Radon transform on symmetric matrix domains

Abstract

Let $\bbK=\mathbb R, \mathbb C, \mathbb H$ be the field of real, complex or quaternionic numbers and $M_{p, q}(\bbK)$ the vector space of all $p\times q$-matrices. Let $X$ be the matrix unit ball in $M_{n-r, r}(\bbK)$ consisting of contractive matrices. As a symmetric space, $X=G/K=O(n-r,r)/O(n-r)\times O(r)$, $U(n-r,r)/U(n-r)\times U(r)$ and respectively $Sp(n-r,r)/Sp(n-r)\times Sp(r)$. The matrix unit ball $y_0$ in $M_{r^{\prime }-r,r}$ with $r^{\prime }\le n-1$ is a totally geodesic submanifold of $X$ and let $Y$ be the set of all $G$-translations of the submanifold $y_0$. The set $Y$ is then a manifold and an affine symmetric space. We consider the Radon transform $\mathcal{R}f(y)$ for functions $f\in C_0^{\infty }(X)$ defined by integration of $f$ over the subset $y$, and the dual transform ${\mathcal{R}}^{t}F(x),x\in X$ for functions $F(y)$ on $Y$. We find inversion formulas by constructing explicit certain invariant differential operators.

Trans. Amer. Math. Soc., to appear

References in corpus (1)

• The Radon transform of functions of matrix argument