Inverse spectral results for non-abelian group actions

In this paper we will extend to non-abelian groups inverse spectral results, proved by us in an earlier (Guillemin and Wang, 2016), for compact abelian groups, i.e. tori. More precisely, Let G be a Lie group acting isometrically on Riemannian manifold X. We show that the Schrödinger operator ??2?+V with V?C?(X)G, potential function V is, some interesting examples, determined G-equivariant spectrum. The key ingredient proof is generalized Legendrian relation between Lagrangian manifolds Graph(dV) Graph(dF), where F invariant defined open subset of positive Weyl chamber.