Families of affine Grassmannians

Abstract

This chapter studies families of affine Grassmannians. In the geometric case, if X is a smooth curve over a field k, Beilinson-Drinfeld defined a family of affine Grassmannians whose fiber parametrizes G-torsors on X. If one fixes a coordinate at x, this gets identified with the affine Grassmannian considered previously. Over fibers with distinct points xi, one gets a product of n copies of the affine Grassmannian, while over fibers with all points xi = x equal, one gets just one copy of the affine Grassmannian: This is possible as the affine Grassmannian is infinite-dimensional. However, sometimes it is useful to remember more information when the points collide. The chapter then discusses the convolution affine Grassmannian in the setting of the previous lecture.