Geometric Quantizations Related to the Laplace Eigenspectra of Compact Riemannian Symmetric Spaces via Borel-Weil-Bott Theory

Authors

Camilo Montoya, Florida International UniversityFollow

Document Type

Dissertation

Degree

Doctor of Philosophy (PhD)

Major/Program

Applied Mathematical Sciences

First Advisor's Name

Gueo Grantcharov

First Advisor's Committee Title

Committee Chair

Second Advisor's Name

Miroslav Yotov

Second Advisor's Committee Title

Committee member

Third Advisor's Name

Philippe Rukimbira

Third Advisor's Committee Title

Committee member

Fourth Advisor's Name

Rajamani Narayanan

Fourth Advisor's Committee Title

Committee member

Keywords

differential geometry, Riemmanian geometry, symmetric spaces, quantizations, eigenfunctions

Date of Defense

6-24-2021

Abstract

The purpose of this thesis is to suggest a geometric relation between the Laplace-Beltrami spectra and eigenfunctions on compact Riemannian symmetric spaces and the Borel-Weil theory using ideas from symplectic geometry and geometric quantization. This is done by associating to each compact Riemannian symmetric space, via Marsden-Weinstein reduction, a generalized flag manifold which covers the space parametrizing all of its maximal totally geodesic tori. In the process we notice a direct relation between the Satake diagram of the symmetric space and the painted Dynkin diagram of its associated flag manifold. We consider in detail the examples of the classical simply-connected spaces of rank one and the space SU(3)/SO(3).

We briefly present the necessary background material and also provide detailed study of examples of rank 2 symmetric spaces and possible decomposition of their eigenspaces into irreducible subspaces. In the last part of the thesis, with the aid of harmonic polynomials, we induce Laplace-Beltrami eigenfunctions on the symmetric space from holomorphic sections of the associated line bundle on the generalized flag manifold. We consider a generalization of a method of constructing explicit representations of the Laplace-Beltrami eigenfunction using homogeneous harmonic polynomials (under some mild conditions) as the (proper) restrictions in some ambient space, as opposed to the known implicit integral representations of these eigenfunctions \cite{H2,Gi}. We apply this method to the examples of the simply connected rank one space $\H P^n$ and maximal rank 2 space $SU(3)/SO(3)$, moreover applying the connection to the Borel-Weil theorem we show that our construction produces the explicit representation of all of the eigenfunctions.

Identifier

FIDC010250

ORCID

https://orcid.org/0000-0002-3086-0832

Creative Commons License

This work is licensed under a Creative Commons Public Domain Dedication 1.0 License.

Recommended Citation

Montoya, Camilo, "Geometric Quantizations Related to the Laplace Eigenspectra of Compact Riemannian Symmetric Spaces via Borel-Weil-Bott Theory" (2021). FIU Electronic Theses and Dissertations. 4719.
https://digitalcommons.fiu.edu/etd/4719