Jacobi–Madden equation

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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 ^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 $k-\mathrm{A}\mathrm{l}\mathrm{g}$ and $\mathrm{S}\mathrm{e}\mathrm{t}$ 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 $X:k-\mathrm{A}\mathrm{l}\mathrm{g}\to \mathrm{S}\mathrm{e}\mathrm{t}$ 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 $\mathcal{𝒦}=k((t))$ the field of formal Laurent series over k, and by $\mathcal{𝒪}=k[[t]]$ the ring of formal power series over k. By choosing a trivialization of E over all of Spec $\mathcal{𝒪}$, the set of k-points of Gr_G is identified with the coset space $G(\mathcal{𝒦})/G(\mathcal{𝒪})$.

References

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