Non-Abelian Fourier Analysis on $\mathbf{\Gamma}\backslash SE(d)$

Abstract

This paper presents a systematic study for the general theory of non-Abelian Fourier series of integrable functions on the homogeneous space $\mathbf{\Gamma}\backslash SE(d)$, where $SE(d)$ is the special Euclidean group in dimension $d$, and $\mathbf{\mathbf{\Gamma}}$ is a discrete and co-compact subgroup of $SE(d)$. Suppose that $\mu$ is the finite $SE(d)$-invariant measure on the right coset space $\mathbf{\Gamma}\backslash SE(d)$, normalized with respect to Weil's formula. The analytic aspects of the proposed method works for any given orthonormal basis of the Hilbert function space $L^2(\mathbf{\Gamma}\backslash SE(d),\mu)$. The paper is concluded with some convolution and Plancherel formulas.