104 research outputs found
Harmonic analysis for real spherical spaces
We give an introduction to basic harmonic analysis and representation theory
for homogeneous spaces attached to a real reductive Lie group . A
special emphasis is made to the case where is real spherical.Comment: Shortened title, typos fixed, more details on dual smooth Frobenius
reciprocity (now Lemma 6.6). 38 pages, lecture notes for the Sanya meeting on
spherical varieties. To appear in Acta Math. Sinic
A local Paley-Wiener theorem for compact symmetric spaces
The Fourier coefficients of a smooth -invariant function on a compact
symmetric space are given by integration of the function against the
spherical functions. For functions with support in a neighborhood of the
origin, we describe the size of the support by means of the exponential type of
a holomorphic extension of the Fourier coefficient
Holomorphic Extension of Eigenfunctions
Let X be a Riemannian symmetric space of non-compact type. We prove a theorem
of holomorphic extension for eigenfunctions of the Laplace-Beltrami operator on
X, by techniques from the theory of partial differential equations.Comment: 8p, no figure
Representation theory, Radon transform and the heat equation on a Riemannian symmetric space
Let X=G/K be a Riemannian symmetric space of the noncompact type. We give a
short exposition of the representation theory related to X, and discuss its
holomorphic extension to the complex crown, a G-invariant subdomain in the
complexified symmetric space X_\C=G_\C/K_\C. Applications to the heat transform
and the Radon transform for X are given
Eigenspaces of the Laplacian on hyperbolic spaces: Composition series and integral transforms
AbstractLet X be a projective real, complex, or quaternion hyperbolic space, realized as the pseudo-Riemannian symmetric space X ≅ GH 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
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