Lie theory and the Chern-Weil homomorphism

Details Lie theory and the Chern-Weil homomorphism Lie theory and the Chern-Weil homomorphism

2003-08-14

arxiv.org/abs/math/0308135v1

Ann. Sci. Ecole Norm. Sup. (4) 38 (2005), no. 2, 303--338

Mathematics

Representation Theory

38 pages

Scientific paper

We introduce a canonical Chern-Weil map for possibly non-commutative g-differential algebras with connection. Our main observation is that the generalized Chern-Weil map is an algebra homomorphism ``up to g-homotopy''. Hence, the induced map from invariant polynomials to the basic cohomology is an algebra homomorphism. As in the standard Chern-Weil theory, this map is independent of the choice of connection. Applications of our results include: a conceptually easy proof of the Duflo theorem for quadratic Lie algebras, a short proof of a conjecture of Vogan on Dirac cohomology, generalized Harish-Chandra projections for quadratic Lie algebras, an extension of Rouviere's theorem for symmetric pairs, and a new construction of universal characteristic forms in the Bott-Shulman complex.

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