Cellular pavings of fibers of convolution morphisms

TL;DR

This paper demonstrates that fibers of convolution morphisms in affine flag varieties are paved by affine lines and their punctured variants, with extensions over integers relating to geometric Satake and motives.

Contribution

It establishes cellular pavings of fibers for convolution morphisms over arbitrary fields and extends these results over integers, connecting to geometric Satake and motives.

Findings

Fibers are paved by products of affine lines and punctured lines.

Results apply to affine Grassmannian and convolution morphisms in geometric Satake.

Extensions over $\ ext{\mathbb Z}$ relate to motives and alternative proofs for recent work.

Abstract

This article proves, in the case of split groups over arbitrary fields, that all fibers of convolution morphisms attached to parahoric affine flag varieties are paved by products of affine lines and affine lines minus a point. This applies in particular to the affine Grassmannian and to the convolution morphisms in the context of the geometric Satake correspondence. The second part of the article extends these results over $\mathbb Z$. Those in turn relate to the recent work of Cass-van den Hove-Scholbach on the geometric Satake equivalence for integral motives, and provide some alternative proofs for some of their results.