Connections to Schubert Calculus Learning Seminar

The Borel-Weil-Bott theorem gives a method for constructing the irreducible representations of a connected compact Lie group on cohomology spaces associated to line bundles on flag varieties.  In this lecture, we review the necessary background from the representation theory of compact Lie groups and complex geometry, and, in the realization of the theorem using global differential forms and Dolbeault cohomology, we explicitly construct the associated harmonic cocycles and verify the harmonic property.   For such a construction, we lift the explicit Lie algebra cocycles described in Kostant’s theorem using the Peter-Weyl theorem.