ar X iv : 0 90 5 . 11 53 v 2 [ m at h . K T ] 2 7 M ay 2 00 9 EQUIVARIANT CORRESPONDENCES AND THE BOREL - BOTT - WEIL THEOREM

We formulate and prove the special case of Serre duality involved in the Borel-Bott-Weil theorem in the language of equivariant Kasparov theory. The method is to combine the Atiyah-Singer index theorem and the framework of equivariant correspondences developed in another paper by the first author and Ralf Meyer. The twisted Dolbeault cohomology groups of the flag variety figuring in the Borel-Bott-Weil theorem are interpreted as the equivariant analytic indices of an appropriate family of equivariant elliptic operators. These analytic indices are equal to their topological indices by the Aityah-Singer theorem and we prove our result by by purely topological calculations with the topological indices. The key point in the proof is the construction of a family of equivariant self-correspondences parameterised by the Weyl group. These intertwine the topological indices up to the sign change and shift factor predicted by the Borel-Bott-Weil theorem.