# The Theta Correspondence

In Siegel modular forms, we mentioned that one could construct Siegel modular forms from elliptic modular forms (see Modular Forms) via a process called “lifting”. In this post, we discuss a more general method that produces new automorphic forms (which generalize modular forms, and are also more properly a part of representation theory, see also Automorphic Forms) out of old ones. There is also a local version that deals with representations of p-adic Lie groups. Both of these form the theory of the (global and local) theta correspondence.

We begin with the local theory. Let $F$ be a nonarchimedean local field of characteristic zero (e.g. $\mathbb{Q}_{p}$ or a finite extension of $\mathbb{Q}_{p}$). Let $E$ be a quadratic etale $F$-algebra. Let $V$ be a vector space over $E$ equipped with a Hermitian form $\langle-,-\rangle_{V}$, and let $W$ be a vector space over $E$ equipped with a skew-Hermitian form $\langle -,-\rangle_{W}$. Their respective groups of isometries are the unitary groups $\mathrm{U}(V)$ and $\mathrm{U}(W)$. These two groups form an example of a reductive dual pair. The theory of the local theta correspondence relates representations of one of these groups to representations of the other.

Now the tensor product $V\otimes_{E} W$ can be viewed as a vector space over $F$ and we can equip it with a symplectic form $(-,-)=\mathrm{Tr}(\langle-,-\rangle_{V}\otimes\langle -,-\rangle_{W})$. We have a map

$\displaystyle \mathrm{U}(V)\times\mathrm{U}(W)\to\mathrm{Sp}(V\otimes_{E} W)$

The symplectic group $\mathrm{Sp}(V\otimes_{E} W)$ has a cover called the metaplectic group; we will describe it in more detail shortly, but we say now that the importance of it for our purposes is that it has a special representation called the Weil representation, which as we shall shortly see will be useful in relating representations of $\mathrm{U}(V)$ to $\mathrm{U}(W)$, and vice-versa.

We first need to construct the Heisenberg group $H(V\otimes_{E} W)$. Its elements are given by $(V\otimes_{E} W)\oplus F$, and we give it the group structure

$\displaystyle (x_{1},t_{1})\cdot (x_{2},t_{2})=\left(x_{1}+x_{2},t_{1}+t_{2}+\frac{1}{2}(x_{1},x_{2})\right)$

The Stone-von Neumann theorem tells us that, for every nontrivial character $\psi:F\to\mathbb{C}^{\times}$ the Heisenberg group has a unique irreducible representation $\omega_{\psi}$ with central character $\psi$. Furthermore, the representation $\omega_{\psi}$ is unitary.

If $V\otimes_{E} W=X\oplus Y$ is a Lagrangian decomposition, we can realize the representation $\omega_{\psi}$ on the vector space of Schwarz functions on either $X$ or $Y$. Let us take it to be $Y$. In particular, we can express $\omega_{\psi}$ as follows. We first extend the character $\psi$ to $H(X)$ (defined to be the subgroup $X\oplus F$ of $H(V\otimes_{E}W)$) and then define $\omega_{\psi}$ as the induced representation (see also The Local Langlands Correspondence for General Linear Groups for another example of an induced representation)

$\displaystyle \omega_{\psi}=\mathrm{c-Ind}_{H(X)}^{H(V\otimes_{E}W)}\psi$

The symplectic group $\mathrm{Sp}(V\otimes_{E} W)$ acts on the Heisenberg group $H(V\otimes_{E}W)$ by $g\cdot (x,t)=(g\cdot x,t)$ for $g\in \mathrm{Sp}(V\otimes_{E} W)$ and $(x,t)\in H(V\otimes_{E}W)$. We can compose this action with the representation $\omega_{\psi}$ to get another representation ${}^{g}\omega_{\psi}=\omega_{\psi}\circ g^{-1}$ of $H(W)$. Now since the action of $\mathrm{Sp}(V\otimes_{E} W)$ on $H(V\otimes_{E}W)$ has trivial center, the central characters of ${}^{g}\omega_{\psi}$ and $\omega_{\psi}$ are the same. By the Stone-von Neumann theorem, ${}^{g}\omega_{\psi}$ and $\omega_{\psi}$ have to be isomorphic.

What this means now, is that for every $g\in \mathrm{Sp}(V\otimes_{E}W)$, we have a linear transformation $A_{\psi}(g)$ of the underlying vector space $\mathcal{S}$ of the representation $\omega_{\psi}$, so that

$\displaystyle A_{\psi}(g)\circ {}^{g}\omega_{\psi}=\omega_{\psi}\circ A_{\psi}(g)$

This action however is only defined up to a factor of $\mathbb{C}^{\times}$. Since $\omega_{\psi}$ is unitary, we can also require $A_{\psi}$ to be unitary, and so the action becomes well-defined up to $S^{1}$. All in all, this means that we have a representation

$\displaystyle A_{\psi}:\mathrm{Sp}(V\otimes W)\to \mathrm{GL}(\mathcal{S})/S^{1}$

Now if we pull back the map $\mathrm{GL}(\mathcal{S})\to\mathrm{GL}(\mathcal{S})/S^{1}$ by the map $A_{\psi}:\mathrm{Sp}(V\otimes_{E} W)\to \mathrm{GL}(\mathcal{S})/S^{1}$, we get a map $\widetilde{A}_{\psi}:\mathrm{Mp}(V\otimes_{E} W)\to \mathrm{GL}(\mathcal{S})$, where the group $\mathrm{Mp}(V\otimes_{E} W)$ is an $S^{1}$-cover of $\mathrm{Sp}(V\otimes_{E} W)$. This group $\mathrm{Mp}(V\otimes_{E} W)$ is the metaplectic group mentioned earlier.

Our construction allows us to extend the representation $\omega_{\psi}$ of $H(V\otimes_{E }W)$ to the semidirect product $\mathrm{Mp}(V\otimes_{E} W)\ltimes H(V\otimes_{E} W)$. This representation of $\mathrm{Mp}(V\otimes_{E} W)\ltimes H(V\otimes_{E} W)$ is called the Heisenberg-Weil representation. The representation of $\mathrm{Mp}(V\otimes_{E} W)$ obtained by restriction is called the Weil representation.

Recall that we have a map $\mathrm{U}(V)\times \mathrm{U}(W)\to \mathrm{Sp}(V\otimes_{E}W)$. If we could lift this to a map $\mathrm{U}(V)\times \mathrm{U}(W)\to \mathrm{Mp}(V\otimes_{E}W)$, then we could obtain a representation of $\mathrm{U}(V)\times \mathrm{U}(W)$ by restricting the Weil representation $\omega_{\psi}$ from $\mathrm{Mp}(V\otimes_{E}W)$ to $\mathrm{U}(V)\times \mathrm{U}(W)$. It turns out such a lifting can be defined and is determined by a pair $(\chi_{V},\chi_{W})$ of characters of $E^{\times}$ satisfying certain conditions. Once we have this lifting, we denote the resulting representation of $\mathrm{U}(V)\times \mathrm{U}(W)$ by $\Omega$.

Now let $\pi$ be an irreducible representation of $V$. We consider the maximal $\pi$-isotypic quotient of $\Omega$, which is its quotient by the intersection of all the kernels of morphisms of representations of $U(V)$ from $\Omega$ to $\pi$. This quotient is of the form $\pi\otimes\theta(\pi)$, where $\Theta(\pi)$ is a representation of $U(W)$ called the big theta lift of $\pi$. The maximal semisimple quotient of $\Theta(\pi)$ is denoted $\theta(\pi)$, and is called the small theta lift of $\pi$.

Let us now look at the global picture. Let $k$ be a number field and let $k_{v}$ be the completion of $k$ at one of its places $v$. Let $E$ be a quadratic extension of $k$. Now we let $V$ and $W$ be vector spaces over $E$ equipped with Hermitian and skew-Hermitian forms $\langle-,-\rangle_{B}$ and $\langle--\rangle_{W}$, as in the local case, and consider the tensor product $V\otimes_{E} W$ as a vector space over $k$, and equip it with the symplectic form $\mathrm{Tr}(\langle-,-\rangle_{V}\otimes\langle-,-\rangle_{W})$. We have localizations $(V\otimes_{E} W)_{v}$ for every $v$, and we have already seen that in this case we can construct the metaplectic group $\mathrm{Mp}((V\otimes_{E} W)_{v})$. We want to put each of these together for every $v$ to construct an “adelic” metaplectic group.

First we take the restricted product $\prod_{v}'\mathrm{Mp}((V\otimes_{E} W)_{v})$. “Restricted” means that all but finitely many of the factors in this product belong to the hyperspecial maximal compact subgroup $K_{v}$ of $\mathrm{Sp}((V\otimes_{E} W)_{v})$, which is also a compact open subgroup of $\mathrm{Mp}((V\otimes_{E} W)_{v})$. This restricted product contains $\bigoplus_{v}S^{1}$ as a central subgroup. Now if we quotient out the restricted product $\prod_{v}'\mathrm{Mp}((V\otimes_{E} W)_{v})$ by the central subgroup $Z$ given by the set of all $(z_{v})\in\bigoplus_{v}S^{1}$ such that $\prod_{v}z_{v}=1$, the resulting quotient is the “adelic” metaplectic group $\mathrm{Mp}(V\otimes_{E} W)(\mathbb{A})$ that we are looking for.

We have a representation $\bigotimes_{v}'\omega_{\psi_{v}}$ of $\prod_{v}'\mathrm{Mp}((V\otimes_{E} W)_{v})$ which acts trivially on the central subgroup $Z$ defined above and therefore gives us a representation $\omega_{\psi}$ of $\mathrm{Mp}(V\otimes_{E} W)(\mathbb{A})$.

What is the underlying vector space of the representation $\omega_{\psi}$? If $V\otimes_{E}W=X\oplus Y$ is a Lagrangian decomposition, we have seen that we can realize the local Weil representation $\omega_{\psi_{v}}$ on $\mathcal{S}(Y_{v})$, the vector space of Schwarz functions of $Y_{v}$ (the corresponding localization of $Y$). Likewise we can also realize the global Weil representation $\omega_{\psi}$ as functions on the vector space $\mathcal{S}(Y_{\mathbb{A}})$, defined to be the restricted product $\bigotimes'\mathcal{S}(Y_{v})$.

So now we have the global Weil representation $\omega_{\psi}$, which is a representation of the group $\mathrm{Mp}(V\otimes_{E}W)(\mathbb{A})$ on the vector space $\mathcal{S}(Y_{\mathbb{A}})$. But suppose we want an automorphic representation, i.e. one realized on the vector space of automorphic forms for $\mathrm{Mp}(V\otimes_{E}W)(\mathbb{A})$ (recall that one of our motivations in this post is to “lift” automorphic forms from one group to another). This is accomplished by the formation of theta functions $\theta(f)(g)$, so-called because it is a generalization of the Jacobi theta function discussed in Sums of squares and the Jacobi theta function. Let $f$ be a vector in the underlying vector space of the Weil representation. Then the theta function $\theta(f)(g)$ is obtained by summing the evaluations of the output of the action of Weil representation on $f$ over all rational points $y\in Y(k)$:

$\displaystyle \theta(f)(g)=\sum_{y\in Y(k)}(\omega_{\psi}(g)\cdot f)(y)$

Now suppose we have a pair of characters $\chi_{1},\chi_{2}$ of $E^{\times}$, so that we have a lifting of $U(V)(\mathbb{A})\times U(W)(\mathbb{A})$ to $\mathrm{Mp}(V\otimes_{E}W)(\mathbb{A})$. This lifting sends $U(V)(k)\times U(W)(k)$ to $\mathrm{Mp}(V\otimes_{E}W)(k)$, which means that we can consider $\theta(f)(g)$ as an automorphic form for $U(V)(\mathbb{A})\times U(W)(\mathbb{A})$.

Now we can perform our lifting. Let $f$ be a cuspidal automorphic form for $U(V)$, and let $\varphi$ be a vector in the underlying vector space of the Weil representation. We can now obtain an automorphic form $\theta(\varphi,f)(g)$ on $U(W)$ as follows:

$\displaystyle \theta(\varphi,f)(g)=\int_{[\mathrm{U}(V)]}\theta(\varphi)(g,h)\cdot \overline{f(h)}dh$

The space generated in this way, for all vectors $f$ in a cuspidal automorphic representation $\pi$ of $U(V)$, and all vectors $\varphi$ in the in the underlying vector space of the Weil representation, is called the global theta lift of $\pi$, denoted $\Theta(\pi)$. It is an automorphic representation of $U(W)$.

There is also an analogue of all that we discussed for the reductive dual pair $\mathrm{SO}(V)\times \mathrm{Sp}(W)$ when $V$ and $W$ are vector space over some field, equipped with a quadratic form and symplectic form respectively.

Many cases of “lifting”, for instance the Saito-Kurokawa lift from elliptic modular forms to Siegel modular forms, can be considered special cases of the global theta lift (in particular for the reductive dual pair $\mathrm{SO}(V)\times \mathrm{Sp}(W)$). The theory of theta lifting is itself part of the theory of Langlands functoriality (see also Trace Formulas). More aspects and examples of the theta correspondence will be discussed in future posts on this blog.

