Affine Grassmannian

In mathematics, the term affine Grassmannian has two distinct meanings. In one meaning the affine Grassmannian (manifold) is the manifold of all "k"-dimensional subspaces of a finite dimensional vector space, while (described here) the affine Grassmannian of an algebraic group "G" over a field "k" is defined in one of two ways:
* As the coset space "G"("K")/"G"("O"), where "K" = "k"(("t")) is the field of formal Laurent series over "k" and "O" = "k""t" is the ring of formal power series;
* As the ind-scheme Gr_"G" which is described as a functor by the following data: to every "k"-algebra "A", Gr_"G"("A") is 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 "X"_"A" and "φ" a trivialization on ("X" − "x")_"A".That Gr_"G" is an ind-scheme does not follow trivially from the definition. Its "k"-points can be identified with the coset space in the first definition, by choosing a trivialization of "E" over all of Spec "O". When "G" is a reductive group, Gr_"G" is in fact ind-projective, i.e., an inductive limit of projective schemes.