Langlands

David Nadler

Our goal is to extend our work down the sequence

G reductive group/ℂ  problem  numbers, rings, …, categories

We want to describe numbers, rings, etc. in terms of combinatorics. A solution should be a Chevalley description such as root data.

The geometric Satake is a geometric construction of the duual group G^{∨}. Consider the affine Grassmannian

Gr_{G} = G(ℂ((t)))/G(ℂ[[t]]) =: G(K)/G(O),

an infinite-dimensional variety. Note that G(ℂ((t))) is the loop group and G(ℂ[[t]]) is the arc group. We have the Cartan decomposition

G(O)\Gr_{G} ≃ Λ_{G}/W_{G} ≃ Λ_{G}^{+} ≃^{!} Λ̌_{G^{∨}}^{+},

where ≃ denotes equivalence, Λ_{G}^{+} is a set of dominant weights for G, and Λ̌_{G^{∨}}^{+} is a set of dominant weights for the dual group G^{∨}. The combinatorics of the representation theory of G^{∨}, denoted Rep(G^{∨}), is encoded in

convolution  ⟷  ⊗
MV cycles  ⟷  weight multiplicities

All of this can be encoded as a categorical statement—namely, that the perverse sheaves Perv_{G(O)}(Gr_{G}) are equivalent, at least as abelian categories, to the representations of G^{∨} Rep(G^{∨}):

Perv_{G(O)}(Gr_{G}) ≃ Rep(G^{∨})

as abelian categories. What goes in the ellipses above are abelian categories.

Example. Let G = GL_{1}.

Then Gr_{G} ≅ ℤ as a topological space.

punctured disk  G = ℂ^{×}

Since we only need to worry about the order of the pole,

⋯  •  • −1  •0  •1  •2  •3  •  ⋯

we assign a (finite-dimensional) vector space to each point.

Perv_{GL_{1}(O)}(Gr_{G}) ≃ ℤ − graded vector spaces ≃^{!} Rep(G)

But we’re supposed to be doing G(O)-equivariant geometry on Gr_{G}. In other words, we should be doing geometry on G(O)\Gr_{G} (note that as a set, G(O)\Gr_{G} is Λ_{G}^{+}). What should replace the ℤ-graded vector spaces? For GL_{1}, it should be GL_{1}(O)\ℤ (stack, groupoid, “Artin orbifold,” …). To a topologist who has never met an algebraic geometer, GL_{1}(O) = ℂ^{×} ≃ S^{1}, so

GL_{1}(O)\•  =  S^{1}\•  ≃  S^{1}\S^{∞}  =  ℂP^{∞} 0

What is ℂP^{∞}?

• ℂP^{0} ℂP^{1}  ℂP^{2}^{...} ℂP^{∞}

We replace ℤ with

⋯  • ℂP^{∞}  • ℂP^{∞}  • ℂP^{∞}  • ℂP^{∞}  • ℂP^{∞} ⋯

A vector space is a ℂ-module. We had

•  algebra ℂ = H^{∗}(pt)  category of ℂ-modules

In the new picture,

ℂP^{∞}  H^{∗}(ℂP^{∞}) = ℂ[u] deg u = 2  category of differential graded ℂ[u]-modules

We thus have a new Satake category: differential graded (dg) ℂ[u]-modules (two gradings). What is the spectral description of this? In other words, what should we consider instead of Rep(GL_{1})? Earlier, we had Rep(GL_{1}). But now we have Λ = ℂ ⊕ ℂ · λ_{−1} (= H_{∗} (S^{1})), where deg λ_{−1} = −1 and λ_{−1}^{2} = 0. Put ℂ[u] =: 𝕊. graded (dg) Λ-modules

( · ) ( · ) ( · ) ( · ) ( · )

The “derived” geometric Satake

graded (dg) Λ-modules ≃ graded (dg) 𝕊-modules
Λ ↦ trivial module
trivial module ↤ 𝕊

Note that the above equivalence is equivalence of categories.

In general (Drinfeld: “The Exercise”):

D_{G(O)}(Gr_{G}) ≃ G^{∨}-equivariant dg Λ(𝔤^{∨}[1])-modules,

where D_{G(O)}(Gr_{G}) is the set of all G(O)-equivariant sheaves on the Grassmannian.

Remarks.

1. This is a local statement.

2. The degree of the set of all G^{∨}-equivariant dg Λ(𝔤^{∨}[1])-modules is −1.

3. In the case of GL_{1},

(a) D_{G(O)}(Gr_{G}) is 𝕊,

(b) the set of all G^{∨}-equivariant dg Λ(𝔤^{∨}[1])-modules is Λ.

Where is Λ coming from? The role of the geometric Satake in the geometric Langlands:

geometric Langlands  harmonic analysis on locally symmetric spaces homological mirror symmetry (topological field theory)

Locally symmetric space: Let C be a Riemann surface

× × ×  ×  × ×

and Bun_{G}(C) the space of all principle G-bundles on C (when G = GL_{1}, then Bun_{G}(C) is the space of all line bundles on C). Adelic description (Weil): remove enough points so that the G-bundle is trivial.

∏_{p∈C} G(O_{p})\ ∏_{p∈C} G(K_{p})/G(ℂ(C)) = Bun_{G}(C)

Remarks.

1. G(K_{p}) is the loop group at p

2. Bun_{G}(C) is a global object

In terms of points, the above is

gauge on disks\gluing data/gauge on C generic.

Now we want to do harmonic analysis. Let the operator K\G/K act on K\G/Γ. Then for each p ∈ C,

G(O_{p})\G(K_{p})/G(O_{p}).

What is the conjectured spectral space for Bun_{G}(C) with the Satake action?

•_{p} G(O_{p})\G(K_{p})/G(O_{p})

eigenvalue: Rep(G^{∨}) ⟶ vector spaces

V ⟼ P ×_{G^{∨}} V

(P a principle G^{∨}-bundle on the point)

The spectral space is Conn_{G^{∨}}(C), the set of all G^{∨}-connections on C.

Conn_{G^{∨}}(C)

• potential eigenvalue

Finally, what is the Rep(G^{∨})-action on this side? Also, where does Λ for GL_{1} come from? \underline{Recall}: The Satake action on Bun_{G}(ℂ) at a point p ∈ C was “regluing.” Let’s mimic this for connections.

\begin{aligned}& • \\ & p \\ ℂ \end{aligned} •

•

𝔻 double punctured disk

Consider Conn_{G^{∨}}(𝔻).

•  • + • −  • p_{+}  p_{−}

Conn_{G^{∨}}(D)  Conn_{G^{∨}}(D) Conn_{G^{∨}}(𝔻) p_{+}^{∗}  p_{−}^{∗}

Thus,

Conn_{G^{∨}}(𝔻) = {0} ⋂_{𝔤^{∨}} {0} = Spec (ℂ_{0} ⨂_{Sym(𝔤)}^{derived}ℂ_{0}) = Spec (Λ^{−∗} 𝔤^{∨}) ,

where {0} ∩_{𝔤^{∨}} {0} is the trivial connection and Sym(𝔤) denotes the symmetric algebra. Bridgeland: bases for categories, mutations