Geometric Satake equivalence

Statement

Let $k$ be an algebraically closed field, let $G$ be a connected reductive group over $k$ , and choose a characteristic-zero coefficient field $E$ and an appropriate sheaf theory. The geometric Satake equivalence identifies the convolution tensor category of $G\lbrack\!\lbrack t\rbrack\!\rbrack$ -equivariant $E$ -perverse sheaves on the affine Grassmannian $\operatorname{Gr}_G$ with

\operatorname{Rep}_E(\widehat G_E),

the tensor category of finite-dimensional algebraic $E$ -representations of the split reductive $E$ -group with root datum dual to that of $G$ , the Langlands dual group.

Under these characteristic-zero hypotheses, the intersection-cohomology complex of the affine Schubert variety indexed by a dominant coweight $\lambda$ corresponds to the irreducible $\widehat G$ -representation of highest weight $\lambda$ .

Coefficients and versions

For classical sheaves over $k=\mathbb C$ , one may take $E=\mathbb C$ . In the étale setting, a standard choice is $E=\overline{\mathbb Q}_\ell$ with $\ell\ne\operatorname{char}k$ . The classical statement uses perverse sheaves with characteristic-zero coefficients and the fiber functor of global cohomology. There are integral, modular, motivic, and derived versions with distinct technical hypotheses and sometimes a non-semisimple representation category. The tensor structure comes from convolution, not pointwise tensor product.

Role in geometric Langlands

Geometric Satake turns local modifications of $G$ -bundles into Hecke functors indexed by representations of $\widehat G$ .

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.