Affine Grassmannian - Knowlpedia

Definition

Let $k$ be an algebraically closed field and $G$ a connected reductive group over $k$ . Its affine Grassmannian is the ind-projective ind-scheme

\operatorname{Gr}_G=LG/L^+G, \qquad LG(R)=G(R((t))),\quad L^+G(R)=G(R\lbrack\!\lbrack t\rbrack\!\rbrack)

for $k$ -algebras $R$ , where $LG$ is the loop group, $L^+G$ is the positive loop group, and the quotient is fpqc-sheafified. Functorially, it classifies a principal $G$ -bundle on the formal disc $\operatorname{Spec}k\lbrack\!\lbrack t\rbrack\!\rbrack$ together with a trivialization on the punctured disc $\operatorname{Spec}k((t))$ .

Orbits

The $G\lbrack\!\lbrack t\rbrack\!\rbrack$ -orbits are indexed by dominant coweights $\lambda$ . Their closures are affine Schubert varieties. The coweight records the relative position of a Hecke modification.

Langlands role

With a specified sheaf theory and coefficients satisfying the hypotheses of geometric Satake, the spherical equivariant sheaf category on $\operatorname{Gr}_G$ is identified with representations of the Langlands dual group. This supplies the labels for geometric Hecke functors.

References

Ivan Mirković and Kari Vilonen, “Geometric Langlands duality and representations of algebraic groups over commutative rings,” Annals of Mathematics 166 (2007), 95–143. arXiv.