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

TL;DR

This paper establishes an explicit Macdonald's formula for the spherical Hecke algebra of an almost split Kac-Moody group over a local field, linking geometric and algebraic methods.

Contribution

It provides the first explicit formula for basis elements of the Hecke algebra in this setting, extending Macdonald's formula to Kac-Moody groups over local fields.

Findings

Derived an explicit Macdonald's formula for Kac-Moody groups

Connected geometric masure arguments with algebraic Cherednik tools

Enhanced understanding of the Satake isomorphism in this context

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.