Pre-Talbot seminar, lecture 3
Sheel Ganatra - The Geometric Satake Correspondence
Let G be a reductive algebraic group, and let G∨ be its Langlands
dual - it has the dual root datum. The theorem for today is that there
is an equivalence of categories
Rep(G∨ ) ∼= PG[[t]] (Gr),where we write Gr for the affine Grassmannian for G. For the first halfof the talk, I will explain the right side, and for the second half, I will
explain the proof.
Affine Grassmannian
There is an object called the “loop group of G,” written LG or G((t)).
It is an ind-scheme, whose complex points are in natural bijection withG(C((t))). There is also a subscheme of “positive loops,” written L+ Gor G[[t]], whose complex points are G(C[[t]]). L+ G acts on LG on theleft and right, by restricting the multiplication maps. We define theaffine Grassmannian Gr = LG/L+ G. This is an ind-scheme.
c× → C× }Example 1: Let G = GL1 = C × . Then G(C((t))) = {f : C
This is the group of invertible
formal Laurent series, and elements
P∞
ican be written f (z) = −∞ ai z . G(C[[t]]) is the group of invertible
formal Taylor series, namely those series supported on non-negative
exponents with nonzero constant term. Then Gr(C) ∼= Z, because we
can uniquely get a monomial representative z n by multiplication.
Example 2: G = T a torus. Then Gr = X∗ (T ) = Hom(C× , T ), i.e.,the coweight lattice.
An interesting exercise is the affine Grassmannian of SL2 .
Structure of Gr: Fix a triangular decomposition T ⊂ B ⊂ G. Pickλ ∈ X∗ (T ). We can associate to λ an element tλ ∈ Gr in the followingway. λ is a map C× → T . We postcompose with T ,→ G and precompose with the completion at zero Spec C((t)) ,→ C× . We get a map
Spec C((t)) → G, i.e., a point teλ ∈ G((t)), and we let tλ := teλ · L+ G.
L+ G acts on Gr on the left, and we define Grλ := L+ G · tλ to be theorbit. This has nice properties:(1) dimC Grλ = 2ρ(λ), for λ dominant integral.
S(2) Grλ = µ≤λ Grµ .
(3) Any point in Gr is in some Grλ .
This gives us a stratification of Gr by L+ G-orbits.
Definition: Let S be a poset. A Whitney stratification of a space X
is a collection of locally finite disjoint subspaces Sα , α ∈ S such that12S(1) α Sα = X.
(2) All Sα are smooth.
(3) Sα ∩ Sβ 6= 0 ⇔ α ≤ b ⇔ Sα ⊂ Sβ .
(4) technical conditions for containments.
We want to look at sheaves that behave nicely with respect to astratification.
• A sheaf is locally constant with respect to S if we can associate
to any path on a stratum an isomorphism on the stalks, suchthat the isomorphism depends only on the homotopy class ofthe path.
• A complex of sheaves is constructible if– The cohomology is locally constant with respect to S.
– The cohomology has finitely generated stalks.
• Dcb (X) is the bounded derived category of constructible sheaves.
Objects are constructible complexes, and morphisms are given
by certain zig-zags.
• The category PS (X) of perverse sheaves is the full subcategory
whose objects are A• ∈ Dcb (X) satisfying:– (support) H k (jα∗ A) = 0 for k > −dimC Sα .
– (cosupport) H k (jα! A) = 0 for k < dimC Sα .
The conditions on perverse sheaves control the failure of transversalityof cycles with substrata. Note that the two conditions are Verdier dual,
so perverse sheaves are self-dual. In fact, the extension of Poincaré
duality to singular spaces was the initial motivation for perversity.
We have the following properties:(1) PS (X) is abelian.
(2) There is a “truncation” functor p H 0 : Dcb (X) → PS (X)(3) Given a stratified map j : (X, S) → (Y, T ), there is a functorpj∗ : PS (X) → PT (X).
p(4) j∗ := p H 0 Rj∗ .
In PS (X), there is a unique “simple object” ICX = p H 0 (C[dimC X]). Itis called the intersection cohomology sheaf. Since each Grλ is stratified,
we have ICGrλ ∈ PS (Grλ ). For j : Grλ ,→ Gr, we define ICλ =pj∗ (ICGrλ ).
Part 2: the proof
We now have a bijection between irreducible representations Vλ of∨G and simple perverse sheaves ICλ on Gr. We’d like to promote thisto an equivalence of tensor categories. To do this, we use the Tannakian
formalism so nicely developed by John last week. [John says, “I think
Deligne might have played a bigger role.”] If we can construct a faithful3

