Radon transform on real, complex, and quaternionic Grassmannians

Abstract

Let $G_{n,k}(\mathbb{K})$ be the Grassmannian manifold of $k$-dimensional $\mathbb{K}$-subspaces in ${\mathbb{K}}^n$, where $\mathbb{K}=\mathbb{R},\mathbb{C},\mathbb{H}$ is the field of real, complex, or quaternionic numbers. For $1\le k<k^{\text{'}}\le n-1$, we define the Radon transform $(Rf)(\eta )$, $\eta \in G_{n,k^{\text{'}}}(\mathbb{K})$, for functions $f(\xi )$ on $G_{n,k}(\mathbb{K})$ as an integration over all $\xi \subset \eta$. When $k+k^{\text{'}}\le n$, we give an inversion formula in terms of the Gårding-Gindikin fractional integration and the Cayley-type differential operator on the symmetric cone of positive ($k\times k$)-matrices over $\mathbb{K}$. This generalizes the recent results of Grinberg and Rubin [4] for real Grassmannians