THE ANNIHILATING IDEAL OF THE FISHER INTEGRAL

Abstract

The Fisher integral is the normalizing constant of a statistical model on the special orthogonal group. In this paper, we discuss a system of differential equations for the Fisher integral. Especially, we explicitly give a set of linear differential operators which generates the annihilating ideal of the Fisher integral, and we show that the annihilating ideal is a maximal left ideal of the ring of differential operators with polynomial coefficients. The Fisher integral for diagonal matrices is related to the hypergeometric function of matrix arguments. We also give a new approach to get differential operators annihilating the Fisher integral for the diagonal matrix.