Non-Abelian Fourier Analysis on Γ\ SE(d)

This paper presents a systematic study for the general theory of non-Abelian Fourier series of integrable functions on the homogeneous space Γ\ SE(d), where SE(d) is the special Euclidean group in dimension d, and Γ is a discrete and co-compact subgroup of SE(d). Suppose that μ is the finite SE(d)-invariant measure on the right coset space Γ\ 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(Γ\ SE(d),μ). The paper is concluded with some convolution and Plancherel formulas.