In mathematics, the affine Grassmannian of an algebraic group G over a field k is an ind-schemeâÂÂa colimit of finite-dimensional schemesâÂÂwhich 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 <sup>L</sup>G through what is known as the geometric Satake correspondence.
Let k be a field, and denote by and the category of commutative k-algebras and the category of sets respectively. Through the Yoneda lemma, a scheme X over a field k is determined by its functor of points, which is the functor which takes A to the set X(A) of A-points of X. We then say that this functor is representable by the scheme X. The affine Grassmannian is a functor from k-algebras to sets which is not itself representable, but which has a filtration by representable functors. As such, although it is not a scheme, it may be thought of as a union of schemes, and this is enough to profitably apply geometric methods to study it.
Let G be an algebraic group over k. The affine Grassmannian Gr<sub>G</sub> is the functor that associates to a k-algebra A the set of isomorphism classes of pairs (E, ÃÂ), where E is a principal homogeneous space for G over Spec A and àis an isomorphism, defined over Spec A((t)), of E with the trivial G-bundle G àSpec A((t)). By the BeauvilleâÂÂLaszlo 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 X<sub>A</sub> and àa trivialization on (X â x)<sub>A</sub>. When G is a reductive group, Gr<sub>G</sub> is in fact ind-projective, i.e., an inductive limit of projective schemes.
Let us denote by 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 Gr<sub>G</sub> is identified with the coset space .