Affine Grassmannian

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.

Definition of Gr via functor of points

Let k be a field, and denote by [math]\displaystyle{ k\text{-Alg} }[/math] and [math]\displaystyle{ \mathrm{Set} }[/math] 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 [math]\displaystyle{ X:k\text{-Alg} \to \mathrm{Set} }[/math] 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[[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. When G is a reductive group, Gr_G is in fact ind-projective, i.e., an inductive limit of projective schemes.

Definition as a coset space

Let us denote by [math]\displaystyle{ \mathcal K = k((t)) }[/math] the field of formal Laurent series over k, and by [math]\displaystyle{ \mathcal O = kt }[/math] the ring of formal power series over k. By choosing a trivialization of E over all of [math]\displaystyle{ \operatorname{Spec}\mathcal O }[/math], the set of k-points of Gr_G is identified with the coset space [math]\displaystyle{ G(\mathcal K) / G(\mathcal O) }[/math].

References

• Alexander Schmitt (11 August 2010). Affine Flag Manifolds and Principal Bundles. Springer. pp. 3–6. ISBN 978-3-0346-0287-7. https://books.google.com/books?id=xrPoBAdiVdQC&pg=PA1. Retrieved 1 November 2012.