References:

Theta correspondence on Wikipedia

Heisenberg group on Wikipedia

Metaplectic group on Wikipedia

Saito-Kurokawa lift on Wikipedia

Automorphic forms and the theta correspondence by Wee Teck Gan

A brief survey of the theta correspondence by Dipendra Prasad

Non-tempered Arthur packets of G2 by Wee Teck Gan and Nadia Gurevich

A quaternionic Saito-Kurokawa lift and cusp forms on G2 by Aaron Pollack

# The Geometrization of the Local Langlands Correspondence

In The Global Langlands Correspondence for Function Fields over a Finite Field, we introduced the global Langlands correspondence for function fields over a finite field, and Vincent Lafforgue’s work on the automorphic to Galois direction of the correspondence. In this post we will discuss the work of Laurent Fargues and Peter Scholze which uses similar ideas but applies it to the local Langlands correspondence (and this time it works not only for “equal characteristic” cases like Laurent series fields $\mathbb{F}_{q}((t))$ but also for “mixed characteristic” cases like finite extensions of $\mathbb{Q}_{p}$). Note that instead of having complex coefficients like in The Local Langlands Correspondence for General Linear Groups, here we will use $\ell$-adic coefficients.

### I. The Fargues-Fontaine Curve

Let us briefly discuss the idea of “geometrization” and what is meant by Fargues and Scholze making use of V. Lafforgue’s work. Recall that V. Lafforgue’s work concerns the global Langlands correspondence for function fields over a finite field $\mathbb{F}_{q}$, which on one side concerns the space of cuspidal automorphic forms, which are certain functions on $\mathrm{Bun}_{G}(\mathbb{F}_{q})$, which in turn parametrizes $G$-bundles on some curve $X$ over $\mathbb{F}_{q}$, and on the other side concerns representations (or more precisely L-parameters) of the etale fundamental group of $X$ (which can also be phrased in terms of the Galois group of its function field).

Perhaps the first question that comes to mind is, what is the analogue of the curve $X$ in the case of the local Langlands correspondence when the field is not a function field (or more correctly a power series field, since it has to be local) over $\mathbb{F}_{q}$, but some finite extension of $\mathbb{Q}_{p}$? Let $E$ be this finite extension of $\mathbb{Q}_{p}$. Since the absolute Galois group of $E$ is also the etale fundamental group of $\mathrm{Spec}(E)$, perhaps we should take $\mathrm{Spec}(E)$ to be our analogue of $X$.

However, in the traditional formulation of the local Langlands correspondence, it is the Weil group that appears instead of the absolute Galois group itself. Considering the theory of the Weil group in Weil-Deligne Representations, this means that we will actually want $\pi_{1}(\mathrm{Spec}(\breve{E})/\mathrm{Frob}^{\mathbb{Z}})$, where $\breve{E}$ is the maximal unramified extension of $E$ and $\mathrm{Frob}$ is the Frobenius, instead of $\pi_{1}(E)$.

Now, we want to “relativize” this. For instance, in The Global Langlands Correspondence for Function Fields over a Finite Field, we considered $\mathrm{Bun}_{G}(\mathbb{F}_{q})$, which parametrizes $G$-bundles on the curve $X$ over $\mathbb{F}_{q}$. But we may also want to consider say $\mathrm{Bun}_{G}(R)$, where $R$ is some $\mathbb{F}_{q}$-algebra; this would parametrize $G$-bundles on $X\times_{\mathrm{Spec}(\mathbb{F}_{q})}\mathrm{Spec}(R)$ instead. In fact, we need this “relativization” to properly define $\mathrm{Bun}_{G}$ as a stack (see also Algebraic Spaces and Stacks).

The problem with transporting this to the case of $E$ a finite extension of $\mathbb{Q}_{p}$ is that we do not have an “base” like $\mathbb{F}_{q}$ was for the function field case (unless perhaps if we have something like an appropriate version of the titular object in The Field with One Element, which is at the moment unavailable). The solution to this is provided by the theory of adic spaces and perfectoid spaces (see also Adic Spaces and Perfectoid Spaces).

For motivation, let us consider first the case where our field is $\mathbb{F}_{q}((t))$. Let $S=\mathrm{Spa}(R,R^{+})$ be a perfectoid space over $\overline{\mathbb{F}}_{q}$ with pseudouniformizer $\varpi$. Consider the product $S\times_{\mathrm{Spa}(\mathbb{F}_{q})}\mathrm{Spa}(\mathbb{F}_{q}((t)))$. We may look at this as the punctured open unit disc over $S$. It sits inside $\mathrm{Spa}(R^{+})\times_{\mathrm{Spa}(\mathbb{F}_{q})}\mathrm{Spa}(\mathbb{F}_{q}[[t]])$ as the locus where the pseudo-uniformizer $\pi$ of $R$ and the uniformizer $t$ of $\mathbb{F}_{q}[[t]]$ is invertible (or “nonzero”).

In the case where our field is $E$, a finite extension of $\mathbb{Q}_{p}$, as mentioned earlier we have no “base” like $\mathbb{F}_{q}$ was for $\mathbb{F}_{q}((t))$. So we cannot form the fiber products analogous to $S\times_{\mathrm{Spa}(\mathbb{F}_{q})}\mathrm{Spa}(\mathbb{F}_{q}((t)))$ or $\mathrm{Spa}(R^{+})\times_{\mathrm{Spa}(\mathbb{F}_{q})}\mathrm{Spa}(\mathbb{F}_{q}[[t]])$. However, notice that

$\displaystyle \mathrm{Spa}(R^{+})\times_{\mathrm{Spa}(\mathbb{F}_{q})}\mathrm{Spa}(\mathbb{F}_{q}[[t]])\cong \mathrm{Spa}(R^{+}[[t]])$.

This has an analogue in the mixed-characteristic, given by the theory of Witt vectors (compare, for instance $\mathbb{F}_{p}[[t]]$ and its “mixed-characteristic analogue” $\mathbb{Z}_{p}=W(\mathbb{F}_{p})$)! If $\kappa$ is the residue field of $\mathcal{O}_{E}$, we define the ramified Witt vectors $W_{\mathcal{O}_{E}}(R^{+})$ to be $W(R^{+})\otimes_{W(\kappa)}\mathcal{O}_{E})$. This is the analogue of $\mathrm{Spa}(R^{+})\times_{\mathrm{Spa}(\mathbb{F}_{q})}\mathrm{Spa}(\mathbb{F}_{q}[[t]])$. Now all we have to do to find the analogue of $S\times_{\mathrm{Spa}(\mathbb{F}_{q})}\mathrm{Spa}(\mathbb{F}_{q}((t)))$ that we are looking for is to define it as the locus in $W_{\mathcal{O}_{E}}(R^{+})$ where both the uniformizer $\varpi$ of $R^{+}$ and the uniformizer $\pi$ of $\mathcal{O}_{E}$ are invertible!

We denote this locus by $Y_{S}$. But recall again our discussion earlier, that due to the local Langlands correspondence being phrased in terms of the Weil group, we have to quotient out by the powers of Frobenius. Therefore we define the Fargues-Fontaine curve $X_{S}$ to be $Y_{S}/\mathrm{Frob}^{\mathbb{Z}}$.

Aside from our purpose of geometrizing the local Langlands correspondence, the Fargues-Fontaine curve $X_{S}$ is in itself a very interesting mathematical object. For instance, when $S$ is a complete algebraically closed nonarchimedean field over $\mathbb{F}_{q}$, the classical points of $X_{S}$ (i.e. maximal ideals of the rings $B$ such that $X_{S}$ is locally $\mathrm{Spa}(B,B^{+})$) correspond to untilts of $S$ (modulo the action of Frobenius)!

There is also a similar notion for more general $S$. To explain this we need the concept of diamonds, which will also be very important for the rest of the post. A diamond is a pro-etale sheaf on the category of perfectoid spaces over $\mathbb{F}_{p}$, which is the quotient of some perfectoid space $X$ over $\mathrm{Spa}(\mathbb{F}_{p})$ by a pro-etale equivalence relation $R \subset X\times X$ (we also say that the diamond is a coequalizer). An example of a diamond is given by $\mathrm{Spd}(\mathbb{Q}_{p})$. Note that $\mathbb{Q}_{p}$ is not perfectoid, but is the quotient of a perfectoid field we denoted $\mathbb{Q}_{p}^{\mathrm{cycl}}$ in Adic Spaces and Perfectoid Spaces by the action of $\mathbb{Z}_{p}^{\times}$. Now we can take the tilt $(\mathbb{Q}_{p}^{\mathrm{cycl}})^{\flat}$ and quotient out by $\underline{\mathbb{Z}_{p}}^{\times}$ (the underline notation will be explained later – for now we think of this as making the group $\mathbb{Z}_{p}^{\times}$ into a perfectoid space) – this is the diamond $\mathrm{Spd}(\mathbb{Q}_{p})$. More generally, if $X$ is an adic space over $\mathrm{Spa}(\mathbb{Z}_{p})$ satisfying certain conditions (“analytic”), we can define the diamond $X^{\diamond}$ to be such that $X^{\diamond}(S)$, for $S$ a perfectoid space over $\mathrm{Spa}(\mathbb{F}_{p})$, is the set of isomorphism classes of pairs $(S^{\#},S^{\#}\to X)$, $S^{\#}$ being the untilt of $S$. If $X=\mathrm{Spa}(R,R^{+})$, we also use $\mathrm{Spd}(R)$ to denote $X^{\diamond}$. Note that if $X$ is already perfectoid, $X^{\diamond}$ is just the same thing as the tilt $X^{\flat}$.

Now recall that $Y_{S}$ was defined to be the locus in $W_{\mathcal{O}_{E}}(S)$ where the uniformizer $\varpi$ of $S$ and the uniformizer $\pi$ of $E$ were invertible. We actually have that $Y_{S}^{\diamond}=S\times \mathrm{Spd}(E)$, and, for the Fargues-Fontaine curve $X_{S}$, we have that $X_{S}^{\diamond}=S\times \mathrm{Spd}(E)/(\mathrm{Frob}^{\mathbb{Z}}\times\mathrm{id})$.

Our generalization of the statement that the points of $X_{S}$ parametrize untilts of $S$ is now as follows. There exists a three-way bijection between sections of the map $Y^{\diamond}\to S$, maps $S\to\mathrm{Spd}(E)$, and untilts $S^{\#}$ over $E$ of $S$. Given such an untilt $S^{\#}$, this defines a closed Cartier divisor on $Y_{S}$, which in turn gives rise to a closed Cartier divisor on $X_{S}$. By the bijection mentioned earlier, these closed Cartier divisors on $X$ will be parametrized by maps $S\to \mathrm{Spd}(E)/\mathrm{Frob}^{\mathbb{Z}}$.

The closed Cartier divisors that arise in this way will be referred to as closed Cartier divisors of degree $1$. We have seen that they are parametrized by the following moduli space we denote by $\mathrm{Div}^{1}$ (this will also become important later on):

$\mathrm{Div}^{1}=\mathrm{Spd}(E)/\mathrm{Frob}^{\mathbb{Z}}$

Now that we have discussed the Fargues-Fontaine curve $X_{S}$ and some of its properties, we can define $\mathrm{Bun}_{G}$ as the stack that assigns to any perfectoid space $S$ over $\overline{\mathbb{F}}_{q}$ the groupoid of $G$-bundles on $X_{S}$.

When $G=\mathrm{GL}_{n}$, our $G$-bundles are just vector bundles. In this case we shall also denote $\mathrm{Bun}_{\mathrm{GL}_{n}}$ by $\mathrm{Bun}_{n}$.

### II. Vector Bundles on the Fargues-Fontaine Curve

Let us now try to understand a little bit more about vector bundles on the Fargues-Fontaine curve. They turn out to be related to another important thing in arithmetic geometry – isocrystals – and this will allow us to classify them completely.

Let $\breve{E}$ be the completion of the maximal unramified extension of $E$. Letting $\kappa$ denote the residue field of $\mathcal{O}_{E}$, $\breve{E}$ may also be expressed as the fraction field of $W(\kappa)$. It is equipped with a Frobenius lift $\mathrm{Frob}$. An isocrystal $V$ over $\breve{E}$ is defined to be a vector space over $\breve{E}$ equipped with a $\mathrm{Frob}$-semilinear automorphism.

Given an isocrystal $V$ over $\breve{E}$, we can obtain a vector bundle $\mathcal{E}$ on the Fargues-Fontaine curve $X_{S}$ by defining $\mathcal{E}=(V\times Y_{S})/\mathrm{Frob}^{\mathbb{Z}}$. It turns out all the vector bundles over $X_{S}$ can be obtained in this way!

