Fourier Analysis on Matrix Space

This paper treats two related problems of harmonic analysis on [italic]n x [italic]n real matrix space [italic]M([italic]n,ℝ). The first concerns the explicit decomposition of the additive Fourier operator with respect to the group representation theory of [italic]G = [italic]GL([italic]n,ℝ). The second concerns the analytic continuation and functional equation of certain zeta-functions defined on [italic]G. The solution of both these problems involves the generalized gamma functions of Gelfand and Graev.