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・ Affine action
・ Affine algebra
・ Affine arithmetic
・ Affine braid group
・ Affine bundle
・ Affine cipher
・ Affine combination
・ Affine connection
・ Affine coordinate system
・ Affine curvature
・ Affine differential geometry
・ Affine focal set
・ Affine gauge theory
・ Affine geometry
・ Affine geometry of curves
Affine Grassmannian
・ Affine Grassmannian (manifold)
・ Affine group
・ Affine Hecke algebra
・ Affine hull
・ Affine involution
・ Affine Lie algebra
・ Affine logic
・ Affine manifold
・ Affine manifold (disambiguation)
・ Affine monoid
・ Affine plane
・ Affine plane (incidence geometry)
・ Affine pricing
・ Affine q-Krawtchouk polynomials


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Affine Grassmannian : ウィキペディア英語版
Affine Grassmannian

In mathematics, the term affine Grassmannian has two distinct meanings; the concept treated in this article is the affine Grassmannian of an algebraic group ''G'' over a field ''k''. It is an ind-scheme—a limit 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 ''L''''G'' through what is known as the geometric Satake correspondence.
== Definition of Gr via functor of points ==

Let ''k'' be a field, and denote by k\text\mathrm and \mathrm 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 X:k\text\mathrm\to\mathrm 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''G'' 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'' 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''. When ''G'' is a reductive group, Gr''G'' is in fact ind-projective, i.e., an inductive limit of projective schemes.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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