Now the advantage of relating vector bundles on the Fargues-Fontaine curve to isocrystals is that isocrystals are completely classified via the Dieudonne-Manin classification. This says that the category of isocrystals over $\breve{E}$ is semi-simple (so every object is a direct sum of the simple objects), and the form of the simple objects are completely determined by two integers which are coprime, the rank (i.e. the dimension as an $\breve{F}$-vector space) $n$ which must be positive, and the degree (which determines the form of the $\mathrm{Frob}$-semilinear automorphism) $d$. Since these two integers are coprime and one is positive, there is really only one number that completely determines a simple $\breve{E}$-isocrystal – its slope, defined to be the rational number $d/r$. Therefore we shall also often denote a simple $\breve{E}$-isocrystal as $V(d/n)$. Since isocrystals over $\breve{E}$ and vector bundles on the Fargues-Fontaine curve $X_{S}$ are in bijection, if we have a simple $\breve{E}$-isocrystal $V(d/n)$ we shall denote the corresponding vector bundle by $\mathcal{E}(-d/n)$. More generally, an isocrystal is a direct sum of simple isocrystals and they can have different slopes. If an isocrystal only has one slope, we say that it is semistable (or basic). We use the same terminology for the corresponding vector bundle.

More generally, for more general reductive groups $G$, we have a notion of $G$-isocrystals; this can also be thought of functors from the category of representations of $G$ over $E$ to the category of isocrystals over $\breve{E}$. These are in correspondence with $G$-bundles over the Fargues-Fontaine curve. There is also a notion of semistable or basic for $G$-isocrystals, although its definition involves the Newton invariant (one of two important invariants of a $G$-isocrystal, the other being the Kottwitz invariant).

The set of $G$-isocrystals is denoted $B(G)$ and is also called the Kottwitz set. This set is in fact also in bijection with the equivalence classes in $G(\breve{E})$ under “Frobenius-twisted conjugacy”, i.e. the equivalence relation $g\sim \varphi(y)gy^{-1}$. Given an element $b$ of $B(G)$, we can define the algebraic group $G_{b}$ to be such that the elements of $G_{b}(F)$ are the elements $g$ of $G(\breve{F})$ satisfying the condition $\varphi(g)=bgb^{-1}$. If $b=1$, then $G_{b}=G$.

The groups $G_{b}$ are inner forms of $G$ (see also Reductive Groups Part II: Over More General Fields). More precisely, the $G_{b}$ are the extended pure inner forms of $G$, which are all the inner forms of $G$ if the center of $G$ is connected. Groups which are inner forms of each other are in some way closely related under the local Langlands correspondence – for instance, they have the same Langlands dual group. It has been proposed that these inner forms should really be studied “together” in some way, and we shall see that the use of $\mathrm{Bun}_{G}$ to formulate the local Langlands correspondence provides a realization of this approach.

Let us mention one more important part of arithmetic geometry that vector bundles on the Fargues-Fontaine curve are related to, namely p-divisible groups. A p-divisible group (also known as a Barsotti-Tate group) $G$ is an direct limit of group schemes

$\displaystyle G=\varinjlim_{n} G_{n}=(G_{1}\to G_{2}\to\ldots)$

such that $G_{n}$ is a finite flat commutative group scheme which is $p^{n}$-torsion of order $p^{nh}$ and such that the inclusion $G_{n}\to G_{n+1}$ induces an isomorphism of $G_{n}$ with $G_{n+1}[p^{n}]$ (the kernel of the multiplication by $p^{n}$ map in $G_{n+1}$). The number $h$ is called the height of the p-divisible group.

An example of a p-divisible group is given by $\mu_{\infty}=\varinjlim_{n} \mu_{p^{n}}$. This is a p-divisible group of height $1$. Given an abelian variety of dimension $g$, we can also form a p-divisible group of height $2g$ by taking the direct limit of its $p$-torsion.

We can also obtain p-divisible groups from formal group laws (see also The Lubin-Tate Formal Group Law) by taking the direct limit of its $p^{n}$-torsion. In this case we can then define the dimension of such a p-divisible group to be the dimension of the formal group law it was obtained from. More generally, for any p-divisible group over a complete Noetherian local ring of residue characteristic $p$, the connected component of its identity always comes from a formal group law in this way, and so we can define the dimension of the p-divisible group to be the dimension of this connected component.

Now it turns out p-divisible groups can also be classified by a single number, the slope, defined to be the dimension divided by the height. If the terminology appears suggestive of the classification of isocrystals and vector bundles on the Fargues-Fontaine curve, that’s because it is! Isocrystals (and therefore vector bundles on the Fargues-Fontaine curve) and p-divisible groups are in bijection with each other, at least in the case where the slope is between $0$ and $1$. This is quite important because the cohomology of deformation spaces of p-divisible groups (such as that obtained from the Lubin-Tate group law) have been used to prove the local Langlands correspondence before the work of Fargues and Scholze! We will be revisiting this later.

### III. The Geometry of $\mathrm{Bun}_{G}$

Let us now discuss more about the geometry of $\mathrm{Bun}_{n}$. It happens that $\mathrm{Bun}_{G}$ is a small v-sheaf. A v-sheaf is a sheaf on the category of perfectoid spaces over $\overline{\mathbb{F}}_{q}$ equipped with the v-topology, where the covers of $X$ are any maps $X_{i}\to X$ such that for any quasicompact $U\subset X$ there are finitely many $U_{i}$ which cover $U$. A v-sheaf is small if it admits a surjective map from a perfectoid space. In particular being a small v-sheaf implies that $\mathrm{Bun}_{G}$ has an underlying topological space $\vert \mathrm{Bun}_{G}\vert$. The points of this topological space are going to be in bijection with the elements of the Kottwitz set $B(G)$.

If $G$ is a locally profinite topological group, we define $\underline{G}$ to be the functor from perfectoid spaces over $\mathbb{F}_{q}$ which sends a perfectoid space $S$ over $\mathbb{F}_{q}$ to the set $\mathrm{Hom}_{\mathrm{top}}(\vert S\vert,\vert G\vert)$. We let $[\ast/\underline{G}]$ be the classifying stack of $G$-bundles; this means that we can obtain any $\underline{G}$-bundle on any perfectoid space $S$ over $\mathbb{F}_{q}$ by pulling back a universal $\underline{G}$-bundle on $[\ast/\underline{G}]$.

We write $\vert \mathrm{Bun}_{G}^{\mathrm{ss}}\vert$ for the locus in $\vert \mathrm{Bun}_{G}\vert$ corresponding to the $G$-isocrystals that are semistable. We let $\mathrm{Bun}_{G}^{ss}$ the substack of $\mathrm{Bun}_{G}$ whose underlying topological space is $\vert\mathrm{Bun}_{G}^{\mathrm{ss}}\vert$. It turns out that we have a decomposition

$\displaystyle \mathrm{Bun}_{G}^{\mathrm{ss}}\cong\coprod_{b\in B(G)_{\mathrm{basic}}}[\ast/\underline{G_{b}(E)}]$

More generally, even is $b$ is not basic, we have an inclusion

$\displaystyle j:[\ast/\underline{G_{b}(E)}]\hookrightarrow \mathrm{Bun}_{G}$

Let us now look at some more of the properties of $\mathrm{Bun}_{G}$. In particular, $\mathrm{Bun}_{G}$ satisfies the conditions for an analogue of an Artin stack (see also Algebraic Spaces and Stacks) but with locally spatial diamonds instead of algebraic spaces and schemes.

A diamond $X$ is called a spatial diamond if it is quasicompact quasiseparated, and its underlying topological space $\vert X\vert$ is generated by $\vert U\vert$, where $U$ runs over all sub-diamonds of $X$ which are quasicompact. A diamond is called a locally spatial diamond if it admits an open cover by spatial diamonds.

Now we recall from Algebraic Spaces and Stacks that to be an Artin stack, a stack must have a diagonal that is representable in algebraic spaces, and it has charts which are representable by schemes. It turns out $\mathrm{Bun}_{G}$ satisfies analogous properties – its diagonal is representable in locally spatial diamonds, and it has charts which are representable by locally spatial diamonds.

We can now define a derived category (see also Perverse Sheaves and the Geometric Satake Equivalence) of sheaves on the v-site of $\mathrm{Bun}_{G}$ with coefficients in some $\mathbb{Z}_{\ell}$-algebra $\Lambda$. If $\Lambda$ is torsion (e.g. $\mathbb{F}_{\ell}$ or $\mathbb{Z}/\ell^{n}\mathbb{Z}$), this can be the category $D_{\mathrm{et}}(\mathrm{Bun}_{G},\Lambda)$, which is the subcategory of $D(\mathrm{Bun}_{G,v},\Lambda)$ whose pullback to any strictly disconnected perfectoid space $S$ lands in $D(S_{\mathrm{et}},\Lambda)$ (here the subscripts $v$ and $\mathrm{et}$ denote the v-site and the etale site respectively). If $\Lambda$ is not torsion (e.g. $\mathbb{Z}_{\ell}$ or $\mathbb{Q}_{\ell}$) one needs the notion of solid modules (which was further developed in the work of Clausen and Scholze on condensed mathematics) to construct the right derived category.

If $X$ is a spatial diamond and $j:U\to X$ is a pro-etale map expressible as a limit of etale maps $j_{i}U_{i}\to X$, we can construct the sheaf $\widehat{\mathbb{Z}}[U]$ as the limit $\varprojlim_{i}j_{i!}\widehat{\mathbb{Z}}$. We say that a sheaf $\mathcal{F}$ on $X$ is solid if $\mathcal{F}(U)$ is isomorphic to $\mathrm{Hom}(\widehat{\mathbb{Z}}[U],\mathcal{F})$. We can extend this to small v-stacks – if $X$ is a small v-stack and $\mathrm{F}$ is a v-sheaf on $X$, we say that $\mathcal{F}$ is solid if for every map from a spatial diamond $Y$ to $X$ the pullback of $\mathcal{F}$ to $Y$ coincides with the pullback of a solid sheaf from the quasi-pro-etale site of $Y$. We denote by $D_{\blacksquare}(X,\widehat{\mathbb{Z}})$ the subcategory of $D(X_{v},\widehat{\mathbb{Z}})$ whose objects have cohomology sheaves which are solid. Now if we have a solid $\widehat{\mathbb{Z}}$-algebra $\Lambda$, we can consider $D(X_{v},\Lambda)$ inside $D(X_{v},\widehat{\mathbb{Z}})$, and we denote by $D_{\blacksquare}(X,\Lambda)$ the subcategory of objects of $D(X_{v},\Lambda)$ whose image in $D(X_{v},\widehat{\mathbb{Z}})$ is solid.

This category $D_{\blacksquare}(X,\Lambda)$ is still too big for our purposes. Therefore we cut out a subcategory $D_{\mathrm{lis}}(X,\widehat{\mathbb{Z}})$ as follows. If we have a map of v-stacks $f:X\to Y$, we have a pullback map $f^{*}:D_{\blacksquare}(Y,\Lambda)\to D_{\blacksquare}(X,\Lambda)$. This pullback map has a left-adjoint $f_{\natural}:D_{\blacksquare}(X,\Lambda)\to D_{\blacksquare}(Y,\Lambda)$. We define $D_{\mathrm{lis}}(X,\Lambda)$ to be the smallest triangulated subcategory stable under direct sums that contain $f_{\natural}\Lambda$, for all $f:X\to Y$ which are separated, representable by locally spatial diamonds, and $\ell$-cohomologically smooth. If $\Lambda$ is torsion, then $D_{\mathrm{lis}}(X,\Lambda)$ coincides with $D_{\mathrm{et}}(X,\Lambda)$.

Let $D(G_{b}(E),\Lambda)$ be the derived category of smooth representations of the group $G_{b}(E)$ over $\Lambda$. We have

$\displaystyle D_{\mathrm{lis}}(\ast/\underline{G_{b}(E)},\Lambda)\cong D(G_{b}(E),\Lambda)$

Now taking the pushforward of this derived category of sheaves through the inclusion $j$, and using the isomorphism above, we get

$\displaystyle j_{!}:D(G_{b}(E),\Lambda)\to D_{\mathrm{lis}}(\mathrm{Bun}_{G},\Lambda)$

Now we can see that this derived category $D_{\mathrm{lis}}(\mathrm{Bun}_{G},\Lambda)$ of sheaves on $\mathrm{Bun}_{G}$ encodes the representation theory of $G$, which is one side of the local Langlands correspondence, but more than that, it encodes the representation theory of all the extended pure inner forms of $G$ altogether.

The properties of $\mathrm{Bun}_{G}$ mentioned earlier, in particular its charts which are representable by locally spatial diamonds, allow us to define properties of objects in $D_{\mathrm{lis}}(\mathrm{Bun}_{G},\Lambda)$ which translate into properties of interest in $D(G_{b}(E),\Lambda)$. For example, we have a notion of $D_{\mathrm{lis}}(\mathrm{Bun}_{G},\Lambda)$ being compactly generated, and this translates into a notion of compactness for $D(G_{b}(E),\Lambda)$. We also have a notion of Bernstein-Zelevinsky duality for $D_{\mathrm{lis}}(\mathrm{Bun}_{G},\Lambda)$, which translates into Bernstein-Zelevinsky duality for $D(G_{b}(E),\Lambda)$, and finally, we have a notion of universal local acyclicity in $D_{\mathrm{lis}}(\mathrm{Bun}_{G},\Lambda)$, which translates into being admissible for $D(G_{b}(E),\Lambda)$.

### IV. The Hecke Correspondence and Excursion Operators

Now let us look at how the strategy in The Global Langlands Correspondence for Function Fields over a Finite Field works for our setup. We will be working in the “geometric” setting (i.e. sheaves or complexes of sheaves instead of functions) mentioned at the end of that post, so there will be some differences from the work of Lafforgue that we discussed there, although the motivations and main ideas (e.g. excursion operators) will be somewhat similar.

