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The Plancherel measure for p-forms in real hyperbolic spaces

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Publication details

JournalJournal of Geometry and Physics
DatePublished - 1994
Issue number1
Number of pages38
Pages (from-to)57-94
Original languageEnglish


The Plancherel measure is calculated for antisymmetric tensor fields (p-forms) on the real hyperbolic space HN. The Plancherel measure gives the spectral distribution of the eigenvalues ωλ of the Hodge-de Rham operator Δ=dδ+δd. The spectrum of Δ is purely continuous except for N even and p= 1 2N. For N odd the Plancherel measure μ(λ) is a polynomial in λ2. 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 SN 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 SN and HN 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 Δ and are related to the discrete series of SO0(2k, 1). We also give a group-theoretic derivation of μ(λ) based on the Plancherel formula for the Lorentz group SO0(N, 1).

    Research areas

  • hyperbolic space, Plancherel measure

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