Let $(M, g)$ be a Riemannian manifold admitting a Lie group action. I will explain how the symmetry induces an extra structure in the spectrum of Laplace-like operators, and how to apply ideas from symplectic geometry to study the induced equivariant spectrum. Then I will discuss some of our recent results on the equivariant spectrum of Kahler manifolds admitting a large symmetry. 

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