Just like in The Global Langlands Correspondence for Function Fields over a Finite Field, we will have a Hecke stack $\mathrm{Hck}_{G}$ that parametrizes modifications of $G$-bundles over the Fargues-Fontaine curve. This means that $\mathrm{Hck}_{G}(S)$ is the groupoid of triples $(\mathcal{E},\mathcal{E}',\phi)$ where $\mathcal{E}$ and $\mathcal{E}'$ are $G$-bundles over $X_{S}$ and $\phi_{D_{S}}:\mathcal{E}\vert_{X_{S}\setminus D_{S}}\xrightarrow{\sim}\mathcal{E}'\vert_{X_{S}\setminus D_{S}}$ is an isomorphism of vector bundles meromorphic on some degree $1$ Cartier divisor $D_{S}$ on $X_{S}$ (which is part of the data of the modification). Note that we have maps $h^{\leftarrow}:\mathrm{Hck}_{G}\to\mathrm{Bun}_{G}$ and $h^{\rightarrow}:\mathrm{Hck}_{G}\to\mathrm{Bun}_{G}\times\mathrm{Div}^{1}$ which sends the triple $(\mathcal{E},\mathcal{E}'\phi_{D_{S}})$ to $\mathcal{E}$ and $(\mathcal{E}',D_{S})$ respectively.

Now we need to bound the relative position of the modification. Recall that this is encoded via (conjugacy classes of) cocharacters $\mu:\mathbb{G}_{m}\to G$. The way this is done in this case is via the local Hecke stack $\mathcal{H}\mathrm{ck}_{G}$, which parametrizes modifications of $G$-bundles on the completion of $X_{S}$ along $D_{S}$ (compare the moduli stacks denoted $\mathcal{M}_{I}$ inThe Global Langlands Correspondence for Function Fields over a Finite Field). The local Hecke stack admits a stratification into Schubert cells labeled by conjugacy classes of cocharacters $\mu:\mathbb{G}_{m}\to G$. We can now pull back a Schubert cell $\mathcal{H}\mathrm{ck}_{G,\mu}$ to the global Hecke stack $\mathrm{Hck}_{G}$ to get a substack $\mathrm{Hck}_{G,\mu}$ with maps $h^{\leftarrow,\mu}$ and $h^{\rightarrow,\mu}$, and define a Hecke operator as

$\displaystyle Rh_{*}^{\rightarrow,\mu}h^{\leftarrow,\mu *}:D_{\mathrm{lis}}(\mathrm{Bun}_{G},\Lambda)\to D_{\mathrm{lis}}(\mathrm{Bun}_{G}\times\mathrm{Div}^{1},\Lambda)$

More generally, to consider compositions of Hecke operators we need to consider modifications at multiple points. For this we will need the geometric Satake equivalence.

Let $S$ be an affinoid perfectoid space over $\mathbb{F}_{q}$. For each $i$ in some indexing set $I$, we let $D_{i}$ be a Cartier divisor on $X_{S}$. Let $B^{+}(S)$ be the completion of $\mathcal{O}_{X_{S}}$ along the union of the $D_{i}$, and let $B(S)$ be the localization of $B$ obtained by inverting the $D_{i}$. For our reductive group $G$, we define the positive loop group $LG^{+}$ to be the functor which sends an affinoid perfectoid space $S$ to $G(B^{+}(S))$, and we define the loop group $LG$ to be the functor which sends $S$ to $G(B(S))$.

We define the Beilinson-Drinfeld Grassmannian $\mathrm{Gr}_{G}^{I}$ to be the quotient $LG^{+}/LG$. We further define the local Hecke stack $\mathcal{H}\mathrm{ck}_{G}^{I}$ to be the quotient $LG\backslash\mathrm{Gr}_{G}^{I}$.

The geometric Satake equivalence tells us that the category $\mathrm{Sat}_{G}^{I}(\Lambda)$ of perverse sheaves on $\mathcal{H}\mathrm{ck}_{G}^{I}$ satisfying certain conditions (quasicompact over $\mathrm{Div}^{1})^{I}$, flat over $\Lambda$, universally locally acyclic) is equivalent to the category of representations of $(\widehat{G}\rtimes W_{E})^{I}$ on finite projective $\Lambda$-modules.

Let $V$ be such a representation of representations of $(\widehat{G}\rtimes W_{E})^{I}$. Let $\mathcal{S}_{V}$ be the corresponding object of $\mathrm{Sat}_{G}^{I}(\Lambda)$. The global Hecke stack $\mathrm{Hck}_{G}^{I}$ has a map $q$ to the local Hecke stack $\mathcal{H}\mathrm{ck}_{G}^{I}$. It also has maps $h^{\leftarrow}$ to $h^{\rightarrow}$ to $\mathrm{Bun}_{G}$ and $\mathrm{Bun}_{G}\times(\mathrm{Div}^{1})^{I}$ respectively. We can now define the Hecke operator $T_{V}$ as follows:

$\displaystyle T_{V}=Rh_{*}^{\rightarrow}(h^{\leftarrow *}\otimes_{\Lambda}^{\mathbb{L}}q^{*}\mathcal{S}_{V}):D_{\mathrm{lis}}(\mathrm{Bun}_{G},\Lambda)\to D_{\mathrm{lis}}(\mathrm{Bun}_{G}\times(\mathrm{Div}^{1})^{I},\Lambda)$

Once we have the Hecke operators, we can then consider excursion operators and apply the strategy of Lafforgue discussed in The Global Langlands Correspondence for Function Fields over a Finite Field. We set $\Lambda$ to be $\overline{\mathbb{Q}}_{\ell}$. Let $(I,V,\alpha,\beta,(\gamma_{i})_{i\in I})$ be an excursion datum, i.e. $I$ is a finite set, $V$ is a representation of $(\widehat{G}\rtimes Q)^{I}$, $\alpha:1\to V$, $\beta:V \to 1$, and $\gamma_{i}\in W_{E}$ for all $i\in I$. An excursion operator is the following composition:

$\displaystyle A=T_{1}(A)\xrightarrow{\alpha} T_{V}(A)\xrightarrow{(\gamma_{i})_{i\in I}} T_{V}(A)\xrightarrow{\beta} T_{1}(V)=A$

Now this composition turns out to be the same as multiplication by the scalar determined by the following composition:

$\displaystyle \overline{\mathbb{Q}}_{\ell}\to V\xrightarrow{\varphi(\gamma_{i})_{i\in I}} V\to \overline{\mathbb{Q}}_{\ell}$

And the $\varphi$ that appears here is precisely the L-parameter that we are looking for. This therefore gives us the “automorphic to Galois” direction of the local Langlands correspondence.

### V. Relation to Local Class Field Theory

It is interesting to look at how this all works in the case $G=\mathrm{GL}_{1}$, i.e. local class field theory. There is historical precedent for this in the work of Pierre Deligne for what we might now call the $\mathrm{GL}_{1}$ case of the (geometric) global Langlands correspondence for function fields over a finite field, but which might also be called geometric class field theory.

Let us go back to the setting in The Global Langlands Correspondence for Function Fields over a Finite Field, where we are working over a function field of some curve $X$ over the finite field $\mathbb{F}_{q}$. Since we are considering $G=\mathrm{GL}_{1}$, our $\mathrm{Bun}_{G}$ in this case will be the Picard group $\mathrm{Pic}_{X}$, which parametrizes line bundles on $X$. The statement of the geometric Langlands correspondence in this case is that there is an equivalence of character sheaves on $\mathrm{Pic}_{X}$ (see the discussion of Grothendieck’s sheaves to functions dictionary at the end of The Global Langlands Correspondence for Function Fields over a Finite Field) and $\overline{\mathbb{Z}}_{\ell}$-local systems of rank $1$ on $X$ (these are the same as one-dimensional representations of $\pi_{1}(X)$).

We have an Abel-Jacobi map $\mathrm{AJ}: X\to \mathrm{Pic}_{X}$, sending a point $x$ of $X$ to the corresponding divisor $x$ in $\mathrm{Pic}_{X}$. More generally we can define $\mathrm{AJ}^{d}:X^{(d)}\to\mathrm{Pic}_{X}^{d}$, where $X^{(d)}$ is the quotient of $X^{d}$ by the symmetric group on its factors, and $\mathrm{Pic}_{X}^{d}$ is the degree $d$ part of $\mathrm{Pic}_{X}$.

Now suppose we have a rank $1$ $\overline{\mathbb{Z}}_{\ell}$-local system on $X$, which we shall denote by $\mathcal{F}$. We can form a local system $\mathcal{F}^{\boxtimes d}$ on $X^{d}$. We can push this forward to $X^{(d)}$ and get a sheaf $\mathcal{F}^{(d)}$ on $X^{(d)}$. What we hope for is that this sheaf $\mathcal{F}^{(d)}$ is the pullback of the character sheaf on $\mathrm{Pic}_{X}^{d}$ that we are looking for via $\mathrm{AJ}^{(d)}$. This is in fact what happens, and what makes this possible is that the fibers of $\mathrm{AJ}^{(d)}$ are simply connected for $d>2g-2$, by the Riemann-Roch theorem. So for this $d$, by taking fundamental groups of the fiber sequence, we have that $\pi_{1}(X^{(d)})\cong\pi_{1}(\mathrm{Pic}_{X}^{d})$. So representations of $\pi_{1}(X^{(d)})$ give rise to representations of $\pi_{1}(\mathrm{Pic}_{X}^{d})$, and since representations of the fundamental group are the same as local systems, we see that there must be a local system on $\mathrm{Pic}_{X}^{d}$, and furthermore the sheaf $\mathcal{F}^{(d)}$ is the pullback of this local system. There is then an inductive method to extend this to $d\leq 2g-2$, and we can check that the local system is a character sheaf.

Now let us go back to our case of interest, the local Langlands correspondence. Instead of the curve $X$ we will use $\mathrm{Div}^{1}$, the moduli of degree $1$ Cartier divisors. It will be useful to have an alternate description of $\mathrm{Div}^{1}$ in terms of Banach-Colmez spaces.

For any perfectoid space $T$ over $S$ and any vector bundle $\mathcal{E}$ over $X_{S}$, the Banach-Colmez space $\mathcal{BC}(\mathcal{E})$ is the locally spatial diamond such that $\mathcal{BC}(\mathcal{E})(S)=H^{0}(X_{T},\mathcal{E}\vert_{X_{T}})$. We define $\mathcal{BC}(\mathcal{E})\setminus \lbrace 0\rbrace$ to be such that $\mathcal{BC}(\mathcal{E})\setminus \lbrace 0\rbrace (S)$ are the sections in $H^{0}(X_{T},\mathcal{E}\vert_{X_{T}})$ which are nonzero fiberwise on $S$.

There is a map from $\mathcal{BC}(\mathcal{O}(1))\setminus \lbrace 0\rbrace$ to $\mathrm{Div}^{1}$ which sends a section $f$ to $V(f)$, which in turn induces an isomorphism $(\mathcal{BC}(\mathcal{O}(1))\setminus \lbrace 0\rbrace)/\underline{E^{\times}}\cong \mathrm{Div}^{1}$. A more explicit description of this map is given by Lubin-Tate theory (see also The Lubin-Tate Formal Group Law). After choosing a coordinate, the Lubin-Tate formal group law $\mathcal{G}$ with an action of $\mathcal{O}_{E}$, over $\mathcal{O}_{E}$, is isomorphic to $\mathrm{Spf}(\mathcal{O}_{E}[[x]])$. We can form the universal cover $\widetilde{\mathcal{G}}$ which is isomorphic to $\mathrm{Spf}(\mathcal{O}_{E}[[x^{1/q^{\infty}}]])$. Now let $S=\mathrm{Spa}(R,R^{+})$ be a perfectoid space with tilt $S^{\#}=\mathrm{Spa}(R^{\#},R^{\#+})$. We have $\widetilde{\mathcal{G}}(R^{\#+})=R^{\circ\circ}$, where $R^{\circ\circ}$ is the set of topologically nilpotent elements in $R$, and the map which sends a topologically nilpotent element $x$ to the power series $\sum_{i}\pi^{i}[x^{q^{-i}}]$ gives a map to $H^{0}(Y_{S},\mathcal{O}(1))$, which upon quotienting out by the action of Frobenius gives an isomorphism between $\widetilde{\mathcal{G}}(R^{\#+})$ and $H^{0}(X_{S},\mathcal{O}(1))$.

What this tells us is that $H^{0}(X_{S},\mathcal{O}(1))\cong \mathrm{Spd}(\mathbb{F}[[x^{1/p^{\infty}}]])$. Defining $E_{\infty}$ to be the completion of the union over all $n$ of the $\pi^{n}$-torsion points of $\mathcal{G}$ in $\overline{E}$, we have that $H^{0}(X_{S},\mathcal{O}(1))\cong \mathrm{Spd}(E_{\infty})$. This is an $\underline{\mathcal{O}_{E}^{\times}}$-torsor over $\mathrm{Spd}(E)$, and then quotienting out by the action of Frobenius we obtain our map to $\mathrm{Div}^{1}$.

