Sam Raskin: Introduction to GLC

Setup:

• $X/\mathbb{k}$ a smooth projective curve over $\mathbb{k}$ (geometrically connected).
• Fix a “sheaf theory” $\operatorname{Lisse}(\mathscr{Y})\subseteq \operatorname{Shv}(\mathscr{Y})$ for any prestack $\mathscr{Y}$ over $\mathbb{k}$.

Geometric class field theory

Construction: for $\sigma$ rank 1 local system on $X$, one can construct $\chi_ {\sigma}\in \operatorname{Shv}(\operatorname{Bun}_ {\mathbb{G}_ {m}})$ with some properties, and $\chi_ {\sigma}$ is the prototypical “eigensheaf”.

The stack $\operatorname{Bun}_ {\mathbb{G}_ {m}}$ = moduli stack of line bundles on $X$, which is non-canonically isomorphic to $\operatorname{Jac}(X)\times B\mathbb{G}_ {m}\times \mathbb{Z}$. Let $$\operatorname{Bun}_ {\mathbb{G}_ {m}}^{n}=\{\mathscr{L}:\operatorname{deg}(\mathscr{L})=n\},$$ then $$\operatorname{Bun}_ {\mathbb{G}_ {m}}\cong \sqcup_ {n}\operatorname{Bun}_ {\mathbb{G}_ {m}}^{n}.$$

What do we want?

• normalization: Fixed isomorphism $\chi_ {\sigma}|_ {\text{triv}}\cong e$ our field of coefficients for the chosen sheaf theory.
• $\chi_ {\sigma}|_ {X}$ along $X\xrightarrow{\operatorname{AJ}}\operatorname{Bun}_ {\mathbb{G}_ {m}}$ (sending point $x$ to $\mathscr{O}(x)$), one obtains $\sigma$.

Betti setting

In Betti setting $\mathbb{k}=\mathbb{C}$, then $\operatorname{Jac}(X)\cong H^{1}(X,\mathscr{O})/H^{1}(X,\mathbb{Z})$ as complex manifolds. Then choose $x_ {0}\in X$, one has $$\operatorname{AJ}_ {x_ {0}}: X\rightarrow \operatorname{Jac}(X)$$ sending $$x\mapsto \mathscr{O}(x-x_ {0}).$$ This gives $$\pi_ {1}(X)\rightarrow \pi_ {1}(\operatorname{Jac}(X)) = H^{1}(X,\mathbb{Z}),$$ which factors as $$\pi_ {1}(X)^{\text{ab}}\rightarrow \pi_ {1}(\operatorname{Jac}(X)) = H^{1}(X,\mathbb{Z}).$$ Then can check that $$\pi_ {1}(X)^{\text{ab}} = H_ {1}(X,\mathbb{Z}) \rightarrow \pi_ {1}(\operatorname{Jac}(X))= H^{1}(X,\mathbb{Z})$$ is given by Poincare duality. Then $$\sigma: \pi_ {1}(X)^{\text{ab}}\rightarrow e$$ gives $$\pi_ {1}(\operatorname{Jac}(X))\rightarrow e$$ and thus gives $\chi_ {\sigma}.$

For any $x\in X$, one has Hecke action $$H_ {x}: \operatorname{Bun}_ {\mathbb{G}_ {m}}\xrightarrow{\cong} \operatorname{Bun}_ {\mathbb{G}_ {m}},$$ by $\mathscr{L}\mapsto \mathscr{L}(x).$ Then one has a map \begin{align} H_ {X}: X\times \operatorname{Bun}_ {\mathbb{G}_ {m}}&\rightarrow \operatorname{Bun}_ {\mathbb{G}_ {m}}, \\ (x,\mathscr{L}) & \mapsto \mathscr{L}(x). \end{align}

Then $H_ {X}^{\ast}(\chi_ {\sigma})\cong \sigma\boxtimes \chi_ {\sigma}\in \operatorname{Shv}(X\times \operatorname{Bun}_ {\mathbb{G}_ {m}})$ and has compatibilities (Hecke property).

