Abstract
Section 1 of the paper gives three applications of a basic principle that eigenfunctions of invariant differential operators are eigenfunetions of invariant integral operators. The first application is a derivation of a noneuclidean analogue of the Poisson summation formula, the second is the evaluation of an integral of Muirhead which has been of interest in multivariate statistics, the third is the evaluation of gammaa type integrals arising in the theory of Eisenstein series for the general linear group. Section 2 of the paper concerns K-Bessel functions for the general linear group. It relates such functions with gamma functions for the symplectic group and applies the theory to the estimation of Fourier coefficients of automorphic forms for ${\operatorname{GL}}(3,\mathbb{Z})$.
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