Pursuing the double affine Grassmannian I: transversal slices via instantons on A_k-singularities

TL;DR

This paper explores the geometric structure of affine Grassmannian singularities related to affine Kac-Moody groups, proposing a conjecture linking their intersection cohomology to integrable representations, supported by initial checks.

Contribution

It constructs a model for certain singularities in the affine Grassmannian and formulates a conjecture connecting their intersection cohomology to dual affine Kac-Moody representations.

Findings

Constructed a model for singularities using instantons on A_k-singularities.

Formulated a conjecture relating intersection cohomology to Langlands dual representations.

Validated the conjecture in several specific cases.

Abstract

This paper is the first in a series that describe a conjectural analog of the geometric Satake isomorphism for an affine Kac-Moody group. In this paper we construct a model for the singularities of some would-be Schubert varieties in the affine Grassmannian for an affine Kac-Moody group. We formulate a conjecture describing the (local) intersection cohomology of these varieties in terms of integrable representations of the Langlands dual affine Kac-Moody group and check this conjecture in a number of cases. Roughly speaking the above singularities are constructed by looking at the Uhlenbeck space of instantons on the quotient of the affine plane by a finite cyclic subgroup of SL(2).