Idea: there exists at least one such $\chi_ {\sigma}$ for trivial reasons. hecke property says : $\chi_ {\sigma}|_ {\mathscr{L}(x)} \cong \sigma_ {x}\otimes \chi|_ {\mathscr{L}}.$ In particular, for $\mathscr{L}\cong \mathscr{O}(D)$, where $D=\sum_ {i}n_ {i}x_ {i}$ a divisor, then $\chi_ {\sigma}|_ {\mathscr{L}}\cong \otimes_ {i}\sigma_ {x_ {i}}^{\otimes n_ {i}}$, i.e. $\chi_ {\sigma}$ is “overdetermined”.

Souped up version: let $$\operatorname{Sym}^{n}X=\{\text{effective divisors on $X$ of degree $n$}\}=\{\mathscr{L}+s\in \Gamma(\mathscr{L})-\{0\},\text{deg}(\mathscr{L})=n\}.$$

Then one has a map $p_ {n}: \operatorname{Sym}^{n}X\rightarrow \operatorname{Bun}_ {\mathbb{G}_ {m}}^{n}$ and the fiber at $\mathscr{L}$ is $\Gamma(\mathscr{L})-\{0\}$.

Let $\sigma^{(n)}\in \operatorname{Shv}(\operatorname{Sym}^{n}X)$ as following: we have a map $X^{n}\xrightarrow{\text{add}}\operatorname{Sym}^{n}X$ and $\text{add}_ {\ast}(\sigma\boxtimes\cdots\boxtimes \sigma)$ is acted by $S_ {n}$ and we define $\sigma^{(n)}$ to be its $S_ {n}$-invariants. Namely, $\sigma^{(n)}$ has fiber $\otimes \sigma_ {x_ {i}}^{\otimes n_ {i}}$ at $D=\sum_ {i}n_ {i}x_ {i}$.

By desiderata: $$p_ {n}^{\ast}(\chi_ {\sigma})\cong \sigma^{(n)}$$ compatible with Hecke property in a natural sense. Claim: there exists $\leq 1$ such $\chi_ {\sigma}$.

Idea: $\chi_ {\sigma}$ is determined by $\chi_ {\sigma}|_ {\operatorname{Bun}_ {\mathbb{G}_ {m}}}^{\geq n}$ for any $n$ with Hecke property. For $n>\!>0$ (actually $n>2g-2$ suffices), by Riemann-Roch, $$\operatorname{Sym}^{n}X\rightarrow \operatorname{Bun}_ {\mathbb{G}_ {m}}^{n}$$ is smooth and surjective with simply connected fibers and then $\sigma^{(n)}$ descends to $\operatorname{Bun}_ {\mathbb{G}_ {m}}^{(n)}$.

Therefore, $p_ {n}^{\ast}(\chi_ {\sigma})$ determines $\chi_ {\sigma}|_ {\operatorname{Bun}_ {\mathbb{G}_ {m}}^{n}}$. Thusly one obtains the uniqueness. The existence is proved by descent argument above.

This is geometric Langlands for $\operatorname{GL}_ {1}$.

From now on, we mainly focus on the case of $\operatorname{PGL}_ {2}$

Geometry of $Bun_ {G}$ and Constant term

A $\operatorname{PGL}_ {2}$-bundle is same as a vector bundle of rank $2$ well-defined up to $-\otimes\mathscr{L}$. Then $\operatorname{Bun}_ {\operatorname{PGL}_ {2}}$ is the moduli stack of $\operatorname{PGL}_ {2}$-bundles on $X$.

Let $\mathscr{E}$ be a vector bundle of rank 2, and then $\text{deg}(\mathscr{E})\in \mathbb{Z}$ and $\text{deg}(\mathscr{E}\otimes \mathscr{L}) = \text{deg}(\mathscr{E})+2\text{deg}(\mathscr{L})$. Therefore, $\text{deg}(\mathscr{P}) \operatorname{mod} 2$ is well-defined for any $\mathscr{P}\in \operatorname{Bun}_ {\mathrm{PGL}_ {2}}$.

