The Plancherel measure for p-forms in real hyperbolic spaces

TY - JOUR

T1 - The Plancherel measure for p-forms in real hyperbolic spaces

AU - Camporesi, Roberto

AU - Higuchi, Atsushi

PY - 1994

Y1 - 1994

N2 - 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).

AB - 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).

KW - hyperbolic space

KW - Plancherel measure

UR - http://www.scopus.com/inward/record.url?scp=0001789025&partnerID=8YFLogxK

U2 - 10.1016/0393-0440(94)90047-7

DO - 10.1016/0393-0440(94)90047-7

M3 - Article

AN - SCOPUS:0001789025

SN - 0393-0440

VL - 15

SP - 57

EP - 94

JO - Journal of Geometry and Physics

JF - Journal of Geometry and Physics

IS - 1

ER -