More generally, we have an isomorphism $(\mathcal{BC}(\mathcal{O}(d))\setminus\lbrace 0\rbrace)/\underline{E^{\times}}\cong \mathrm{Div}^{d}$, where $\mathrm{Div}^{d}$ parametrized degree $d$ relative Cartier divisors on $X_{S,E}$.

Now that we have our description of $\mathrm{Div}^{1}$ (and more generally $\mathrm{Div}^{d}$) in terms of Banach-Colmez spaces, let us now see how we can translate the strategy of Deligne to the local case. Once again we have an Abel-Jacobi map

$\displaystyle \mathrm{AJ}^{d}:\mathcal{BC}(\mathcal{O}(d))\setminus\lbrace 0\rbrace\to\mathrm{Pic}^{d}$

Given a local system on $\mathcal{BC}(\mathcal{O}(d))$, we want to have a character sheaf on $\mathrm{Pic}^{d}$ whose pullback to $\mathcal{BC}(\mathcal{O}(d))$ is precisely this local system. Again what our strategy hinges will be whether $\mathcal{BC}(\mathcal{O}(d))\setminus\lbrace 0\rbrace$ will be simply connected. And in fact this is true for $d\geq 3$, and by a result called Drinfeld’s lemma for diamonds this will actually be enough to prove the local Langlands correspondence for $\mathrm{GL}_{1}$ (i.e. it is not needed for $d<3$ – in fact this is false for $d=1$!). The fact that $\mathcal{BC}(\mathcal{O}(d))\setminus\lbrace 0\rbrace$ is simply connected for $d\geq 3$ is a result of Fargues, and, at least for the characteristic $p$ case, follows from expressing $\mathcal{BC}(\mathcal{O}(d))\setminus\lbrace 0\rbrace=\mathrm{Spa}(\mathbb{F}_{q}[[x_{1}^{1/p^{\infty}},\ldots,x_{d}^{1/p^{\infty}}]])\setminus V(x_{1},\ldots x_{d})$, whose category of etale covers is the same as that of $\mathrm{Spa}(\mathbb{F}_{q}[[x_{1},\ldots,x_{d}]])\setminus V(x_{1},\ldots x_{d})$. Then Zariski-Nagata purity allows one to reduce this to showing that $\mathrm{Spa}(\mathbb{F}_{q}[[x_{1},\ldots,x_{d}]])$ is simply connected, which it is by Hensel’s lemma.

### VI. The Cohomology of Local Shimura Varieties

Many years before the work of Fargues and Scholze, the $\mathrm{GL}_{n}$ case of the local Langlands correspondence (see also The Local Langlands Correspondence for General Linear Groups) was originally proven using the cohomology of the Lubin-Tate tower (which we shall denote by $\mathcal{M}_{\infty}$) which parametrizes deformations of the Lubin-Tate formal group law (see also The Lubin-Tate Formal Group Law) with level structure, together with the cohomology of Shimura varieties. Let us now investigate how the cohomology of the Lubin-Tate tower can be related to what we have just discussed.

It turns out that because of the relationship between Lubin-Tate formal group laws, p-divisible groups, and vector bundles on the Fargues-Fontaine curve, the Lubin-Tate tower is also a moduli space of modifications of vector bundles on the Fargues-Fontaine curve, but of a very specific kind! Namely, it parametrizes modifications where we fix the two vector bundles, and furthermore one has to be the trivial bundle $\mathcal{O}^{n}$ and the other a degree $1$ bundle $\mathcal{O}(1/n)$, and so the only thing that varies is the isomorphism between them (as opposed to the Hecke stack, where the vector bundles can also vary) away from a point. So we see that the Lubin-Tate tower is a part of the Hecke stack (we can think of it as the fiber of the Hecke stack above $(\mathcal{E}_{1},\mathcal{E}_{b})\in \mathrm{Bun}_{G}\times\mathrm{Bun}_{G}$).

More generally, the Lubin-Tate tower is a special case of a local Shimura variety at infinite level, which is itself related to a special case of a moduli stack of local shtukas. These parametrize modifications of $G$-bundles $\mathcal{E}_{1}$ and $\mathcal{E}_{b}$, which are bounded by some cocharacter $\mu:\mathbb{G}_{m}\to G(E)$. This moduli stack of local shtukas, denoted $\mathrm{Sht}_{G,b,\mu,\infty}$, is an inverse limit of locally spatial diamonds $\mathrm{Sht}_{G,b,\mu,K}$ with “level structure” given by some compact open subgroup $K$ of $G(E)$. In the case where the cocharacter $\mu$ is miniscule, the data $(G,b,\mu)$ is called a local Shimura datum, and we define the local Shimura variety at infinite level, denoted $\mathcal{M}_{G,b,\mu,\infty}$, to be such that $\mathrm{Sht}_{G,b,\mu,\infty}=\mathcal{M}_{G,b,\mu,\infty}^{\diamond}$. It is similarly a limit of local Shimura varieties at finite level $K$, denoted $\mathcal{M}_{G,b,\mu,K}$, and for each $K$ we have $\mathrm{Sht}_{G,b,\mu,K}=\mathcal{M}_{G,b,\mu,K}^{\diamond}$.

Let us now see how the cohomology of the moduli stack of local shtukas is related to our setup. We will consider the case of finite level, i.e. $\mathrm{Sht}_{G,b,\mu,K}$, since the cohomology at infinite level may be obtained as a limit. Consider the inclusion $j_{1}:[\ast/\underline{G(E)}]\hookrightarrow \mathrm{Bun}_{G}$. Now consider the object $A=j_{1!}\mathrm{c-ind}_{K}^{G(E)}\mathbb{Z}_{\ell}$ of $D_{\mathrm{lis}}(\mathrm{Bun_{G}},\mathbb{Z}_{\ell})$. Now for our cocharacter $\mu:\mathbb{G}_{m}\to G(E)$, we have a Hecke operator $T_{\mu}$, and we apply this Hecke operator to obtain $T_{\mu}(A)$. Now we pull this back through the inclusion $j_{b}:[\ast/\underline{G_{b}(E)}]\hookrightarrow \mathrm{Bun}_{G}$, to get an object $j_{b}^{*}T_{\mu}(A)$ of $D_{\mathrm{lis}}(\ast/\underline{G_{b}(E)},\mathbb{Z}_{\ell})$. We can think of all this happening not on the entire Hecke stack, but only on $\mathrm{Sht}_{G,b,\mu,K}$, since we are specifically only considering this very special kind of modification parametrized by $\mathrm{Sht}_{G,b,\mu,K}$. But the derived pushforward from $D_{\mathrm{lis}}(\mathrm{Sht}_{G,b,\mu,K})$ to a point gives $R\Gamma(\mathrm{Sht}_{G,b,\mu,K},\mathbb{Z}_{\ell})$ (from which we can compute the cohomology).

This relationship between the cohomology of the moduli stack of local shtukas and sheaves on $\mathrm{Bun}_{G}$, as we have just discussed, has been used to obtain new results. For instance, David Hansen, Tasho Kaletha, and Jared Weinstein used this formulation together with the concept of the categorical trace to prove the Kottwitz conjecture.

Let $\rho$ be a smooth irreducible representation of $G_{b}(E)$ over $\overline{\mathbb{Q}}_{\ell}$. We define

$\displaystyle R\Gamma(G,b,\mu)[\rho]=\varinjlim_{K\subset G(E)}R\mathrm{Hom}(R\Gamma_{c}(\mathrm{Sht}_{G,b,\mu,K},\mathcal{S}_{\mu}),\rho)$

Let $S_{\varphi}$ be the centralizer of $\varphi$ in $\widehat{G}$. Given a representation $\pi$ in the L-packet $\Pi_{\varphi}(G)$ and a representation $\rho$ in the L-packet $\Pi_{\varphi}(G_{b})$, the refined local Langlands correspondence gives us a representation $\delta_{\pi,\rho}$ of $S_{\varphi}$. We let $r_{\mu}$ be the extension of the highest-weight representation of $\widehat{G}$ to ${}^{L}G$. The Kottwitz conjecture states that

$\displaystyle R\Gamma(G,b,\mu)[\rho]=\sum_{\pi\in\Pi_{\varphi}(G)}\pi\boxtimes\mathrm{Hom}_{S_{\varphi}}(\delta_{\pi,\rho},r_{\mu}\circ \varphi)$

The approach of Hansen, Kaletha, and Weinstein involve first using a generalized Jacquet-Langlands transfer operator $T_{b,\mu}^{G\to G_{b}}$. We define the regular semisimple elements in $G$ to be the semisimple elements whose connected centralizer is a maximal torus, and we define the strongly regular semisimple elements to be the regular semisimple elements whose centralizer is connected. We denote their corresponding open subvarieties in $G$ by $G_{\mathrm{rs}}$ and $G_{\mathrm{rs}}$ respectively. The generalized Jacquet-Langlands transfer operator $T_{b,\mu}^{G\to G_{b}}: C(G(E)_{\mathrm{sr}}\sslash G(E))\to C(G_{b}(E)_{\mathrm{sr}}\sslash G(E))$ is defined to be

$\displaystyle [T_{b,\mu}^{G\to G_{b}}f](g')=\sum_{(g,g',\lambda)\in\mathrm{Rel}_{b}}f(g)\dim r_{\mu}[\lambda]$

Here the set $\mathrm{Rel_{b}}$ is the set of all triples $(g,g',\lambda)$ where $g\in G(E)$, $g'\in G_{b}(E)$, and $\lambda$ is a certain specially defined element of $X_{*}(T)$ ($T$ being the centralizer of $g$ in $G$) that depends on $g$ and $g'$. When applied to the Harish-Chandra character $\Theta_{\rho}$, we have

$\displaystyle [T_{b,\mu}^{G\to G_{b}}\Theta_{\rho}](g)=\sum_{\pi\in\Pi_{\varphi}(G)}\dim \mathrm{Hom}_{S_{\varphi}}(\delta_{\pi,\rho},r_{\mu})\Theta_{\pi}(g)$

Next we have to relate this to the cohomology of the moduli stack of local shtukas. We first need the language of distributions. We define

$\mathrm{Dist}(G(E),\Lambda)^{G(E)}:=\mathrm{Hom}_{G(F)}(C_{c}(G(E),\Lambda)\otimes \mathrm{Haar}(G,\Lambda),\Lambda$

To any object $A$ of $D(G(E),\Lambda)$, we can associate an object $\mathrm{tr.dist}(A)$ of $\mathrm{Dist}(G(E),\Lambda)^{G(E)}$. We also have “elliptic” versions of these constructions, i.e. an object $\mathrm{tr.dist}_{\mathrm{ell}}(A)$ of the category $\mathrm{Dist}_{\mathrm{ell}}(G(E),\Lambda)^{G(E)}$. Now we can define the action of the generalized Jacquet-Langlands transfer operator on $\mathrm{Dist}_{\mathrm{ell}}(G(E),\Lambda)^{G(E)}$. The hope will be that we will have the following equality:

$\displaystyle T_{b,\mu}^{G\to G_{b}}\mathrm{tr.dist}_{\mathrm{ell}}\rho=\mathrm{tr.dist}_{\mathrm{ell}}R\Gamma(G,b,\mu)[\rho]$

Proving this equality is where the geometry of $\mathrm{Bun}_{G}$ (and the Hecke stack) and the trace formula come into play. The action of the generalized Jacquet-Langlands transfer operator $\displaystyle T_{b,\mu}^{G\to G_{b}}$ on $\mathrm{tr.dist}_{\mathrm{ell}}\rho$ can be described in a similar way to a Hecke operator where we pull back to the moduli of local Shtukas, multiply by a kernel function, and then push forward.

On the other side, one needs to compute $\mathrm{tr.dist}_{\mathrm{ell}}R\Gamma(G,b,\mu)[\rho]$. Here we use that $R\Gamma(G,b,\mu)[\rho]=h_{\rightarrow *}'j^{*}(q^{*}\mathcal{S}_{\mu}\otimes h_{\leftarrow}^{*}i_{b *}\rho)$. This is a version of the expression of the cohomology of the moduli stack of local shtukas that we previously discussed where $q^{*}\mathcal{S}_{\mu}$ is the pullback to the Hecke stack of the sheaf corresponding to $\mu$ provided by the geometric Satake equivalence and before pushing forward via $h_{\rightarrow}$ we are pulling back to the degree $1$ part of the Hecke stack, which is why we have $j^{*}$ (the embedding of this degree $1$ part) and $h_{\rightarrow}'$ denotes that we are pushing forward from this degree $1$ part.

Hansen, Kaletha, and Weinstein then apply a categorical version of the Lefschetz-Verdier trace formula (using a framework developed by Qing Lu and Weizhe Zheng) to be able to relate $\mathrm{tr.dist}_{\mathrm{ell}}R\Gamma(G,b,\mu)[\rho]=\mathrm{tr.dist}_{\mathrm{ell}}h_{\rightarrow *}'j^{*}(q^{*}\mathcal{S}_{\mu}\otimes h_{\leftarrow}^{*}i_{b *}\rho)$ to $\displaystyle T_{b,\mu}^{G\to G_{b}}\mathrm{tr.dist}_{\mathrm{ell}}\rho$.

