Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Hidden symmetries and the generic spectral setting of generalized laplacians on homogeneous spaces

Diego S. De Oliveira, Marcus A. M. Marrocos

TL;DR

The paper develops a comprehensive, representation-theoretic framework for the spectra of generalized Laplacians associated to generic $G$-invariant metrics on compact homogeneous spaces $M=G/K$. It identifies hidden symmetries beyond the prescribed group of isometries, organizing eigenspaces via isotypical components, Casimir eigenvalues, and root-system combinatorics; this leads to a criterion for spectral simplicity and a generalized symmetry group $ ilde{G}$ that governs the generic spectrum. Key contributions include a precise decomposition of $ abla_{g,U^*}$ on $L^2(G,U^*)$ and $L^2(G,K;U^*)$, a transitive action of hidden symmetries on weight spheres $S(a)$, and generic estimates bounding the size of eigenspaces for left-invariant metrics. The results extend the understanding of spectral degeneracies in highly symmetric spaces, provide tools for analyzing Hodge-Laplacians on normal homogeneous spaces, and yield concrete a priori bounds for eigenspace dimensions in terms of root-system data and representation multiplicities.

Abstract

The purpose of this work is to establish the spectral setting of some generalized Laplace operators associated to a generic $G$-invariant metric on a compact homogeneous space $M=G/K$. We show that this generic spectral configuration depends on the $G$-isometries and on some certain hidden symmetries constructed in the adjacent structures of $M$ and of these operators.

Hidden symmetries and the generic spectral setting of generalized laplacians on homogeneous spaces

TL;DR

The paper develops a comprehensive, representation-theoretic framework for the spectra of generalized Laplacians associated to generic -invariant metrics on compact homogeneous spaces . It identifies hidden symmetries beyond the prescribed group of isometries, organizing eigenspaces via isotypical components, Casimir eigenvalues, and root-system combinatorics; this leads to a criterion for spectral simplicity and a generalized symmetry group that governs the generic spectrum. Key contributions include a precise decomposition of on and , a transitive action of hidden symmetries on weight spheres , and generic estimates bounding the size of eigenspaces for left-invariant metrics. The results extend the understanding of spectral degeneracies in highly symmetric spaces, provide tools for analyzing Hodge-Laplacians on normal homogeneous spaces, and yield concrete a priori bounds for eigenspace dimensions in terms of root-system data and representation multiplicities.

Abstract

The purpose of this work is to establish the spectral setting of some generalized Laplace operators associated to a generic -invariant metric on a compact homogeneous space . We show that this generic spectral configuration depends on the -isometries and on some certain hidden symmetries constructed in the adjacent structures of and of these operators.

Paper Structure

This paper contains 18 sections, 15 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Laplace operators
  4. Basic facts on root systems
  5. Basic facts on the quaternion group $Q_8$
  6. The space of $G$-invariant metrics
  7. Operators on each representation
  8. Recent advances in the literature
  9. Generalized Laplacians on Lie groups with left-invariant metrics
  10. The operator $\Delta_{g,U^*}$ on each isotypical component of $L^2(G,U^*)$
  11. Real $G$-properties vs. complex $(Q_8 \times G)$-properties
  12. The main criterium of the section
  13. Generalized Laplacians on normal homogeneous spaces
  14. Representations with the same Casimir eigenvalue
  15. The linear structure on $-\delta$
  16. ...and 3 more sections

Key Result

Theorem 2.1

There exists a $G$-invariant metric $g$ such that the Laplace-Beltrami operator $\Delta_g$ on $M$ has a real $G$-simple spectrum if and only if the items bellow are simultaneously satisfied: Moreover, the existence of a such metric is equivalent to say that the Laplace-Beltrami operator of a generic $G$-invariant metric on $M$ has real $G$-simple spectrum.

Theorems & Definitions (43)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Theorem 2.1
  • Example 1
  • Remark 6
  • Example 2
  • Theorem 3.1
  • ...and 33 more