Spectral correspondences for locally symmetric spaces - the case of exceptional parameters / von Christian Arends, unter der Betreuung durch Prof. Dr. Joachim Hilgert. Paderborn, 2023
Content
Introduction
Ruelle resonances
Reducible principal series
Realizations of principal series representations
Principal series and the first band
Globalizations and infinitesimal character
Reducibility
Poisson transforms
Invariant differential operators and eigensections
Mapping properties of scalar Poisson transforms
Vector-valued Poisson transforms
An example: The case of surfaces
The quantum-classical correspondence
Proof of the quantum-classical correspondence
Vector-valued Poisson transforms
The role of generalized gradients
Mapping properties of vector-valued Poisson transforms
Injectivity of vector-valued Poisson transforms
Minimal K-types and proofs of injectivity results
The role of generalized gradients
Želobenko operators
Gamma-invariant elements
Fourier characterization
Generalized Fourier series
Convergence of generalized Fourier series
Tensor product decompositions
Computations for the Fourier characterization
Properties of generalized gradients
Spectral correspondence
The case of G=SO(n,1), n>=3
The case of G=SU(n,1), n>=2
The case of G=Sp(n,1), n>=2
The case of G=F4(-20)
An example: The real hyperbolic case
Generalized gradients
Fourier characterization
Explicit formulas for generalized gradients
Spectral correspondence
Computations of scalars relating Poisson transforms
The Case of G=SO(n,1), n>=3
The Case of G=SU(n,1), n>=2
The Case of G=Sp(n,1), n>=2
The Case of G=F4(-20)
The scalars relating Poisson transforms
Structure theory of rank one groups
Structure theory of PSL(2,R)
Structure theory of SO(n,1), n>=3
General structure
Decomposition of H mu into spherical harmonics
Composition series of H mu
Tensor products and proof of Proposition 7.3.12
Structure theory of SU(n,1), n!=1
Structure theory of Sp(n,1), n!=1
General structure
Decomposition of H mu as K-representation and description of M-spherical K-representations
Composition series of H mu
Structure theory of F4
Relative discrete series for F4(-20)
An example: G=SU(n,1), n>=2
Generalized gradients
Computations for spherical harmonics
Poisson transforms
Fourier characterization
Spectral correspondence
Bibliography
List of Symbols