Affine Gindikin–Karpelevich Formula via Uhlenbeck Spaces

Abstract

We prove a version of the Gindikin–Karpelevich formula for untwisted affine Kac–Moody groups over a local field of positive characteristic. The proof is geometric and it is based on the results of [Braverman, Finkelberg, and Gaitsgory, Uhlenbeck spaces via affine Lie algebras, Progr. Math., 244, 17–135, 2006] about intersection cohomology of certain Uhlenbeck-type moduli spaces (in fact, our proof is conditioned upon the assumption that the results of [Braverman, Finkelberg, and Gaitsgory, Uhlenbeck spaces via affine Lie algebras, Progr. Math., 244, 17–135, 2006] are valid in positive characteristic; we believe that generalizing [Braverman, Finkelberg, and Gaitsgory, Uhlenbeck spaces via affine Lie algebras, Progr. Math., 244, 17–135, 2006] to the case of positive characteristic should be essentially straightforward but we have not checked the details). In particular, we give a geometric explanation of certain combinatorial differences between finite-dimensional and affine case (observed earlier by Macdonald and Cherednik), which here manifest themselves by the fact that the affine Gindikin–Karpelevich formula has an additional term compared to the finite-dimensional case. Very roughly speaking, that additional term is related to the fact that the loop group of an affine Kac-Moody group (which should be thought of as some kind of “double loop group”) does not behave well from algebro-geometric point of view; however, it has a better behaved version, which has something to do with algebraic surfaces.A uniform (i.e. valid for all local fields) and unconditional (but not geometric) proof of the affine Gindikin–Karpelevich formula is going to appear in [Braverman, Kazhdan, and Patnaik, The Iwahori-Hecke algebra for an affine Kac-Moody group (in preparation)].