Moment Maps in Geometric Representation Theory

A Poisson structure on a manifold M is a Lie bracket on the algebra of smooth functions that satisfies the Leibniz identity. This bracket induces a foliation of M in which each leaf carries a symplectic form, and at each point the transverse structure of this foliation is encoded by the action of a Lie algebra. In this way, Poisson geometry is a crossroads where foliation theory, symplectic geometry, and representation theory meet. When M has a compatible action of a Lie group G, Hamiltonian reduction gives a procedure that "simplifies" M to a smaller Poisson manifold by removing its G-symmetries. I'll survey Hamiltonian reduction from several geometric perspectives, and then show how this tool can be applied to construct interesting Poisson varieties in geometric representation theory. This is based on joint work with Maxence Mayrand.