nLab geometric Satake equivalence

Idea

Let $K$ be a local field with ring of integers $\mathcal{O}$, and let $G$ be a geometrically connected split reductive group over $\mathcal{O}$. The spherical Hecke algebra is the ring $\mathcal{H}$ of $\mathbf{Z}$-valued compactly supported functions on the double coset space $G(\mathcal{O}) \backslash G(K) / G(\mathcal{O})$ under convolution. Although convolution algebras are generally non-commutative, an argument known as Gelfand's trick? implies that $\mathcal{H}$ is commutative.

The Satake isomorphism is a ring isomorphism

$$\mathcal{H} \otimes \mathbf{Z}[q^{\pm 1/2}] \simeq R({}^L G) \otimes \mathbf{Z}[q^{\pm 1/2}]$$

where

• $q$ denotes the cardinality of the residue field of $\mathcal{O}$,

• ${}^L G$ denotes the Langlands dual group of $G$, a split reductive group over $\mathbf{C}$, and

• $R({}^L G) = K_0(\mathrm{Rep}({}^L G))$ denotes the representation ring of ${}^L G$.

The geometric Satake equivalence categorifies the Satake isomorphism in the setting of local geometric Langlands. To be precise, let now $G$ denote a reductive group over $\mathbf{C}$ with Langlands dual group ${}^L G$ also over $\mathbf{C}$. Let $\mathcal{L}G$ denote the loop group of $G$ and let $\mathcal{L}^+G$ denote the arc group of $G$. Then one has an equivalence of symmetric monoidal abelian categories

$$\mathrm{DMod}(\mathrm{Gr}_G)^{\mathcal{L}^+G} \simeq \mathrm{Rep}({}^L G)$$

where:

• $\mathrm{Gr}_G = \mathcal{L}G/\mathcal{L}^+G$ is the affine Grassmannian of $G$,

• $\mathrm{DMod}(\mathrm{Gr}_G)^{\mathcal{L}^+G}$ denotes the category of $\mathcal{L}^+G$-equivariant D-modules, with the convolution monoidal structure, and

• $\mathrm{Rep}({}^L G)$ denotes the category of algebraic representations of ${}^L G$.

The construction of the correct commutativity constraint on the left hand side of this equivalence is subtle–we warn that this is an additional data and not merely a property of the monoidal structure.

Related concepts

• geometric Langlands correspondence

• categorification via groupoid schemes– section on geometric Satake

References

• Wikipedia, Satake isomorphism

• Benedict Gross, On the Satake isomorphism (pdf)

• Dennis Gaitsgory, Graduate Seminar. Geometric Representation theory. Fall 2009–Spring 2010, Feb. 23: Affine Grassmannian-II: the Satake equivalence (Ryan Reich) (pdf)

• example 1.7 in these rough notes, apparently taken in some talk by Jacob Lurie: Talk by Jacob Lurie