Spectral correspondences for locally symmetric spaces - the case of exceptional parameters / von Christian Arends, unter der Betreuung durch Prof. Dr. Joachim Hilgert. Paderborn, 2023
Inhalt
- 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