Abstract
We prove a vanishing theorem for Lie algebra cohomology which constitutes a loop group analogue of Kostant's Lie algebra version of the Borel-Weil-Bott theorem. Consider a complex semi-simple Lie algebra Open image in new window and an integrable, irreducible, negative energy representation ℋ of Open image in new window . Givenn distinct pointszk in ℂ, with a finite-dimensional irreducible representationVk of Open image in new window assigned to each, the Lie algebra Open image in new window of Open image in new window -valued polynomials acts on eachVk, via evaluation atzk. Then, the relative Lie algebra cohomologyH* Open image in new window is concentrated in one degree. As an application, based on an idea of G. Segal's, we prove that a certain “homolorphic induction” map from representations ofG to representations ofLG at a given level takes the ordinary tensor product into the fusion product. This result had been conjectured by R. Bott.
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