Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A note about the generic irreducibility of the spectrum of the Laplacian on homogeneous spaces

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

TL;DR

The paper analyzes the Laplace-Beltrami operator on compact homogeneous spaces $M=G/K$, focusing on when its spectrum is generically real $G$-simple under $G$-invariant metrics. It develops a framework that relates real $G$-simplicity to complex $(Q_8\times G)$-simplicity via a commuting $Q_8$-action arising from the representation theory of $G$, and it explains how root-system symmetries in symmetric spaces of rank $\ge 2$ force shared Casimir eigenvalues among distinct irreducibles. The authors introduce and utilize operator families $D^V(\kappa)$ and their restrictions to $V^K$, along with a root-system potentiation by translating to the origin $-\delta$, to prove a transitive action of a finite subgroup on the set of representations with a given Casimir eigenvalue. Their results recover known rank-1 and product-of-rank-1 cases as real $G$-simple spectra correspondingly yielding complex $(Q_8\times G)$-simple spectra, while higher rank spaces exhibit structural symmetries that obstruct generic simplicity. Overall, the work unifies real/complex spectral perspectives for Laplacians on homogeneous spaces and clarifies how algebraic symmetries control eigenvalue multiplicities in a representation-theoretic setting.

Abstract

Petrecca and Röser (2018, \cite{Petrecca2019}), and Schueth (2017, \cite{Schueth2017}) had shown that for a generic $G$-invariant metric $g$ on certain compact homogeneous spaces $M=G/K$ (including symmetric spaces of rank 1 and some Lie groups), the spectrum of the Laplace-Beltrami operator $Δ_g$ was real $G$-simple. The same is not true for the complex version of $Δ_g$ when there is a presence of representations of complex or quaternionic type. We show that these types of representations induces a $Q_8$-action that commutes with the Laplacian in such way that $G$-properties of the real version of the operator have to be understood as $(Q_8 \times G)$-properties on its corresponding complex version. Also we argue that for symmetric spaces on rank $\geq 2$ there are algebraic symmetries on the corresponding root systems which relates distinct irreducible representations on the same eigenspace.

A note about the generic irreducibility of the spectrum of the Laplacian on homogeneous spaces

TL;DR

The paper analyzes the Laplace-Beltrami operator on compact homogeneous spaces , focusing on when its spectrum is generically real -simple under -invariant metrics. It develops a framework that relates real -simplicity to complex -simplicity via a commuting -action arising from the representation theory of , and it explains how root-system symmetries in symmetric spaces of rank force shared Casimir eigenvalues among distinct irreducibles. The authors introduce and utilize operator families and their restrictions to , along with a root-system potentiation by translating to the origin , to prove a transitive action of a finite subgroup on the set of representations with a given Casimir eigenvalue. Their results recover known rank-1 and product-of-rank-1 cases as real -simple spectra correspondingly yielding complex -simple spectra, while higher rank spaces exhibit structural symmetries that obstruct generic simplicity. Overall, the work unifies real/complex spectral perspectives for Laplacians on homogeneous spaces and clarifies how algebraic symmetries control eigenvalue multiplicities in a representation-theoretic setting.

Abstract

Petrecca and Röser (2018, \cite{Petrecca2019}), and Schueth (2017, \cite{Schueth2017}) had shown that for a generic -invariant metric on certain compact homogeneous spaces (including symmetric spaces of rank 1 and some Lie groups), the spectrum of the Laplace-Beltrami operator was real -simple. The same is not true for the complex version of when there is a presence of representations of complex or quaternionic type. We show that these types of representations induces a -action that commutes with the Laplacian in such way that -properties of the real version of the operator have to be understood as -properties on its corresponding complex version. Also we argue that for symmetric spaces on rank there are algebraic symmetries on the corresponding root systems which relates distinct irreducible representations on the same eigenspace.

Paper Structure

This paper contains 15 sections, 6 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Types of each representation
  4. Laplace operators
  5. Basic facts on root systems of symmetric pairs
  6. Basic facts on the quaternion group $Q_8$
  7. The space of $G$-invariant metrics
  8. Operators on each representation
  9. Recent advances in the literature
  10. Real $G$-properties vs. complex $(Q_8 \times G)$-properties of the spectrum
  11. Structural symmetries for symmetric spaces of any rank
  12. Root systems and representations with the same Casimir eigenvalue
  13. The linear structure on $-\delta$
  14. Representations with the same Casimir eigenvalue
  15. Proof of the Theorem \ref{['th:transitive-action']}

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 $M$ on $M$ has real $G$-simple spectrum.

Theorems & Definitions (22)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 2.1
  • Example 1
  • Example 2
  • Remark 5
  • Theorem 3.1
  • proof
  • ...and 12 more