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 C G we introduce a generating function Z(?p which roughly speaking counts framed G-bundles on P2 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'*fp coincides with Nekrasov's partition function introduced in [23] and studied thoroughly in [22,24] for G = SL(n). In the "opposite case" when P is a Borel subgroup of G we show that -ZgPp is equal (roughly speaking) to the Whittaker matrix coefficient in the universal Verma module for the Lie algebra gaff- -the Langlands dual Lie algebra of gaff. This clarifies somewhat the connection between certain asymptotics of Z^?p (studied in loc. cit. for P = G) and the classical afHne 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, 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 Zf?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 [22,24] by other methods).
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