Noneuclidean Harmonic Analysis

Abstract

Analysis on Lie groups and their homogeneous spaces has seen a recent flowering, thanks to the work of Harish-Chandra, Helgason, Selberg and many others. The purpose of this paper is to give an elementary discussion of Fourier analysis on the noneuclidean upper half plane H; that is, the spectral resolution of the noneuclidean Laplace operator $\Delta $. This leads to a study of the elementary eigenfunctions of $\Delta $. Such eigenfunctions can be viewed as analogues of the trigonometric functions in euclidean Fourier analysis. The elementary eigenfunctions of $\Delta $ on H are Legendre (conical or spherical) functions, K-Bessel functions and Eisenstein series. The invariance of $\Delta $ under the special linear group $G = SL(2,\mathbb{R})$, of $2 \times 2$ real matrices of determinant l, gives a way of recognizing the spectral measure in the noneuclidean Fourier inversion formula, once one knows the asymptotics and functional equations of the elementary eigenfunctions of $\Delta $. In fact, this method allows a unified treatment of the inversion formulas for the standard integral transforms of mathematical physics. The connection between Fourier analysis on H and that on the fundamental domain $\Gamma \backslash H$, $\Gamma = $ a discrete subgroup of G (e.g., $\Gamma = $ the modular group $SL(2,\mathbb{Z})$ of $2 \times 2$ integer matrices of determinant 1) provides an analogue of the Poisson summation formula. Applications to the solution of the heat equation on H and $\Gamma \backslash H$ are considered, as well as the distribution of eigenvalues of the noneuclidean Laplacian on $L^2 (\Gamma \backslash H)$.