Then we have $$\operatorname{Bun}_ {\operatorname{PGL}_ {2}} =\operatorname{Bun}_ {\operatorname{PGL}_ {2}}^{\text{even}}\sqcup \operatorname{Bun}_ {\operatorname{PGL}_ {2}}^{\text{odd}}.$$

Idea: motivated by geometric class field theory and Langlands program, given an irreducible $\operatorname{SL}_ {2}$-local system $\sigma$ on $X$, one desires $\mathscr{F}_ {\sigma}\in \operatorname{Shv}(\operatorname{Bun}_ {\operatorname{PGL}_ {2}})$ canonically.

Warning: $\mathscr{F}_ {\sigma}$ is not a locally system, but is an irreducible (on each connected component) perverse sheaf.

Let $B=\begin{pmatrix} \ast & \ast \\ 0 & \ast \end{pmatrix}\subseteq \operatorname{PGL}_ {2}$ and $T=\mathbb{G}_ {m}$ the Cartan. Then $\operatorname{Bun}_ {B}$ classifies $$0\rightarrow \mathscr{L} \rightarrow \mathscr{E} \rightarrow \mathscr{O}\rightarrow 0$$

and substack $\operatorname{Bun}_ {B}^{n}$ classifies such extensions with $\text{deg}(\mathscr{L})=n$. Let $$\operatorname{Bun}_ {G}\xleftarrow {p}\operatorname{Bun}_ {B}\xrightarrow{q} \operatorname{Bun}_ {T}.$$

Idea: $\operatorname{CT}_ {\ast}$ is an analogue of Jacquet functor.

Philosophy: cuspidal corresponds to irreduciblility on spectral side.

So we expect $\mathscr{F}_ {\sigma}$ should be cuspidal.

Claim: for $n>1$, the map $\operatorname{Bun}_ {G}^{n}\rightarrow \operatorname{Bun}_ {\operatorname{PGL}_ {2}}$ is a locally closed immersion with disjoint images. Let $\mathrm{Bun}_ {G}^{\text{ss}}$ be the semi-stable bundles, i.e. bundles that are not in the image of these maps.

Over $\mathbb{P}^{1}$, every rank 2 vector bundle is isomorphic to some $\mathscr{O}(m)\oplus \mathscr{O}(n)$, then up to $-\otimes\mathscr{L}$, one can write a representative of $\operatorname{PGL}_ {2}$-bundle as $\mathscr{O}(n)\oplus \mathscr{O}$. As $n$ gets larger, the automorphism group gets larger. So we see that $\text{dim}\operatorname{Bun}_ {B}^{n}$ gets smaller as $n$ gets larger.

Now in this case $\operatorname{CT}_ {\ast} = \bigoplus_ {n\in \mathbb{Z}}\operatorname{CT}_ {\ast}^{n}$. There is a correspondence $$\mathrm{Bun}_ {G}\xleftarrow {p_ {n}}\operatorname{Bun}_ {B}^{n}\xrightarrow{q_ {n}}\operatorname{Bun}_ {\mathbb{G}_ {m}}^{n}.$$ For $n>2g-2$, any extension $0\rightarrow \mathscr{L}\rightarrow \mathscr{E}\rightarrow \mathscr{O}\rightarrow 0$ splits since $H^{1}(\mathscr{L})=0$ for $\text{deg}(\mathscr{L})=n$. Therefore, the fiber of $q_ {n}$ at $\mathscr{L}$ is just $\mathbb{B}H^{0}(\mathscr{L})$.

Reminder: $\operatorname{Shv}(\mathbb{G}_ {a}) = \operatorname{Shv}(\text{pt})$.

This is to say that $\mathscr{F}$ vanishes in $!$-sense at $\infty$.

We can also consider $\operatorname{CT}_ {!}=\oplus \operatorname{CT}_ {!}^{n}$ defined by $\operatorname{CT}_ {!}^{n}=q_ {n,!}p_ {n}^{\ast}$.

The main ingredient of the proof is Braden’s hyperbolic localization.

Then $\operatorname{CT}_ {\ast}^{-n}(\mathscr{F})=0$ implies that $\operatorname{CT}_ {!}(\mathscr{F})=0$.

