Peter Scholze and Jared Weinstein
- Published in print:
- 2020
- Published Online:
- January 2021
- ISBN:
- 9780691202082
- eISBN:
- 9780691202150
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691202082.003.0020
- Subject:
- Mathematics, Geometry / Topology
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 ...
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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.Less
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.
