Semi-infinite parabolic IC-sheaf II: the Ran space version

TL;DR

This paper advances the understanding of semi-infinite IC-sheaves on the Ran space version of the affine Grassmannian for a split reductive group, connecting local and global categories and relating to Drinfeld compactifications.

Contribution

It introduces the semi-infinite parabolic IC-sheaf on the Ran space affine Grassmannian and provides multiple descriptions and relations to other geometric objects.

Findings

Descriptions of the semi-infinite parabolic IC-sheaf as colimit and intermediate extension.

Relations established between local and global semi-infinite categories.

Connection made between the IC-sheaf and the intersection cohomology of Drinfeld compactification.

Abstract

This paper is a sequel to arXiv:2310.0638. Let G be a split reductive group. We define the parabolic semi-infinite category on the Ran version of the affine Grassmanian Gr_{G,Ran}. We study the semi-infinite parabolic IC-sheaf in this category. We provide several descriptions of this object, one as certain colimit, another as an intermediate extension in certain category. We relate the global and local semi-infinite categories of sheaves. We also relate the intersection cohomology sheaf of the Drinfeld compactification of Bun_P with the semi-infinite parabolic IC-sheaf.