Isospectral metrics and potentials on classical compact simple Lie groups

Abstract

We prove the existence of multiparameter isospectral deformations of metrics on SO(n) $(n = 9$ and $n\geq 11)$, SU(n) $(n\geq 8)$, and $Sp(n)$ $(n\geq 4)$. For these examples, we follow a metric construction developed by Schueth who had given one-parameter families of isospectral metrics on orthogonal and unitary groups. Our multiparameter families are obtained by a new proof of nontriviality establishing a generic condition for nonisometry of metrics arising from the construction. We also show the existence of non-congruent pairs of isospectral potentials and nonisometric pairs of isospectral conformally equivalent metrics on $Sp(n)$ for $n\geq 6$.