Instanton Counting via Affine Lie Algebras I: Equivariant J-functions of (affine) Flag Manifolds and Whittaker Vectors

Let g be a simple complex Lie algebra, G the corresponding simply connected group; let also gaff be the corresponding untwisted affine Lie algebra. For a parabolic subgroup P ⊂ G we introduce a generating function Z G,P which roughly speaking counts framed G-bundles on P endowed with a P -structure on the horizontal line (the formal definition uses the corresponding Uhlenbeck type compactifications studied in [3]). In the case P = G the function Z G,P coincides with Nekrasov’s partition function introduced in [23] and studied thoroughly in [24] and [22] for G = SL(n). In the ”opposite case” when P is a Borel subgroup of G we show that Z G,P is equal (roughly speaking) to the Whittaker matrix coefficient in the universal Verma module for the Lie algebra ǧaff – the Langlands dual Lie algebra of gaff . This clarifies somewhat the connection between certain asymptotic of Z G,P (studied in loc. cit. for P = G) and the classical affine Toda system. We also explain why the above result gives rise to a calculation of (not yet rigorously defined) equivariant quantum cohomology ring of the affine flag manifold associated with G. In particular, we reprove the results of [13] and [18] about quantum cohomology of ordinary flag manifolds using methods which are totally different from loc. cit. We shall show in a subsequent publication how this allows one to connect certain asymptotic of the function Z G,P with the Seiberg-Witten prepotential (cf. [2], thus proving the main conjecture of [23] for an arbitrary gauge group G (for G = SL(n) it has been proved in [24] and [22] by other methods.