Let us discuss briefly the setting of this categorical trace. We consider a category $\mathrm{CoCorr}$ whose objects are pairs $(X,A)$ where $X$ is an Artin v-stack over $\ast$ and $A\in D_{et}(X,\Lambda)$. The morphisms in this category are given by a pair of maps $c_{1},c_{2}:C\to X$ where $c_{2}$ is smooth-locally representable in diamonds, together with a map $u:c_{1}^{*}A\to c_{2}^{!}A$. We also write $c$ for the pair $(c_{1},c_{2})$. Given an endomorphism $f:(X,A)\to (X,A)$ the categorical trace of $f$ is given by $(\mathrm{Fix}(c),\omega)$ where $\mathrm{Fix}(c)$ is the pullback of $c:C\to X\times X$ and $\Delta_{X}: X\to X\times X$ and $\omega\in H^{0}(\mathrm{Fix}(c),K_{X})$ (here $K_{X}$ is the dualizing sheaf, which may obtained as the right-derived pullback of $\Lambda$ via the structure morphism of $X$). In the special case where the correspondence $c$ arises form an automorphism $g$ of $X$, and $g^{*}A=A$, then one may think of $\mathrm{Fix}(c)$ as the fixed points of $g$ and the categorical trace gives an element of $\Lambda$ (the local term) for each fixed point.

For Hansen, Kaletha, and Weinstein’s application, they consider $f$ to be the identity. The categorical trace is then given by $(\mathrm{In}(X),\mathrm{cc}_{X}(A))$, where $\mathrm{In}(X)=X\times_{X\times X}X$ is the inertia stack, classifying pairs $(x,g)$ with $g$ an automorphism of $x$, and $\mathrm{cc}_{X}(A)\in H^{0}(\mathrm{In}(X),K_{\mathrm{In}(X)})$ is called the characteristic class.

The idea now is that certain properties of the setting we are considering (such as universal local acyclicity) allow us to identify the trace distribution $\mathrm{tr.dist}_{\mathrm{ell}}h_{\rightarrow *}'j^{*}(q^{*}\mathcal{S}_{\mu}\otimes h_{\leftarrow}^{*}i_{b *}\rho)$ as a characteristic class $\mathrm{cc}_{\mathrm{Bun}_{G}^{1}}(h_{\rightarrow *}'j^{*}(q^{*}\mathcal{S}_{\mu}\otimes h_{\leftarrow}^{*}i_{b *}\rho))$. From there we can use properties of the abstract theory to relate it to $\displaystyle T_{b,\mu}^{G\to G_{b}}\mathrm{tr.dist}_{\mathrm{ell}}\rho$ (for instance, we can use a Kunneth formula for the characteristic class to decouple the parts involving $\rho$ and $\mathcal{S}_{\mu}$, and relate the former to pulling back to the moduli stack of local shtukas, and relate the part involving the latter to multiplication by the kernel function).

### VII. The Spectral Action

We have seen that the machinery of excursion operators gives us the automorphic to Galois direction of the local Langlands correspondence. We now describe one possible approach to obtain the other (Galois to automorphic) direction. We are going to use the language of the categorical geometric Langlands correspondence mentioned at the end of in The Global Langlands Correspondence for Function Fields over a Finite Field.

Recall our construction of the moduli stack of local $\ell$-adic Galois representations in Moduli Stacks of Galois Representations. Using the same strategy we can construct a moduli stack of L-parameters, which we shall denote by $Z^{1}(W_{E},\widehat{G})_{\Lambda}/\widehat{G}$. This notation comes from the fact that in Fargues and Scholze’s work the L-parameters can be viewed as 1-cocycles.

Let $D(\mathrm{Bun}_{G},\Lambda)^{\omega}$ denote the subcategory of compact objects in $D(\mathrm{Bun}_{G},\Lambda)^{\omega}$. The categorical local Langlands correspondence in this case is the following conjectural equivalence of categories:

$\displaystyle D(\mathrm{Bun}_{G},\Lambda)^{\omega}\cong D_{\mathrm{coh},\mathrm{Nilp}}^{b,\mathrm{qc}}(Z^{1}(W_{E},\widehat{G})_{\Lambda}/\widehat{G})$

Here the right-hand side is the derived category of bounded complexes on $Z^{1}(W_{E},\widehat{G}$ with quasicompact support, coherent cohomology, and nilpotent singular support. We will leave the definition of these terms to the references, but we will think of $D_{\mathrm{coh},\mathrm{Nilp}}^{b,\mathrm{qc}}(Z^{1}(W_{E},\widehat{G})_{\Lambda}/\widehat{G})$ as being a derived category of coherent sheaves on $Z^{1}(W_{E},\widehat{G})$.

We now outline an approach to proving the categorical local Langlands correspondence. Let $\mathrm{Perf}(Z^{1}(W_{E},\widehat{G})_{\Lambda}/\widehat{G})$ be the category of perfect complexes on $Z^{1}(W_{E},\widehat{G})_{\Lambda}/\widehat{G}$. Then there is an action of $\mathrm{Perf}(Z^{1}(W_{E},\widehat{G})_{\Lambda}/\widehat{G})$ on $\mathcal{D}_{\mathrm{lis}}(\mathrm{Bun}_{G},\Lambda)^{\omega}$, called the spectral action, such that composing with the map $\mathrm{Rep}(\widehat{G})^{I}\to \mathrm{Perf}(Z^{1}(W_{E},\widehat{G})_{\Lambda}/\widehat{G})^{BW^{I}}$ gives us the action of the Hecke operator.

The idea is that the spectral action gives us a functor from $\mathrm{Perf}(Z^{1}(W_{E},\widehat{G})_{\Lambda}/\widehat{G})$ to $\mathcal{D}_{\mathrm{lis}}(\mathrm{Bun}_{G},\Lambda)^{\omega}$, sending an object $M$ of $\mathrm{Perf}(Z^{1}(W_{E},\widehat{G})_{\Lambda}/\widehat{G})$ to the object $M\ast \mathcal{W}_{\psi}$ of $\mathcal{D}_{\mathrm{lis}}(\mathrm{Bun}_{G},\Lambda)^{\omega}$, where $\mathcal{W}_{\psi}$ is the Whittaker sheaf (the sheaf on $\mathrm{Bun}_{G}$ corresponding to the representation $\mathrm{c-Ind}_{U(F)}^{B(F)}\psi$, where $B$ is a Borel subgroup of $G$, $U$ is the unipotent radical of $B$, and $\psi$ is a character of $U$). The hope is then that this functor can be extended from $\mathrm{Perf}(Z^{1}(W_{E},\widehat{G})_{\Lambda}/\widehat{G})$ to all of $D_{\mathrm{coh},\mathrm{Nilp}}^{b,\mathrm{qc}}(Z^{1}(W_{E},\widehat{G})_{\Lambda}/\widehat{G})$, and that it will provide the desired equivalence of categories.

Now we discuss how this spectral action is constructed. Let us first consider the following more general situation. Let $L$ be a field of characteristic $0$, let $H$ be a split reductive group, and let $W$ be a discrete group. We write $BH$ and $BW$ for their corresponding classifying spaces. Let $\mathcal{C}$ be an idempotent-complete, $L$-linear stable $\infty$-category.

For all $I$, a $W$-equivariant, exact tensor action of $\mathrm{Rep}(H)$ on $\mathcal{C}$ is a functor

$\displaystyle \mathrm{Rep}(H^{I})\times \mathcal{C}\to\mathcal{C}^{BW^{I}}$

natural in $I$, exact as an action of $\mathrm{Rep}(H)$ after forgetting the $BW^{I}$-equivariance, and such that the action of $BW^{I}$ is compatible with the tensor product.

Now what we want to show is that a $W$-equivariant, exact tensor action of $\mathrm{Rep}(H)$ on $\mathcal{C}$ is the same as an $L$-linear action of $\mathrm{Perf}(\mathrm{Maps}(BW,BH))$ on $\mathcal{C}$.

To prove the above statement, Fargues and Scholze use the language of higher category theory. Let $\mathrm{An}$ be the $\infty$-category of anima, which is obtained from simplicial sets by inverting weak equivalences. The specific anima that we are interested in is $BW$, which is obtained by taking the nerve of the category $[\ast/W]$. An important property of $\mathrm{An}$ is that it is freely generated under sifted colimits by the full subcategory of finite sets.

We now define two functors $F_{1}$ and $F_{2}$ from $\mathrm{An}^{\mathrm{op}}$ to $\mathrm{An}$. The functor $F_{1}$ sends a finite set $S$ to the exact $L$-linear actions of $\mathrm{Perf}(\mathrm{Maps}(S,BH))$ on $\mathcal{C}$, which is equivalent to the exact $L$-linear monoidal functors from $\mathrm{Perf}(\mathrm{Maps}(S,BH))$ to $\mathrm{End}(\mathcal{C})$. The functor $F_{2}$ sends a finite set $S$ to the $S$-equivariant exact actions of $\mathrm{Rep}(H)$ on $\mathcal{C}$, which is equivalent to natural transformations from the functor $I\mapsto\mathrm{Hom}(S,I)$ to the functor $I\mapsto\mathrm{Fun}(\mathrm{Rep}(H^{I}),\mathrm{End}(\mathcal{C}))$.

There is a natural transformation from $F_{1}$ to $F_{2}$ that happens to be an isomorphism on finite sets. Now since the category $\mathrm{An}$ is generated by finite sets under sifted colimits, all we need is for the functors $F_{1}$ and $F_{2}$ to preserve sifted colimits.

For $F_{2}$ this follows from the fact that $S\mapsto S^{I}$ preserves sifted colimits. For $F_{1}$, this comes from the fact that $\mathrm{Maps}(S,BH)\cong [\mathrm{Spec}(A)/H^{S'}]$ for some animated $L$-algebra $A$ and some set $S'$, and then looking at the structure of $\mathrm{Perf}([\mathrm{Spec}(A)/H^{S'}])$ and $\mathrm{IndPerf}([\mathrm{Spec}(A)/H^{S'}])$.

Now that we have our abstract theory let us go back to our intended application. Let $W_{E}$ be the Weil group of $F$. It turns out that every L-parameter $\varphi:W_{E}\to \widehat{G}$ factors through a quotient $W_{E}/P$, where $P$ is some open subgroup of the wild inertia. This means that $Z^{1}(W_{E},\widehat{G})$ is the union of all $Z^{1}(W_{E}/P,\widehat{G})$ over all such $P$ (compare also with the construction in Moduli Stacks of Galois Representations), and this also means that we can focus our attention on $Z^{1}(W_{E}/P,\widehat{G})$.

We can actually go further and replace $W_{E}/P$ with its subgroup $W$ generated by the elements $\sigma$ and $\tau$ satisfying $\sigma\tau\sigma^{-1}=\tau^{q}$, together with the wild inertia (we have also already considered this in Moduli Stacks of Galois Representations, where we called it $\mathrm{WD}/Q$), and get the same moduli space, i.e. $Z^{1}(W_{E}/P,\widehat{G})\cong Z^{1}(W,\widehat{G})$.

Let $F_{n}$ be the free group on $n$ generators. For every map $F_{n}\to W$, we have a map

$\displaystyle Z^{1}(W,\widehat{G})\to Z^{1}(F_{n},\widehat{G})$

The category $\lbrace (n,F_{n})\rbrace$ is a sifted category, and upon taking sifted colimits, we obtain an isomorphism

$\displaystyle \mathrm{colim}_{n,F_{n\to W}}\mathcal{O}(Z^{1}(F_{n},\widehat{G}))\to\mathcal{O}(Z^{1}(W_{E}/P,\widehat{G}))$

There is also a version of this statement that involves higher category theory. It says that the map

$\displaystyle \mathrm{colim}_{n,F_{n\to W}}\mathcal{O}(Z^{1}(F_{n},\widehat{G}))\to\mathcal{O}(Z^{1}(W_{E}/P,\widehat{G}))$

is an isomorphism in the stable $\infty$-category $\mathrm{IndPerf}(B\widehat{G})$. Furthermore the category $\mathrm{Perf}(B\widehat{G})$ generates $\mathrm{Perf}(Z^{1}(W_{E}/P,\widehat{G})/\widehat{G})$ under cones and retracts, and $\mathrm{IndPerf}(Z^{1}(W_{E}/P,\widehat{G})/\widehat{G})$ identifies with the $\infty$-category of $\mathcal{O}(Z^{1}(W_{E}/P, \widehat{G})$-modules inside $\mathrm{IndPerf}(B\widehat{G})$.

If we take invariants under the action of $\widehat{G}$, we then have

$\displaystyle \mathrm{colim}_{n,F_{n\to W}}\mathcal{O}(Z^{1}(F_{n},\widehat{G}))^{\widehat{G}}\to\mathcal{O}(Z^{1}(W_{E}/P,\widehat{G}))^{\widehat{G}}$

Note that $\mathrm{colim}_{n,F_{n\to W}}\mathcal{O}(Z^{1}(F_{n},\widehat{G}))$ is precisely the same data as the algebra of excursion operators. We can see this using the fact that $(Z^{1}(F_{n},\widehat{G}))$ is isomorphic to $\widehat{G}^{n}$, and $\mathcal{O}(Z^{1}(F_{n},\widehat{G}))^{\widehat{G}}$ is functions on $\widehat{G}^{n}$ which are invariant under the action of $\widehat{G}$. But this is the same as the data of an excursion operator $(I,V,\alpha,\beta,(\gamma_{i})_{i\in I})$ ($I$ here has $n$ elements), because such a function is of the form $\langle \beta,\alpha((\gamma_{i})_{i\in I})\rangle$.

