§ 2.8. Functional analysis on spherical 3-manifolds.

The analysis of functions must respect the homotopy of the manifold. In our analysis we write down the bases of such functions by algebraic projection as linear combinations of Wigner $D$ -functions.

We have seen that functional analysis on the circle $S^{1}$ is done by analysis in the periodic Fourier basis. Here we ask for a basis of functional analysis on a spherical 3-manifold. We begin the construction from a basis on the 3-sphere. The 3-sphere as a manifold can be parametrized by three Euler angles $\alpha,\beta,\gamma$ . A well-known orthogonal set of functions of these is formed by Wigner's [57] $D$ -functions

$D^{j}_{{m_{1},m_{2}}}(\alpha,\beta,\gamma),\: j=0,1/2,1,...,\:-j\leq(m_{1},m_{2})\leq j.$

(18)

The Wigner $D$ -functions eq.18 form a complete orthogonal basis on the 3-sphere and moreover are harmonic, that is, vanish under the Laplacian $\Delta$ on $E^{4}$ . Now we make use of the deck transformation groups. For the two cubic manifolds this group we find a cyclic group $H=C_{8}$ for $N2$ and the quaternion group $H=Q$ for $N3$ . Both groups have $8$ elements since they must generate the 8-cell. We know the representations of these groups when acting on the Wigner polynomials. This allows us to project the linear combinations invariant under $H$ , which then form the basis set for the harmonic analysis on the manifold.

In our work [39] we project harmonic bases for the analysis on any Platonic spherical 3-manifolds with fixed homotopy as linear combinations of Wigner $D$ -functions. The essential requirement on a basis function is its invariance under the application of any deck operation. The projection is done algebraically by acting with representations of the appropriate deck operations. All projected basis functions obey on pairs of faces the boundary conditions for the chosen homotopy.