The Plancherel measure for p-forms in real hyperbolic spaces

Abstract

Abstract The Plancherel measure is calculated for antisymmetric tensor fields ( p -forms) on the real hyperbolic space H N . The Plancherel measure gives the spectral distribution of the eigenvalues ω λ of the Hodge-de Rham operator Δ = dδ + δd . The spectrum of Δ is purely continuous except for N even and p= 1 2 N . For N odd the Plancherel measure μ(λ) is a polynomial in λ 2 . For N even the continuous part μ(λ) of the Plancherel measure is a meromorphic function in the complex λ-plane with simple poles on the imaginary axis. A simple relation between the residues of μ(λ) at these poles and the (known) degeneracies of Δ on the N -sphere is obtained. A similar relation between μ(λ) at discrete imaginary values of λ and the degeneracies of Δ on S N is found for N odd. The p -form ζ-function, defined as a Mellin transform of the trace of the heat kernel, is considered. A relation between the ζ-functions on S N and H N is obtained by means of complex contours. We construct square-integrable harmonic k -forms on H 2k . These k -forms contribute a discrete part to the spectrum of Δ and are related to the discrete series of SO 0 (2 k , 1). We also give a group-theoretic derivation of μ(λ) based on the Plancherel formula for the Lorentz group SO 0 ( N , 1).