R\'esolutions de Demazure affines et formule de Casselman-Shalika g\'eom\'etrique

TL;DR

This paper proves a conjecture relating Fourier coefficients of spherical perverse sheaves on the affine Grassmannian for split reductive groups, extending previous work on GL(n) by analyzing resolutions of Schubert varieties.

Contribution

It extends the proof of a conjecture from GL(n) to general split reductive groups using resolutions of Schubert varieties in the affine Grassmannian.

Findings

Confirmed the conjecture for all split reductive groups.

Developed new geometric techniques involving resolutions of Schubert varieties.

Extended previous proofs from GL(n) to broader classes of groups.

Abstract

We prove a conjecture of Frenkel, Gaitsgory, Kazhdan and Vilonen, related to Fourier coefficients of spherical perverse sheaves on the affine Grassmannian associated to a a split reductive group. Our proof is an extension of the proof given by the first author in the case of GL(n) (see math/9801109); it relies on the study of certain resolutions of Schubert varieties in the affine Grassmannian, built from the so-called minuscule or quasi-minuscule cases.