Affine Grassmannian
The affine Grassmannian is a key ingredient in many talks dealing with the geometric (=automorphic) side of the Geometric Langlands correspondence.
Sources:
Sections 4.5.1-4.5.2 of [BD];
Section 2 of Mirkovic-Vilonen's paper.
Sections 2 and 3 of these notes.
Notes by R. Reich.