Nobukazu Shimeno

• Doctor of Mathematical Sciences
• Professor at Kwansei Gakuin University

We give an extension of Pizzetti’s formula associated with the Dunkl operators. It gives an explicit formula for the Dunkl inner product of an arbitrary function and a homogeneous Dunkl harmonic polynomial on the unit sphere.

We give the inversion formula and the Plancherel formula for the hypergeometric Fourier transform associated with a root system of type $BC$, when the multiplicity parameters are not necessarily nonnegative.

We give an extension of Pizzetti's formula associated with the Dunkl operators. It gives an explicit formula for the Dunkl inner product of an arbitrary function and a homogeneous Dunkl harmonic polynomial on the unit sphere.

For a connected semisimple real Lie group G of non-compact type, Wallach introduced a class of K-types called small. We classify all small K-types for all simple Lie groups and prove except just one case that each elementary spherical function for each small K-type \((\pi ,V)\) can be expressed as a product of hyperbolic cosines and a Heckman–Opdam...

We give an explicit formula for the Harish-Chandra $c$-function for a small $K$-type on a noncompact real split Lie group of type $G_2$. As an application we give an explicit formula for spherical inversion for a small $K$-type.

We generalize classical Hobson's formula concerning partial derivatives of radial functions on a Euclidean space to a formula in the Dunkl analysis. As applications we give new simple proofs of known results involving Maxwell's representation of harmonic polynomials, Bochner–Hecke identity, Pizzetti formula for spherical mean, and Rodrigues formula...

We generalize classical Hobson's formula concerning partial derivatives of radial functions on a Euclidean space to a formula in the Dunkl analysis. As applications we give new simple proofs of known results involving Maxwell's representation of harmonic polynomials, Bochner-Hecke identity, Pizzetti formula for spherical mean, and Rodrigues formula...

We express explicitly the Heckman-Opdam hypergeometric function for the root system of type A with a certain degenerate parameter in terms of the Lauricella hypergeometric function.

For a connected semisimple real Lie group $G$ of non-compact type, Wallach introduced a class of $K$-types called small. We classify all small $K$-types for all simple Lie groups and prove except just one case that each elementary spherical function for each small $K$-type $(\pi,V)$ can be expressed as a product of hyperbolic cosines and a Heckman-...

We study the algebra of invariant differential operators on a certain homogeneous vector bundle over a Riemannian symmetric space of type $A_2$. We computed radial parts of its generators explicitly to obtain matrix-valued commuting differential operators with $A_2$ symmetry. Moreover, we generalize the commuting differential operators with respect...

We study the algebra of invariant differential operators on a certain homogeneous vector bundle over a Riemannian symmetric space of type \(A_2\). We computed radial parts of its generators explicitly to obtain matrix-valued commuting differential operators with \(A_2\) symmetry. Moreover, we generalize the commuting differential operators with res...

We shall investigate two uncertainty principles for the Cherednik transform on the Euclidean space $\mathfrak a$ ; Miyachi’s theorem and Beurling’s theorem. We give an analogue of Miyachi’ theorem for the Cherednik transform and under the assumption that $\mathfrak a$ has a hypergroup structure, an analogue of Beurling’s theorem for the Cheredn...

We characterize the image of the Poisson transform on any distinguished boundary of a Riemannian symmetric space of the noncompact type by a system of differential equations. The system corresponds to a generator system of a two sided ideals of an universal enveloping algebra, which are explicitly given by analogues of minimal polynomials of matric...

We prove that the radial part of the class one Whittaker function on a split semisimple Lie group can be obtained as an appropriate limit of the Heckman-Opdam hypergeometric function.

Formulae of Berezin and Karpelevic for the radial parts of invariant differential operators and the spherical function on a complex Grassmann manifold are generalized to the hypergeometric functions associated with root system of type $BC_n$ under condition that the multiplicity of the middle roots is zero or one.

We discuss three topics, confluence, restrictions, and real forms for the Heckman-Opdam hypergeometric functions.

We prove a Fatou-type theorem on a Riemannian symmetric space of the noncompact type and characterize the image of the Poisson transform of Lp -functions on the maximal boundary as a Hardy-type space.

A theorem of Hardy asserts that a function and its Fourier transform cannot both be very small. We prove analogues of Hardy’s theorem for the Harish- Chandra transform for spherical functions on a non-compact semisimple Lie group and the Helgason transform on a Riemannian symmetric space of the non-compact type.

A theorem of Hardy asserts that a function on the real line and its Fourier transform cannot both be very small. We generalize Hardy’s theorem for the Heckman-Opdam transform associated with hypergeometric functions.

A Note on the Uncertainty Principle for the Dunkl Transform

The image of the Poisson transform on a principal series representations on a boundary component of a Hermitian symmetric space is considered. We prove that the image is characterized by a covariant differential operator on a homogeneous line bundle on the symmetric space.

In this article the images of the Poisson transform on the degenerate series representations attached to the Shilov boundary of a bounded symmetric domain of tube type are considered. We characterize the images by means of second-order differential equations.

A realization of a $\ep$-family of semisimple symmetric spaces $\{G/H_\ep\}$ in a compact real analytic manifold $\X$ is constructed. The realization $\X$ has the following properties: a) The action of $G$ on $\X$ is real analytic; b) There exist open $G$-orbits that are isomorphic to $G/H_\ep$ for each signature of roots $\ep$; c) The system $\Cal...

We generalize the Harish-Chandra inversion formula for the spherical transform on a Riemannian symmetric space to homogeneous line bundles on a Hermitian symmetric space. We determine the Plancherel measure explicitly.