  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 [1] about intersection cohomology
of certain Uhlenbeck-type moduli spaces (in fact, our proof is conditioned upon
the assumption that the results of [1] are valid in positive characteristic).
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 roughy speaking, that additional term is related
to the fact that the loop group of an affine Kac-Moody group (which roughly
speaking 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 [2].
