TY - JOUR

T1 - Eigenspaces of the Laplacian on hyperbolic spaces

T2 - Composition series and integral transforms

AU - Schlichtkrull, Henrik

N1 - Funding Information: * Partially supported by NSF grant DMS 80-01854 at Cornell University.

PY - 1987/1

Y1 - 1987/1

N2 - Let X be a projective real, complex, or quaternion hyperbolic space, realized as the pseudo-Riemannian symmetric space X ≅ G H with G = O(p, q), U(p, q), or Sp(p,q) (these are the classical isotropic symmetric spaces). Let Δ be the G-invariant Laplace-Beltrami operator on X. A complete description (by K-types), for each χ ∈ C, of all closed G-invariant subspaces of the eigenspace {f ∈ C∞(X)|Δf = χf} is given. The eigenspace representations are compared with principal series representations, using "Poisson-transformations". Similar results are obtained also for the exceptional isotropic symmetric space. The Langlands parameters of the spherical discrete series representations are determined.

AB - Let X be a projective real, complex, or quaternion hyperbolic space, realized as the pseudo-Riemannian symmetric space X ≅ G H with G = O(p, q), U(p, q), or Sp(p,q) (these are the classical isotropic symmetric spaces). Let Δ be the G-invariant Laplace-Beltrami operator on X. A complete description (by K-types), for each χ ∈ C, of all closed G-invariant subspaces of the eigenspace {f ∈ C∞(X)|Δf = χf} is given. The eigenspace representations are compared with principal series representations, using "Poisson-transformations". Similar results are obtained also for the exceptional isotropic symmetric space. The Langlands parameters of the spherical discrete series representations are determined.

UR - http://www.scopus.com/inward/record.url?scp=0011276711&partnerID=8YFLogxK

U2 - 10.1016/0022-1236(87)90130-3

DO - 10.1016/0022-1236(87)90130-3

M3 - Journal article

AN - SCOPUS:0011276711

VL - 70

SP - 194

EP - 219

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 1

ER -