On the Characteristic Function of the Von Mises-fisher Matrix Distribution

Characteristic functions provide a useful method for analyzing probability distributions. In most instances, the domain is taken to be euclidean space and although the integral transforms may not have a simple expression, some qualitative features about the underlying probability distribution can often be extracted. Although there is some research eeorts in spaces more complicated then the euclidean case, very little is available on characteristic functions for probability distributions on these general domains. In this paper we will calculate the characteristic function of a probability distribution on a more general domain, in particular, the SO(N)?version of the von Mises-Fisher matrix distribution. It is found that when Fourier transforms of the latter are taken with respect to the irreducible representations of SO(N), a concrete expression for the characteristic function is obtained.