Affine Grassmannian Explained

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

k -Alg and Set 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 -Alg \to Set 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.

Definition as a coset space

Let us denote by

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

References

• Book: Alexander Schmitt. Affine Flag Manifolds and Principal Bundles. 1 November 2012. 11 August 2010. Springer. 978-3-0346-0287-7. 3–6.