pp-forms on dd-spherical tessellations

The spectral properties of pp-forms on the fundamental domains of regular tessellations of the dd-dimensional sphere are discussed. The degeneracies for all ranks, pp, are organised into a double Poincaré series which is explicitly determined. In the particular case of coexact forms of rank (d−1)/2(d−1)/2, for odd dd, it is shown that the heat-kernel expansion terminates with the constant term, which equals (−1)p+1/2(−1)p+1/2, and that the boundary terms also vanish, all as expected. As an example of the double-domain construction, it is shown that the degeneracies on the sphere are given by adding the absolute and relative degeneracies on the hemisphere, again as anticipated. The eta invariant on S3/ΓS3/Γ is computed to be irrational.The spectral counting function, N(λ)N(λ), is calculated and the accumulated degeneracy given exactly. The Weyl conjecture is shown not to be valid for the exact N(λ)N(λ) but does hold for an averaged form, N¯(λ). A generalised Weyl–Polya conjecture for pp-forms is suggested and verified and the exact Polya conjecture is tested numerically on the hemisphere.