Consider the 3-dimensional N = 4 supersymmetric gauge theory associated with a compact Lie group G c and its quaternionic representation M. Physicists study its Coulomb branch, which is a noncompact hyper-Kähler manifold with an SU(2)-action, possibly with singularities. We give a mathematical definition of the Coulomb branch as an affine algebraic variety with C × -action when M is of a form N ⊕ N * , as the second step of the proposal given in [Nak16]. Temkin for the useful discussions. We also thank J. Hilburn and B. Webster for pointing out mistakes in an earlier version of the proof of Theorem 4.1, and the formula in (4.7) respectively.

This is a companion paper of [Part II]. We study Coulomb branches of unframed and framed quiver gauge theories of type ADE. In the unframed case they are isomorphic to the moduli space of based rational maps from P 1 to the flag variety. In the framed case they are slices in the affine Grassmannian and their generalization. In the appendix, written jointly with Joel Kamnitzer, Ryosuke Kodera, Ben Webster, and Alex Weekes, we identify the quantized Coulomb branch with the truncated shifted Yangian.

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 for simplicity we consider only untwisted and simply connected case). The usual geometric Satake isomorphism for a reductive group G identifies the tensor category Rep(G ∨ ) of finitedimensional representations of the Langlands dual group G ∨ with the tensor category. As a byproduct one gets a description of the irreducible G(O)-equivariant intersection cohomology sheaves of the closures of G(O)-orbits in GrG in terms of q-analogs of the weight multiplicity for finite dimensional representations of G ∨ . The purpose of this paper is to try to generalize the above results to the case when G is replaced by the corresponding affine Kac-Moody group G aff (we shall refer to the (not yet constructed) affine Grassmannian of G aff as the double affine Grassmannian). More precisely, in this paper we construct certain varieties that should be thought of as transversal slices to various G aff (O)-orbits inside the closure of another G aff (O)-orbit in GrG aff . We present a conjecture that computes the IC sheaf of these varieties in terms of the corresponding q-analog of the weight multiplicity for the Langlands dual affine group G ∨ aff and we check this conjecture in a number of cases. Some further constructions (such as convolution of the corresponding perverse sheaves, analog of the Beilinson-Drinfeld Grassmannian etc.) will be addressed in another publication.