That is to say $\mathscr{F}$ vanishes in $\ast$-sense as well.

Therefore, $\mathscr{F}$ is cleanly extended from $\operatorname{Bun}_ {G}^{\leq 2g-2}$.

How to construct $\mathscr{F}$?

Digression: Inspiration from number theory.

Take $\rho: W_ {\mathbb{Q}}\rightarrow \rightarrow \operatorname{SL}_ {2}(\mathbb{C})$ unramified irreducible, where $W_ {\mathbb{Q}}$ is the Weil group, and then there exists some $f_ {\sigma}(q)=\sum_ {n\geq 0}a_ {n}q^{n}$ modular form attached to $\sigma$, $q=\operatorname{exp}(2\pi \sqrt{-1}\tau)$, $\operatorname{Im}(\tau)>0$. There is a symmetry $f(\tau)=(\text{some factor})\cdot f(\frac{-1}{\tau})$.

How to construct $f_ {\sigma}$?

• $a_ {0}=0$ (cuspidality).
• $a_ {nm}=a_ {n}a_ {m}$ for $n,m$ coprime.
• For $p$ prime, $\operatorname{Fr}_ {p}\in W_ {\mathbb{Q}}$, then $a_ {p}=\operatorname{tr}(\sigma(\operatorname{Fr}_ {p}))$ and $a_ {p^{n+1}} = a_ {p} a_ {p^{n}} - p^{\text{archimedean factor}}\cdot a_ {p^{n-1}}$ for any $n\geq 1$.

Langlands says in this case: $f_ {\sigma}$ is modular.

Definition of $\operatorname{coeff}_ {D}$

Goal: Write $\mathscr{F}_ {\sigma}$ in a similar way.

Analogy: $$\boxed{\mathbb{C}\xleftarrow {f\mapsto a_ {0}(f)}\{\text{modular forms}\} \xrightarrow{f\mapsto a_ {n},n\geq 1} \mathbb{C}} \Leftrightarrow \boxed{\operatorname{Shv}(\operatorname{Bun}_ {\mathbb{G}_ {m}})\xleftarrow {\operatorname{CT}_ {\ast}} \operatorname{Shv}(\operatorname{Bun}_ {G})\xrightarrow{\text{coeff}_ {D}} \operatorname{Vect}}.$$

Note that $\{n\geq 1\}\cong \{p_ {1}^{r_ {1}}\cdots p_ {n}^{r_ {n}}\} = \operatorname{Div}^{\text{eff}}(\operatorname{Spec}\mathbb{Z})$. We should expect that our Fourier coeffecients to be indexed by effective divisors $D$ on $X$.

For $D=0$, we construct $\text{coeff}=\text{coeff}_ {0}$ as following. Let $$\operatorname{Bun}_ {\mathbb{G}_ {a}}^{\Omega} = \{0\rightarrow \Omega \rightarrow \mathscr{E}\rightarrow \mathscr{O}\}$$ and consider the correspondence $$\operatorname{Bun}_ {G}\xleftarrow{p}\operatorname{Bun}_ {\mathbb{G}_ {a}}^{\Omega}\xrightarrow{\psi}\mathbb{A}^{1}=H^{1}(X,\Omega).$$ Then we define $$\operatorname{coeff}(\mathscr{F}) = C^{\bullet}(\operatorname{Bun}_ {\mathbb{G}_ {a}}^{\Omega}, p^{!}\mathscr{F}\otimes^{!} \psi^{!}\operatorname{exp}),$$ where $\operatorname{exp}(\mathbb{A}^{1})$ is the exp $\mathscr{D}$-module or AS sheaf or Kir.

