Macdonald's formula for Kac-Moody groups over local fields

Abstract

For an almost split Kac-Moody group G over a local non-archimedean field, the last two authors constructed a spherical Hecke algebra H (over the complex numbers C, say) and its Satake isomorphism with the commutative algebra of Weyl invariant elements in some formal series algebra C[[Y]].In this article, we prove a Macdonald's formula, i.e. an explicit formula for the image of a basis element of H. The proof involves geometric arguments in the masure associated to G and algebraic tools, including the Cherednik's representation of the Bernstein-Lusztig-Hecke algebra (introduced in a previous article) and the Cherednik's identity between some symmetrizers.