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

Nicole Bardy-Panse (1) , Stéphane Gaussent (2) , Guy Rousseau (1)

1 IECL - Institut Élie Cartan de Lorraine
2 AGL - Algèbre, géométrie, logique

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.

Mots clés: Kac--Moody group; Bernstein-Lusztig relations; Hecke algebras; Satake isomorphism; ultrametric local field; Masure; building; Macdonald's formula

Domaines: Théorie des représentations [math.RT]

Nicole Bardy-Panse, Stéphane Gaussent, Guy Rousseau. Macdonald's formula for Kac-Moody groups over local fields. Proceedings of the London Mathematical Society, 2019, ⟨10.1112/plms.12231⟩. ⟨hal-01588194⟩