Spectral uniqueness of bi-invariant metrics on symplectic groups

Abstract

In this short note, we prove that a bi-invariant Riemannian metric on Sp(n) is uniquely determined by the spectrum of its Laplace–Beltrami operator within the class of left-invariant metrics on 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.