I study harmonic analysis on Lie Groups and on their homogeneous spaces. In particular, I am interested in analysis on a Gelfand pair $(G, K)$ which consists of a Lie group $G$ and a compact subgroup $K$ of $G$, and on the homogeneous space $G/K$ corresponding to $(G, K)$. For example, the sphere $S^2$ and the plane ${\mathbb{R}}^2$ are realized as the homogeneous spaces $G/K$ corresponding to Gelfand pairs $(G, K)$ $=({\rm{SO}}(3)$, ${\rm{SO}}(2))$ and $({\rm{SO}}(2)\ltimes {\mathbb{R}}^2,$ ${\rm{SO}}(2))$, respectively. A Gelfand pair $(G, K)$ is an object which has a sort of commutativity even if $G$ is not commutative. The main themes of the study of mine are to construct spherical representations and to calculate spherical functions for a Gelfand pair $(G, K)$. They are very important when we study harmonic analysis on the homogeneous space $G/K$. Moreover, to compute spherical functions concretely, I study invariant polynomials and invariant differential operators for a multiplicity-free action $(K, V)$ of a compact Lie group $K$ on a vector space $V$ over the complex number field ${\mathbb{C}}$.
I am interested in harmonic analysis on a Gelfand pair $(G, K)$ of a finite group $G$ and its subgroup $K$, and in applications to analysis on graphs, too.