This is analogous to $f\mapsto \int_ {\mathbb{R}/\mathbb{Z}} f(\tau)\text{exp}(-2\pi\sqrt{-1}\tau)d\tau.$

More generally, for $D\geq 0$, consider $$\operatorname{Bun}_ {G}\xleftarrow{p_ {D}}\operatorname{Bun}_ {\mathbb{G}_ {a}}^{\Omega(-D)}\xrightarrow{\psi_ {D}}\mathbb{A}^{1}=H^{1}(X,\Omega),$$ where $\operatorname{Bun}_ {\mathbb{G}_ {a}}^{\Omega(-D)} = \{0\rightarrow \Omega(-D)\rightarrow \mathscr{E}\rightarrow \mathscr{O}\rightarrow 0\}$ and $\psi_ {D}:\operatorname{Bun}_ {\mathbb{G}_ {a}}^{\Omega(-D)}\rightarrow \operatorname{Bun}_ {\mathbb{G}_ {a}}^{\Omega}\rightarrow \mathbb{A}^{1}.$

Then one similarly defines $\operatorname{coeff}_ {D}:=C^{\bullet}(p_ {D}^{!}(-)\otimes^{!}\psi_ {D}^{!}\operatorname{exp})$.

Variant: for $d\geq 0$, $$\operatorname{coeff}_ {d}:\operatorname{Shv}(\operatorname{Bun}_ {G})\rightarrow \operatorname{Shv}(\operatorname{Sym}^{d}X)$$ inverse version of the above in the sense $$\operatorname{coeff}_ {d}(\mathscr{F})|_ {D}^{!} = \operatorname{coeff}_ {D}(\mathscr{F}).$$

Expectation for $\mathscr{F}_ {\sigma}$:

• $\operatorname{coeff}(\mathscr{F}_ {\sigma}) = e$ (our coeffecient field).
• More generally, $\operatorname{coeff}_ {d}(\mathscr{F}_ {\sigma}) \cong \sigma^{(d)}$, where $\sigma^{(d)}\in \operatorname{Shv}(\operatorname{Sym}^{d}(X))$ is $(\operatorname{add}_ {\ast}(\sigma\boxtimes\cdots\boxtimes\sigma))^{S_ {n}}$, whose fiber at $D=\sum r_ {i}x_ {i}$ is $\otimes_ {i} \operatorname{Sym}^{r_ {i}}\sigma_ {x_ {i}}$.

Question: Do the coeffecients of $\mathscr{F}$ determine $\mathscr{F}$?

Examples: If $\mathscr{F}=e_ {\operatorname{Bun}_ {G}}$ the constant sheaf, then $\operatorname{coeff}_ {d}(\mathscr{F})=0$ for any $d$.

What about cuspidal sheaves?

• Sort of yes: $\operatorname{coeff}_ {d}(\mathscr{F})$ and $\operatorname{coeff}_ {d+1}(\mathscr{F})$ for some $d >\!>0$ uniquely determines cuspidal perverse $\mathscr{F}$.
• Sort of no: $\oplus_ {d\geq 0}\operatorname{coeff}_ {d}:\operatorname{Shv}(\operatorname{Bun}_ {G})_ {\text{cusp}}\rightarrow \prod \operatorname{Shv}(\operatorname{Sym}^{d}(X))$ is not fully faithful for trivial reasons.

Recap: $\sigma$ is an irreducible $\operatorname{SL}_ {2}$-local system on $X$, we want $\mathscr{F}_ {\sigma}$ on $\mathrm{Bun}_ {G}$ for $G=\mathrm{PGL}_ {2}$ cuspidal irreducible perverse sheaf (on each connected component), We have constructed $$\operatorname{coeff}_ {d}: \mathrm{Shv}(\mathrm{Bun}_ {G})\rightarrow \mathrm{Shv}(\mathrm{Sym}^{d}X)$$ analoguous to pullback along Abel-Jacobi and to $q$-expansion.

Motivated by number theory, $\operatorname{coeff_ {d}}(\mathscr{F}_ {\sigma}) = \sigma^{(d)}\in \mathrm{Shv}(\mathrm{Sym}^{d}X)$. Now we want to explain why this pins down $\mathscr{F}_ {\sigma}$.

Another description of $\operatorname{coeff}_ {d}$ (special to $\mathrm{GL}_ {n}$) and Uniqueness of Hecke Eigensheaf

Express $\operatorname{coeff}_ {d}$ by Fourier transform. Recall that we have Fourier-Deligne tranform, we have $\mathrm{Shv}(V)\cong \mathrm{Shv}(V^{\vee})$ using $\operatorname{pair}^{!}(\operatorname{exp})$ as a Kernel where $V\times V^{\vee}\xrightarrow{\operatorname{pair}} \mathbb{A}^{1}$.

