#
Peter Kramer

## Harmonic polynomials on Poincare's dodecahedral 3-manifold

Poincare's 3-manifold arises from a spherical dodecahedron by roation/gluing of opposite faces. Its universal cover is the sphere S³ with symmetry group SO(4,R). The fundamental group and the group of deck transformations are isomorphic to the binary icosahedral group (cal H)_3. We study the subduction of
irreps for the group/subgroup pair SO(4,R)> (cal H)_3.
>From an (cal H)_3-invariant polynomial f(z_1,z_2),
analytic in two complex variables and due to Felix Klein,
a generalized Casimir operator K is constructed. K is an hermitian, symmetrized and (cal H)_3-invariant operator-valued
polynomial in the enveloping Lie algebra of SO(4,R). The degeneracies in the spectrum of K are resolved by two commuting operators. The eigenstates and eigenvalues of K characterize completely the subduction. The selected eigenstates belonging to the identity irrep of (cal H)_3
respect the rotation/gluing conditions of the Poincare 3-manifold. They form a complete orthogonal basis of dodecahedral harmonic polynomials. Klein's invariant polynomial becomes part of a degenerate set of 13 eigenstates of lowest degree 12.
If the Poincare 3-manifold is taken to model the space part of the cosmos, the observed temperature fluctuations in the cosmic microwave background must admit an expansion in harmonic polynomials characterized by K.

Preprint 3 Feb 2005