The real affine Grassmannian and quantum SL(2)

TL;DR

This paper establishes a deep connection between the category of equivariant perverse sheaves on the affine Grassmannian of PGL(2, R) and Lusztig's quantum SL(2), revealing new structural insights.

Contribution

It proves the highest weight structure of the category of equivariant perverse sheaves and constructs projective objects, linking geometric and quantum algebraic categories.

Findings

The category of equivariant perverse sheaves is highest weight.

Constructed explicit projective objects in the category.

Showed equivalence to the principal block of quantum SL(2) at a root of unity.

Abstract

We prove that the category of equivariant perverse sheaves on the affine Grassmannian of PGL(2, R) is highest weight and we construct the projective objects. Moreover we prove that the category of perverse sheaves on the odd component is equivalent to the principal block of Lusztig's quantum SL(2) at a primitive fourth root of unity.