Ditto for vector bundle over some base. We have

Fix $d\geq 0$ large enough.

Dual bundle $E^{\vee}=\{0\rightarrow \Omega\rightarrow \mathscr{E}\mathscr{L}\rightarrow 0:\mathscr{L}\in \mathrm{Bun}_ {\mathbb{G}_ {m}}^{d}\} \cong \mathrm{Bun}_ {B}^{-d+2g-2}.$

Then $\mathrm{Bun}_ {\mathbb{G}_ {m}}^{d}\xrightarrow{0} E=\{\mathscr{L}\text{ deg }d,s\in \Gamma(\mathscr{L})\}\xleftarrow {j\text{ open}} \{\mathscr{L}, \text{non-zero }(s\in\Gamma(\mathscr{L}))\}\cong \operatorname{Sym}^{d}.$

Claim: $\text{coeff}_ {d}(\mathscr{F})$ is calculatable via

1. $!$-pullback of $\mathscr{F}$ along $p$,
2. Fourier transform,
3. restrict along $j$.

Geometry of $\operatorname{Bun}_ {G}$:

1. $n>1$, $\mathrm{Bun}_ {B}^{n}\rightarrow \mathrm{Bun}_ {G}$ is locally closed (HN-strata),
2. $n<\!<0$, $\operatorname{Bun}_ {B}^{n}\rightarrow \operatorname{Bun}_ {G}$ is smooth (tangent space calculation + Riemann-Roch).

Claim: we can revover $\mathscr{F}_ {\sigma}$ (up to even vs odd) from $\operatorname{coeff}_ {d}(\mathscr{F}_ {\sigma})$ for $d>\!>0$.

Idea:

1. We can recover $\mathscr{F}_ {\sigma}$ from $p^{!}_ {-d+2g-2}(\mathscr{F}_ {\sigma})$, this is a general act about perverse sheaves and smooth maps with connected fibers and the fact that $p_ {-d+2g-2}$ is smooth for $d>\!>0.$
2. $\operatorname{FT}$ is an equivalence of categories, so we recover $\mathscr{F}_ {\sigma}$ from $\mathrm{FT}(p^{!}_ {-d+2g-2}\mathscr{F}_ {\sigma})$.
3. This Fourier transform is cleanly extended from $\operatorname{Sym}^{d}(X)$, therefore, we can recover $\mathscr{F}_ {\sigma}$ from $j^{!}\mathrm{FT}(p^{!}_ {-d+2g-2}\mathscr{F}_ {\sigma}).$ This is because $i^{!}\mathrm{FT}\cong \ast-\text{pushforward of }p^{!}_ {-d+2g-2}(\mathscr{F})$ along $q$, which is exactly $\mathrm{CT}_ {\ast}^{-d+2g-2}(\mathscr{F}_ {\sigma})=0.$

Existence of Hecke eigensheaf

One can reverse the procedure: first clean extension to $E$, and then Fourier transform, and then descent to $\mathrm{Bun}_ {G}$. The hard part is to show descent.

Here we consider $\operatorname{PGL}_ {2}$ instead of $\operatorname{GL}_ {2}$ to ignore determinant twist. The main procedure and areguments are the same. For general $n$, it needs more techniques.

Generalized recently in work of Arinkin-Beraldo-Compbell-Chen-Faergeman-Gaitsgory-Lin-Raskin-Rozenblyum.

Now let $G$ be a split reductive group over $\mathbb{k}$ with $\operatorname{char}\mathbb{k}=0$. Let $\check{G}$ be its dual group. Let $\sigma$ be an irreducible local system for $\check{G}$ ($\sigma$ does not factor proper parabolic).

Part 3 implies that the generic ran of $\mathscr{F}_ {\sigma}$ is $\prod_ {d_ {i}\text{ exp of }G}d_ {i}^{(g-1)(2d_ {i}-1)}.$

A crucial difficulty is that no Fourier transform for general split reductive group.