# Spectral uniqueness of bi-invariant metrics on symplectic groups

## TL;DR

This paper proves that on symplectic groups, bi-invariant metrics are uniquely identified by their Laplace spectrum among left-invariant metrics, highlighting spectral rigidity within this class.

## Contribution

It establishes spectral uniqueness of bi-invariant metrics on symplectic groups using an elementary proof and a strong spectral obstruction.

## Key findings

- Bi-invariant metrics are spectrally unique among left-invariant metrics on $ ext{Sp}(n)$.
- Non-bi-invariant left-invariant metrics are not isospectral to bi-invariant metrics.
- Elementary proof leveraging a spectral obstruction by Gordon, Schueth, and Sutton.

## Abstract

In this short note, we prove that a bi-invariant Riemannian metric on $\mathrm{Sp}(n)$ is uniquely determined by the spectrum of its Laplace-Beltrami operator within the class of left-invariant metrics on $\mathrm{Sp}(n)$. In other words, on any of these compact simple Lie groups, every left-invariant metric which is not right-invariant cannot be isospectral to a bi-invariant metric. The proof is elementary and uses a very strong spectral obstruction proved by Gordon, Schueth, and Sutton.