Now that we have our description of $\mathcal{O}(Z^{1}(W_{E}/P,\widehat{G}))$ as $\mathrm{colim}_{n,F_{n\to W}}\mathcal{O}(Z^{1}(F_{n},\widehat{G}))$, we can now apply the abstract theory developed earlier to obtain our spectral action.

Let us now focus on the case of $G=\mathrm{GL}_{n}$ and relate the spectral action to the more classical language of Hecke eigensheaves (see also The Global Langlands Correspondence for Function Fields over a Finite Field). Let $L$ be an algebraically closed field over $\mathbb{Q}_{\ell}$. Given an L-parameter $\varphi:W_{E}\to\mathrm{GL}_{n}(L)$, we have an inclusion $i_{\varphi}:\mathrm{Spec}(L)\to Z^{1}(W_{E},\widehat{G})_{L}$ and a sheaf $i_{\varphi *}L$ on $Z^{1}(W_{E},\widehat{G})_{L}$. For any $A\in D(\mathrm{Bun}_{G},\Lambda)$ we can take the spectral action $i_{\varphi *}L \ast A$. This turns out to be a Hecke eigensheaf! However, it is often going to be zero. Still, in work by Johannes AnschĂĽtz and Arthur-CĂ©sar Le Bras, they show that the above construction can give an example of a nonzero Hecke eigensheaf, by relating the spectral action to an averaging functor, which is an idea that comes from the work of Edward Frenkel, Dennis Gaitsgory, and Kari Vilonen on the geometric Langlands program.

### VIII. The p-adic local Langlands correspondence

The work of Fargues and Scholze deals with the “classical” (i.e. $\ell\neq p$) local Langlands correspondence. As we have seen for example in Completed Cohomology and Local-Global Compatibility, the p-adic local Langlands correspondence (i.e. $\ell=p$) is much more complicated and mysterious compared to the classical case. Still, one might wonder whether the machinery we have discussed here can be suitably modified to obtain an analogous “geometrization” of the p-adic local Langlands correspondence.

Since we are dealing with what we might call p-adic, instead of $\ell$-adic, Galois representations, we would have to replace $Z^{1}(W_{E},\widehat{G})$ with the moduli stack of $(\varphi, \Gamma)$-modules (also known as the Emerton-Gee stack, see also Moduli Stacks of (phi, Gamma)-modules).

We still would like to work with the derived category of some sort of sheaves on $\mathrm{Bun}_{G}$. This is because, in work of Pierre Colmez, Gabriel Dospinescu, and Wieslawa Niziol (and also in related work of Peter Scholze which uses a different approach), the p-adic etale cohomology of the Lubin-Tate tower has been used to realize the p-adic local Langlands correspondence, and we have already seen that the Lubin-Tate tower is related to $\mathrm{Bun}_{G}$ and the Hecke stack. Since p-adic etale cohomology is the subject of p-adic Hodge theory (see also p-adic Hodge Theory: An Overview), we might also expect ideas from p-adic Hodge theory to become relevant.

So now have to find some sort of p-adic replacement for $\mathrm{D}_{\mathrm{lis}}(\mathrm{Bun}_{G},\Lambda)$. It is believed that the correct replacement might be the derived category of almost solid modules, whose theory is currently being developed by Lucas Mann. Some of the ideas are similar to that used by Peter Scholze to formulate p-adic Hodge theory for rigid-analytic varieties (see also Rigid Analytic Spaces), but also involves many new ideas. Let us go through each of the meanings of the words in turn.

The “almost” refers to theory of almost rings and almost modules developed by Gerd Faltings (see also the discussion at the end of Adic Spaces and Perfectoid Spaces). For an $R$-module $M$ over a local ring $R$, we say that $M$ is almost zero if it is annihilated by some element of the maximal ideal of $R$. We define the category of almost $R$-modules (or $R^{a}$-modules) to be the category of $R$-modules modulo the category of almost zero modules.

The “solid” refers to the theory of solid rings and solid modules discussed earlier, although we will use the later language developed by Dustin Clausen and Peter Scholze. Let $A$ be a ring. We define the category of condensed $A$-modules, denoted $\mathrm{Cond}(A)$, to be the category of sheaves of $A$-modules on the category of profinite sets. Given a profinite set $S=\varprojlim S_{i}$, we define $A_{\blacksquare}[S]$ to be the limit $\varinjlim_{A'}\varprojlim_{i}A'[S_{i}]$, where $A'$ runs over all finite-type $\mathbb{Z}$-algebras contained in $A$, and we define the category of solid $A$-modules, denoted $A_{\blacksquare}-\mathrm{Mod}$, to be the subcategory of $\mathrm{Cond}(A)$ generated by $A_{\blacksquare}[S]$. The idea of condensed mathematics is to incorporate topology – for instance the category of compactly generated weak Hausdorff spaces, which forms most of the topological spaces we care about, embeds fully faithfully into the category of condensed sets. On the other hand, condensed abelian groups, rings, modules, etc. have nice algebraic properties, for instance when it comes to forming abelian categories, which topological abelian groups, rings, modules, etc. do not have. The solid rings and solid modules corresponds to “completions”, and in particular they have a reasonable “completed tensor product” that will become useful to us later on when forming derived categories.

Finally, the “derived category” refers to the same idea of a category of complexes with morphisms up to homotopy and quasi-isomorphisms inverted, as we have previously discussed, except, however, that we need to actually not completely forget the homotopies; in fact we need to remember not only the homotopies but the “homotopies between homotopies”, and so on, and for this we need to formulate derived categories in the language of infinity category theory. The reason why we need to this is because our definition will involve “gluing” derived categories, and for this we need to remember the homotopies, including the higher ones.

Let us now look at how Mann constructs this derived category of almost solid modules. Let $\mathrm{Perfd}_{pi}^{\mathrm{aff}}$ be the category of affinoid perfectoid spaces $X=\mathrm{Spa}(A,A^{+})$ together with a pseudouniformizer $\pi$ of $A$. We define a functor $X\mapsto D_{\blacksquare}^{a}(\mathcal{O}_{X}^{+}/\pi)$ as the sheafification of the functor $\mathrm{Spa}(A,A^{+})\to D_{\blacksquare}^{a}(A^{+}/\pi)$ (the derived category of almost solid $A^{+}/\pi$-modules) on $\mathrm{Perfd}_{\pi}^{\mathrm{aff}}$ equipped with the pro-etale topology.

If $X=\mathrm{Spa}(A,A^{+})$ is weakly perfectoid of finite type over some totally disconnected space, then $D_{\blacksquare}^{a}(\mathcal{O}_{X}^{+}/\pi)$ is just $D_{\blacksquare}^{a}(A^{+}/\pi)$. More generally, $X$ will gave a pro-etale cover by some $Y$ which is weakly perfectoid of finite type over some totally disconnected space, and $D_{\blacksquare}^{a}(\mathcal{O}_{X}^{+}/\pi)$ can be expressed as the limit $\varprojlim_{n} D_{\blacksquare}^{a}(B_{n}^{+}/\pi)$, where $Y_{n}=\mathrm{Spa}(B_{n},B_{n}^{+})$, and $Y_{n}$ runs over is the degree $n$ part of the Cech nerve of $Y$.

Now let $X$ be a small v-stack. There is a unique hypercomplete (this means it satisfies descent along all hypercovers, which are generalizations of the Cech nerve) sheaf on $X_{v}$ that agrees with the functor $Y\mapsto D_{\blacksquare}^{a}(\mathcal{O}_{Y}^{+}/\pi)$ for every affinoid perfectoid space $Y$ in $X_{v}$. We define $D_{\blacksquare}^{a}(\mathcal{O}_{X}^{+})$ to be the global sections of this sheaf. This is the construction that we want to apply to $X=\mathrm{Bun}_{G}$.

The derived category of almost solid modules comes with a six-functor formalism (see also Perverse Sheaves and the Geometric Satake Equivalence). Let $Y\to X$ be a map. The derived pullback $^{*}:D_{\blacksquare}^{a}(\mathcal{O}_{X}^{+}/\pi)\to D_{\blacksquare}^{a}(\mathcal{O}_{Y}^{+}/\pi)$ is the restriction map of the sheaf $D_{\blacksquare}^{a}$. The derived pushforward $^{*}:D_{\blacksquare}^{a}(\mathcal{O}_{Y}^{+}/\pi)\to D_{\blacksquare}^{a}(\mathcal{O}_{X}^{+}/\pi)$ is defined to be the right adjoint to the derived pullback. The derived tensor product $-\otimes-$ and derived Hom $\underline{\mathrm{Hom}}(-,-)$ are inherited from $D_{\blacksquare}^{a}(A^{+}/\pi)$.

The remaining two functors in the six-functor formalism are the “shriek” functors $f_{!}$ and $f^{!}$. If $f:Y\to X$ is a “nice” enough map, we have a factorization of $f$ into a composition $g\circ j$ where $j:Y\to Z$ is etale and $g:Z\to X$ is proper, and we define

$\displaystyle f_{!}:=g_{*}\circ j_{!}$

where $j_{!}$ is the right-adjoint to $j_{*}$. We then define $f^{!}:D_{\blacksquare}^{a}(\mathcal{O}_{Y}^{+}/\pi)\to D_{\blacksquare}^{a}(\mathcal{O}_{X}^{+}/\pi)$ to be the right-adjoint to $f_{!}$. The six-functor formalism satisfies certain important properties, such as functoriality of $f_{*}, f^{*}, f_{!}, f^{!}$, proper base change for $f_{!}$, and a projection formula for $f_{!}$. In Lucas Mann’s thesis, he uses the six-functor formalism he has developed to prove Poincare duality for a rigid-analytic variety $X$ of pure dimension $d$ over an algebraically closed nonarchimedean field $K$ of mixed characteristic:

$\displaystyle H_{et}^{i}(X,\mathbb{F}_{\ell})\otimes_{\mathbb{F}_{\ell}} H_{et}^{2d-i}(X,\mathbb{F}_{\ell})\to \mathbb{F}_{\ell}(-d)$

As of the moment, there are still many questions regarding a possible geometrization of the p-adic local Langlands program. As more developments are worked out, we hope to be able to discuss them in future posts on this blog, together with the different aspects of the theory that has already been developed, and the many other different future directions that it may lead to.

References:

Geometrization of the Local Langlands Correspondence by Laurent Fargues and Peter Scholze

Geometrization of the Local Langlands Program (notes by Tony Feng from a workshop at McGill University)

The Geometric Langlands Conjecture (notes from Oberwolfach Arbeitsgemeinschaft)

Berkeley Lectures on p-adic Geometry by Peter Scholze and Jared Weinstein

Etale Cohomology of Diamonds by Peter Scholze

On the Kottwitz Conjecture for Local Shtuka Spaces by David Hansen, Tasho Kaletha, and Jared Weinstein

Averaging Functors in Fargues’ Program for GL_n by Johannes AnschĂĽtz and Arthur-CĂ©sar Le Bras

Cohomologue p-adique de la Tour de Drinfeld: le Cas de la Dimension 1 by Pierre Colmez, Gabriel Dospinescu, and WiesĹ‚awa NizioĹ‚

Lectures on Condensed Mathematics by Peter Scholze

# Perverse Sheaves and the Geometric Satake Equivalence

The idea behind “perverse sheaves” originally had its roots in the work of Mark Goresky and Robert MacPherson on “intersection homology”, but has since taken a life of its own after the foundational work of Alexander Beilinson, Joseph Bernstein, and Pierre Deligne and has found many applications in mathematics. In this post, we will describe what perverse sheaves are, and state an important result in representation theory called the geometric Satake equivalence, which makes use of this language.

A perverse sheaf is a certain object of the “derived category of sheaves with constructible cohomology”, satisfying certain conditions. This is quite a lot of new words, but we shall be defining them in this post, starting with “constructible”.

Let $X$ be an algebraic variety with a stratification, i.e. a decomposition

$\displaystyle X=\coprod_{\lambda\in\Lambda}X_{\lambda}$

of $X$ into a finite disjoint union of connected, locally closed, smooth subsets $X_{\lambda}$ called strata, such that the closure of any stratum is a union of strata.

A sheaf $\mathcal{F}$ on $X$ is constructible if its restriction $\mathcal{F}\vert_{X_{\lambda}}$ to any stratum $X_{\lambda}$ is locally constant (for every point $x$ of $X_{\lambda}$ there is some open set $V$ containing $x$ on which the restriction $\mathcal{F}\vert_{V}$ to $U$ is a constant sheaf). A locally constant sheaf which is finitely generated (its stalks are finitely generated modules over some ring of coefficients) is also called a local system. Local systems are quite important in arithmetic geometry – for instance, local sheaves on $X$ correspond to representations of the etale fundamental group $\pi_{1}(X)$. The character sheaves discussed at the end of The Global Langlands Correspondence for Function Fields over a Finite Field are also examples of local systems (in fact, perverse sheaves, which we shall define later in this post, can be viewed as a generalization of local systems and are also important in the geometric Langlands program).

