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 sH (over double-struckC, say) and its Satake isomorphism scriptS with the commutative algebra C[false[Yfalse]]Wv of Weyl invariant elements in some formal series algebra C[[Y]]. In this article, we prove a Macdonald's formula, that is, an explicit formula for the image S(cλ) of a basis element of sH. 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.

On the convergence of the Kac-Moody Correction Factor

“…In this section we will describe the ideas of the proof of our main result. Note that we are going to change into the coroot version of the correction factor in accordance with the formulas in [BKP], [BPGR2], [PP], etc.…”

“…The affine correction factor appears in the affine analogue of the Gindikin-Karpelevich formula [BFK, BGKP], the Macdonald formula [BKP], and the Casselman-Shalika formula [P] in the works towards a generalization of p-adic spherical theory to affine Kac-Moody groups. Subsequently the Kac-Moody correction factor also appeared in the generalizations of the above works to arbitrary p-adic Kac-Moody groups [BPGR1,BPGR2,PP]. The Kac-Moody correction factor m was first studied in [MPW] and later in [LLO].…”