THE PLANCHEREL MEASURE FOR P-FORMS IN REAL HYPERBOLIC SPACES

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 omega(lambda) of the Hodge-de Rham operator DELTA = ddelta + deltad. The spectrum of DELTA is purely continuous except for N even and p = 1/2N. For N odd the Plancherel measure mu(lambda) is a polynomial in lambda2. For N even the continuous part mu(lambda) of the Plancherel measure is a meromorphic function in the complex lambda-plane with simple poles on the imaginary axis. A simple relation between the residues of mu(lambda) at these poles and the (known) degeneracies of DELTA on the N-sphere is obtained. A similar relation between mu(lambda) at discrete imaginary values of lambda and the degeneracies of DELTA on S(N) is found for N odd. The p-form zeta-function, defined as a Mellin transform of the trace of the heat kernel, is considered. A relation between the zeta-functions on S(N) and H(N) is obtained by means of complex contours. We construct square-integrable harmonic k-forms on H2k. These k-forms contribute a discrete part to the spectrum of DELTA and are related to the discrete series of SO0(2k, 1). We also give a group-theoretic derivation of mu(lambda) based on the Plancherel formula for the Lorentz group SO0(N, 1).