e for some
exact tensor functor PS (Gr) → V ect⊗ , then PS (Gr) ∼= Rep Ge Then we show that G
e is reductive, and identify its root datum withG.
∨that of G .
We construct the tensor product in two ways. The first is manifestly
associative, the second is manifestly commutative, and it turns out thatthey coincide.
The convolution tensor structure is given by the following diagram:
G[[t]]pm

Gr × Gr ← G((t)) × Gr → G((t)) × Gr → Gr
Given a sheaf F  G on Gr × Gr, we pull it back to p∗ (F  G) on
G((t))×Gr. This sheaf is G[[t]]×G[[t]]-equivariant, where the first copyof G[[t]] acts on the first factor on the left, and the second copy acts
by left multiplication on the second factor and by right-inverse on thefirst factor. Since the action of the second copy of G[[t]] is free, there isG[[t]]e on G((t)) × Gr. Thea unique G[[t]]-equivariant perverse sheaf F G
elast map m gives us F ∗ G := Rm∗ (F G).
One can show that this isperverse by using the fact that m is a stratified semi-small map. Thisis a technical condition that amounts to counting dimensions.
The fusion tensor structure is given by a global construction due toBeilinson and Drinfeld. We fix a point x on a smooth complex curve
cx , and its field of fractions is Kx .
X. The completed local ring is Ocx ∼Choosing a coordinate at x gives isomorphisms O= Spec C[[t]] and∼Kx = Spec C((t)). We define
Grx := {G-bundles on X, with a trivialization away from x}
Beilinson and Drinfeld showed that we can make this into a family
GrX → X, where GrX parametrizes a point x ∈ X, a G-bundle on
X, and a trivialization of that G-bundle away from x. In fact, we canmake a family over X 2 or X n by choosing more points, so
GrX 2 = {(x1 , x2 ), G-bundle, trivialization on X \ {x1 , x2 }}When X = A1 , the fiber over a diagonal point is just Gr, and thefiber away from the diagonal is Gr × Gr, and the families are trivial
on or away from the diagonal. We have a family of groups
Gx,Ocx/ GX,O
/Xx

4and we can consider PGx,O (GrX ) ∼= PL+ G (Gr). We have a diagramji

U ,→ X × X ←- X, where U is the complement of the diagonal. Givena GX,O × GX,O |U -equivariant sheaf F  G on U , i∗ (j∗! (F  G)) lives inPS (Gr). This is the fusion tensor product, and it is isomorphic to theconvolution tensor product.
Now, we need a tensor functor to V ect. This is given by global
cohomology H(Gr, −). This is a fiber functor, i.e., it is faithful andrespects the tensor product. By the Tannakian formalism, we get
e for some affine algebraic group G.
ePS (Gr) ∼= Rep G
Given a reductive group, we can identify its root data by the weightdecomposition of its irreducibles. We would like to decompose ICλ ina similar way. This is done using MV cycles, which arise from “semiinfinite orbits.” For µ ∈ X∗ (T ), we define Sµ = N ((t))tµ ⊂ Gr, where
N is the unipotent radical of our chosen Borel subgroup B.
Theorem (Mirkovic, Vilonen)
and only if µ appears in the weight• Sµ ∩ Grλ is nonempty ifL
decomposition of Vλ =α Vα , and in this case, it has pure
dimension hρ,Lλ − µi.
• H(Gr, A) = µ H 2ρ(µ) (Sµ , A) for A ∈ PL+ G (Gr). If A = ICλ
Lfor some dominant integral weight λ, then this is α H 2ρ(µ) (Sµ ∩Grα , ICλ ).
Therefore, H(Gr, ICλ ) is a free module generated by the irreducible
components of Sα ∩ Grλ .