ar X iv : 0 90 5 . 11 53 v 1 [ m at h . K T ] 8 M ay 2 00 9 EQUIVARIANT CORRESPONDENCES AND THE BOREL - BOTT - WEIL THEOREM

We show that the special case of Serre duality involved in the Borel-Bott-Weil theorem can be formulated and proved in the context of equi-variant Kasparov theory by combining 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 a flag variety that figure in the Borel-Bott-Weil theorem can be interpreted as the equivariant analytic indices of the corresponding twisted Dolbeault operators. These analytic indices are equal to their topological indices by the Aityah-Singer theorem. We prove our analogue of Serre duality by purely topo-logical calculations with the topological indices. The key point in the proof is the construction of a family of equivariant self-correspondences of the flag variety, 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.