the compact type, in which all root multiplicities are even. This theorem characterizes functions of small support in terms of holomorphic extendability and exponential type of their (discrete) Fourier transforms. We also provide three independent new proofs of the strong Huygens' principle for a suitable constant shift of the wave equation on odd-dimensional spaces from our class.

math.AP↗

arXiv2003

A Paley-Wiener theorem for the $Θ$-spherical transform: the even multiplicity case

The $Θ$-spherical functions generalize the spherical functions on Riemannian symmetric spaces and the spherical functions on non-compactly causal symmetric spaces. In this article we consider the case of even multiplicity functions. We construct a differential shift operator $D_m$ with smooth coefficients which generates the $Θ$-spherical functions from finite sums of exponential functions. We then use this fact to prove a Paley-Wiener theorem for the $Θ$-spherical transfrom.

math.FA↗