Analysis on positive matrices as it might have occurred to Fourier

Abstract

Analysis on matrix groups and their homogeneous spaces is in a period analogous to that of Fourier, thanks to work of Harish-Chandra, Helgason, Langlands, Maass, Selberg, and many others. Here we try to give a simple disccussion of Fourier analysis on the space Pn of positive n×n matrices, as well as on the Minkowski fundamental domain for Pn modulo the discrete group GL(n, Z) of integer matrice of determinant ±1. The main idea is to use the group invariance to see that the Plancherel or spectral measure in the Mellin inversion formula comes from the asymptotics and functional equations of the special functions which appear in the Mellin transform on Pn or Pn/GL(n, Z) as analogues of the power function ys in the ordinary Mellin transform. For Pn, these functions are matrix argument generalizations of K-Bessel and spherical functions. For Pn/GL(N,Z). these special functions are generalizations of Epstein zeta functions known as Eisenstein series.KeywordsZeta FunctionModular FormFourier ExpansionEisenstein SeriesCusp FormThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.