一般線形GL(k;R)の表現と球関数

In 1961, A. T. James [1] introduced the zonal polynomial of real positive definite matrix and he described some properties and a method of calculation of it. Lately, he [2] also showed that it is an eigenfunction of the Laplace-Beltrami operator. The zonal polynomial plays significant roles in distribution problems of eigenvalues related to the normal multivariate distribution [3]. In the present paper, we will describe a representation of the group GL (k ; R) and its spherical functions guided by N. J. Vilenkin [4]. Our assumptions 1 and 2 in the following may be satisfied with zonal polynomials. We also give a definition of (zonal) spherical function of the group GL (k ; R) guided by K. Maurin [5] and we show that our zonal spherical function is in agreement with the latter definition.