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Harish-Chandra computed the center of the enveloping algebra of a semisimple Lie algebra and so determined the possible infinitesimal characters for irreducible representations of semisimple groups. Kirillov proposed a reinterpretation Harish-Chandra's isomorphism that would determine the center of the enveloping algebra for any Lie algebra, and this was eventually proved by Duflo.
Quite recently Alekseev and Meinrenken gave a beautiful new proof the the Duflo isomorphism (for many although not all Lie algebras) by connecting the Kirillov-Duflo map to a non-commutative Chern-Weil homomorphism in equivariant cohomology. I shall present part of their work and consider its implications in representation theory.
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