- Affine Grassmannian
In
mathematics , the term affine Grassmannian has two distinct meanings. In one meaning theaffine Grassmannian (manifold) is the manifold of all "k"-dimensional subspaces of a finite dimensional vector space, while (described here) the affine Grassmannian of analgebraic 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 offormal Laurent series over "k" and "O" = "k""t" is the ring offormal power series ;
* As theind-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 aprincipal 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 theBeauville–Laszlo theorem , it is also possible to specify this data by fixing analgebraic 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 areductive group , Gr"G" is in fact ind-projective, i.e., an inductive limit of projective schemes.
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