Major research activity
My current research topics include (1) geometry of symmetric spaces, (2) harmonic analysis on symmetric spaces, and (3) hypergeometric functions associated with root systems. All of these topics are closely related with symmetry of geometric objects. For example, in plane geometry, isometries consist of reflections, rotations, translations and compositions of these operations. These operations form the motion group of the Euclidean plane. More generally, there is a notion of Lie groups, which represent the theory of continuous symmetry of geometric objects and structures such as homogeneous spaces and symmetric spaces. These spaces include the Poincare model of non-Euclidean geometry. Infinitesimal actions of Lie groups give notion of Lie algebras, which are vector fields on spaces with group actions. On a space with a group action, there are inherited group action on functions, differential equations, and solutions. For example, the Laplace operator on the Euclidean plane is invariant under the action of the Euclidean motion group. Moreover, the space of the radial eigenfunctions of the Laplace operator is invariant under the action of the rotation group. On symmetric spaces, theory of differential equations, special solutions, and related harmonic analysis (or geometric Fourier analysis) have been developed. From an algebraic point of view, representation theory of Lie groups are closely related with our research.
Major relevant publications
- T. Oshima and N. Shimeno, Boundary value problems on Riemannian Symmetric Spaces of the noncompact Type, to appear.
- R DAHER , S L HAMAD, T KAWAZOE, N SHIMENO, Uncertainty principles for the Cherednik transform, Proceedings - Mathematical Sciences, Volume 122, 429-436, (2012)
- T. Oshima and N. Shimeno, Heckman-Opdam hypergeometric functions and their specializations, RIMS Kokyuroku Bessatsu, B20, Research Institue of Mathematics Kyoto University, 2010.
- N. Shimeno, Formula for the hypergeometric functions of type BCn, Pacific J. of Math. Vol. 236, 105-118, (2008)
- N. Shimeno, Boundary Value Problems for Various Boundaries of Hermitian Symmetric Spaces, Journal of Functional Analysis Volume 170, 265–285, (2000)