In mathematics, the affine Grassmannian of an algebraic group G over a field k is an ind-scheme—a colimit of finite-dimensional schemes—which can be thought of as a flag variety for the loop group G(k((t))) and which describes the representation theory of the Langlands dual group LG through what is known as the geometric Satake correspondence.
Let G be an algebraic group over k. The affine Grassmannian GrG is the functor that associates to a k-algebra A the set of isomorphism classes of pairs (E, φ), where E is a principal homogeneous space for G over Spec A[[t]] and φ is an isomorphism, defined over Spec A((t)), of E with the trivial G-bundle G × Spec A((t)).
By the Beauville–Laszlo theorem, it is also possible to specify this data by fixing an algebraic curve X over k, a k-point x on X, and taking E to be a G-bundle on XA and φ a trivialization on (X − x)A.
When G is a reductive group, GrG is in fact ind-projective, i.e., an inductive limit of projective schemes.
the field of formal Laurent series over k, and by
the ring of formal power series over k. By choosing a trivialization of E over all of
, the set of k-points of GrG is identified with the coset space