Spectral uniqueness of the bi-invariant metric on symplectic groups

Emilio A. Lauret

Published 2017-06-27Version 1

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 operator within the class of left-invariant metrics on $\mathrm{Sp}(n)$, for any $n\geq5$ or $n=3$. 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 uses a very strong spectral obstruction proved by Gordon, Schueth and Sutton.