Now let us describe roughly what a derived category is. Given an abelian category (for example the category of abelian groups, or sheaves of abelian groups on some space $X$) $A$, we can think of the derived category $D(A)$ as the category whose objects are the cochain complexes in $A$, but whose morphisms are not quite the morphisms of cochain complexes in $A$, but instead something “looser” that only reflects information about their cohomology.

Let us explain what we mean by this. Two morphisms between cochain complexes in $A$ may be “chain homotopic”, in which case they induce the same morphisms of the corresponding cohomology groups. Therefore, as an intermediate step in constructing the derived category $D(A)$, we first create a category $K(A)$ where the objects are the cochain complexes in $A$, but where the morphisms are the equivalence classes of morphisms of cochain complexes in $A$ where the equivalence relation is that of chain homotopy. The category $K(A)$ is called the homotopy category of cochain complexes (in $A$).

Finally, a morphism of chain complexes in $A$ is called a quasi-isomorphism if it induces an isomorphism of the corresponding cohomology groups. Therefore, since we want the morphisms of $D(A)$ to reflect the information about the cohomology, we want the quasi-isomorphisms of chain complexes in $A$ to actually become isomorphisms in the category $D(A)$. So as our final step, to obtain $D(A)$ from $K(A)$, we “formally invert” the quasi-isomorphisms.

We do not yet have everything we need to define what a perverse sheaf is, but we have mentioned previously that they are an object of the derived category of sheaves on an algebraic variety $X$ with constructible cohomology. We denote this latter category $D_{c}^{b}(X)$ (this is used if there is some stratification of $X$ for which we have this category; if the stratification $\Lambda$ is specified, we say $\Lambda$-constructible instead of constructible, and we denote the corresponding category by $D_{\Lambda}^{b}(X)$).

Let us say a few things about the category $D_{c}^{b}(X)$. Having “constructible cohomology” means that the cohomology sheaves of $D_{c}^{b}(X)$ are complexes of sheaves, we can take their cohomology, and this cohomology is valued in sheaves (these sheaves are what we call cohomology sheaves) which are constructible, i.e. on each stratum $X_{\lambda}$ they are local systems. The category $D_{c}^{b}(X)$ is also equipped with a very useful extra structure (which we will also later need to define perverse sheaves) called the six-functor formalism.

These six functors are $R\mathrm{Hom}$, $\otimes^{\mathbb{L}}$, $Rf_{*}$, $Rf^{*}$, $Rf_{!}$, and $Rf^{!}$, the first four being the derived functors corresponding to the usual operations of Hom, tensor product, pushforward, and pullback, respectively, and the last two are the derived “shriek” functors (see also The Hom and Tensor Functors and Direct Images and Inverse Images of Sheaves). The functor $\otimes^{\mathbb{L}}$ makes $D_{c}^{b}(X)$ into a symmetric monoidal category, and $R\mathrm{Hom}$ is its right adjoint. The functor $Rf_{*}$ is right adjoint to $Rf^{*}$, and similarly $Rf_{!}$ is right adjoint to $Rf^{!}$. In the case that $f$ is proper, $Rf_{!}$ is the same as $Rf_{*}$, and in the case that $f$ is etale, $Rf^{!}$ is the same as $Rf^{*}$. We note that it is quite common in the literature to omit the $R$ from the notation, and to let the reader infer that the functor is “derived” from the context (i.e. it is a functor between derived categories).

A derived category is but a specific instance of the even more abstract concept of a triangulated category, which we have defined already, together with the related concepts of a t-structure and the heart of a t-structure, in The Theory of Motives.

In fact we will need the concept of a t-structure to define perverse sheaves. Let us now define this t-structure on the derived category of constructible sheaves. Let $X=\coprod_{\lambda\in\Lambda} X_{\lambda}$ be an algebraic variety with its stratification, and for every stratum $X_{\lambda}$ let $d_{\lambda}$ denote its dimension. We write $D_{\mathrm{const}}^{b}$ for the subcategory of $D_{\Lambda}^{b}$ whose cohomology sheaves are locally constant, and for any object $\mathfrak{F}$ of some derived category we write $\mathcal{H}^{i}(\mathfrak{F})$ for its $i$-th cohomology sheaf. We define

$\displaystyle ^{p}D_{\lambda}^{\leq 0}=\lbrace\mathfrak{F}\in D_{\mathrm{const}}^{b}(X_{\lambda}):\mathcal{H}^{i}(\mathfrak{F})=0\ \mathrm{for}\ i> d_{\lambda}\rbrace$

$\displaystyle ^{p}D_{\lambda}^{\geq 0}=\lbrace\mathfrak{F}\in D_{\mathrm{const}}^{b}(X_{\lambda}):\mathcal{H}^{i}(\mathfrak{F})=0\ \mathrm{for}\ i< -d_{\lambda}\rbrace$

Now let $i_{\lambda}:X_{\lambda}\to X$ be the inclusion of a stratum $X_{\lambda}$ into $X$. We further define

$\displaystyle ^{p}D^{\leq 0}=\lbrace\mathfrak{F}\in D_{\Lambda}^{b}:Ri_{\lambda}^{*}\frak{F}\in ^{p}D_{\lambda}^{\leq 0}\ \mathrm{for}\ \mathrm{all}\ \lambda\in\Lambda\rbrace$

$\displaystyle ^{p}D^{\geq 0}=\lbrace\mathfrak{F}\in D_{\Lambda}:Ri_{\lambda}^{!}\frak{F}\in ^{p}D_{\lambda}^{\geq 0}\ \mathrm{for}\ \mathrm{all}\ \lambda\in\Lambda\rbrace$

This defines a t-structure, and we define the category of perverse sheaves on $X$, denoted $\mathrm{Perv}(X)$, as the heart of this t-structure.

With the definition of perverse sheaves in hand we can now state the geometric version of the Satake correspondence (see also The Unramified Local Langlands Correspondence and the Satake Isomorphism). Let $k$ be either $\mathbb{C}$ or $\mathbb{F}_{q}$, and let $K=k((t))$, and let $\mathcal{O}=k[[t]]$. Let $G$ be a reductive group. The loop group $LG$ is defined to be the scheme whose $k$-points are $G(K)$ and the positive loop group $L^{+}G$ is defined to be the scheme whose $k$-points are $G(\mathcal{O})$. The affine Grassmannian is then defined to be the quotient $LG/L^{+}(G)$.

The geometric Satake equivalence states that there is equivalence between the category of perverse sheaves $\mathrm{Perv}(\mathrm{Gr}_{G})$ on the affine Grassmannian $\mathrm{Gr}_{G}$ and the category $\mathrm{Rep}(^{L}G)$ of representations of the Langlands dual group $^{L}G$ of $G$. It was proven by Ivan Mirkovic and Kari Vilonen using the Tannakian formalism (see also The Theory of Motives) but we will not discuss the details of the proof further here, and leave it to the references or future posts.

As we have seen in The Global Langlands Correspondence for Function Fields over a Finite Field, the geometric Satake equivalence is important in being able to define the excursion operators in Vincent Lafforgue’s approach to the global Langlands correspondence for function fields over a finite field. It has (in possibly different variants) also found applications in other parts of arithmetic geometry, for example in certain approaches to the local Langlands correspondence, as well as the study of Shimura varieties. We shall discuss more in future posts on this blog.

References:

Perverse sheaf on Wikipedia

Constructible sheaf on Wikipedia

Derived category on Wikipedia

Satake isomorphism on Wikipedia

An illustrated guide to perverse sheaves by Geordie Williamson

Langlands correspondence and Bezrukavnikov’s equivalence by Geordie Williamson and Anna Romanov

Topics in automorphic forms (notes by Chao Li from a course by Jack Thorne)

Perverse sheaves and fundamental lemmas (notes by Chao Li from a course by Wei Zhang)

Perverse sheaves in representation theory (notes by Chao Li from a course by Carl Mautner)

Geometric Langlands duality and representations of algebraic groups over commutative rings by Ivan Mirkovic and Kari Vilonen

# The Unramified Local Langlands Correspondence and the Satake Isomorphism

In The Local Langlands Correspondence for General Linear Groups we gave the statement of the local Langlands correspondence for the groups $\mathrm{GL}_{n}(F)$ for $F$ a p-adic field. In this post we will consider a special case of this correspondence in more detail, called the unramified case (we shall define what this is shortly), and we take the opportunity to introduce an important concept in the theory, that of the Satake isomorphism (which we will state, but not prove).

Let us continue to stick with $\mathrm{GL}_{n}(F)$, although what we discuss here also generalizes to other reductive groups. Let $\mathcal{O}_{F}$ be the ring of integers of $F$.

Let us now explain what “unramified” means for both sides of the correspondence. We say that an irreducible admissible representation of $\mathrm{GL}_{n}(F)$ is unramified if there exists a nonzero vector that is fixed by $\mathrm{GL}_{n}(\mathcal{O}_{F})$. Meanwhile, we say that a Weil-Deligne representation (see also Weil-Deligne Representations) is unramified if it factors as $W_{F}\twoheadrightarrow\mathbb{Z}\hookrightarrow \mathrm{GL}_{2}(\mathbb{C})$, and the monodromy operator $N$ is zero. Let us note that an unramified Weil-Deligne representation is determined by where the Frobenius element (which maps to $1$ under the map to $\mathbb{Z})$ gets sent to, up to conjugacy. Hence the unramified Weil-Deligne representations are in bijection with conjugacy classes of diagonalizable elements in $\mathrm{GL}_{n}(\mathbb{C})$.

Then the unramified local Langlands correspondence is the following statement:

There is a bijection between the set of unramified irreducible admissible representations of $\mathrm{GL}_{n}(F)$ and set of unramified Weil-Deligne representations.

In this post we will prove this statement, assuming the Satake isomorphism. To explain what the Satake isomorphism is, let us first discuss a generalization of the Hecke algebra (see also Hecke Operators).

The spherical Hecke algebra $\mathcal{H}(\mathrm{GL}_{n}(F),\mathrm{GL}_{n}(\mathcal{O}_{F}))$ is the algebra of compactly supported locally constant functions on $\mathrm{GL}_{n}(F)$ which are bi-invariant under the action (invariant under the left and right action) of $\mathrm{GL}_{n}(\mathcal{O}_{F})$. The “multiplication” on this algebra is given by convolution, i.e., given two elements $f_{1}$ and $f_{2}$ of the spherical Hecke algebra, their “product” is given by

$\displaystyle (f_{1}\cdot f_{2})(g)=\int_{\mathrm{GL}_{n}(F)}f_{1}(x)f_{2}(g^{-1}x)dx$

There is an action of the spherical Hecke algebra (more generally there is also an action of compactly supported locally constant functions of $G$, without the bi-invariance condition) on a representation $\pi$ of $\mathrm{GL}_{n}(F)$ as follows. Let $f$ be an element of the spherical Hecke algebra and let $v$ be a vector in the vector space on which the representation $\pi$ acts. Then

$\displaystyle \pi(f)v=\int_{\mathrm{GL}_{n}(F)} f(g)\pi(g)(v) dg$.

This action makes the representation $\pi$ into an $\mathcal{H}(\mathrm{GL}_{n}(F),\mathrm{GL}_{n}(\mathcal{O}_{F}))$-module. The importance of the spherical Hecke algebra to the unramified local Langlands correspondence is that there is a bijection between the set of unramified irreducible admissible representations of $\mathrm{GL}_{n}(F)$ and set of irreducible $\mathcal{H}(\mathrm{GL}_{n}(F),\mathrm{GL}_{n}(\mathcal{O}_{F}))$-modules.

The spherical Hecke algebra $\mathcal{H}(\mathrm{GL}_{n}(F),\mathrm{GL}_{n}(\mathcal{O}_{F}))$ is commutative, and from this it follows that the irreducible $\mathcal{H}(\mathrm{GL}_{n}(F),\mathrm{GL}_{n}(\mathcal{O}_{F}))$-modules are in bijection with maps $\mathcal{H}(\mathrm{GL}_{n}(F),\mathrm{GL}_{n}(\mathcal{O}_{F}))\to\mathbb{C}$.

Let $T$ be a maximal torus in $\mathrm{GL}_{n}(F)$ (see also Reductive Groups Part I: Over Algebraically Closed Fields) and let $X_{\bullet}(T)$ be the set of all cocharacters of $X_{\bullet}(T)$. Let $\mathbb{C}[X_{\bullet}(T)]$ be the ring formed by adjoining the elements of $X_{\bullet}(T)$ as formal variables to $\mathbb{C}$. Recall that the Weyl group $W$ is defined as the quotient of the normalizer of $T$ in $\mathrm{GL}_{n}(F)$ by the centralizer of $T$ in $\mathrm{GL}_{n}(F)$ (see also Reductive Groups Part I: Over Algebraically Closed Fields). The ring $\mathbb{C}[X_{\bullet}(T)]$ has an action of $W$ which comes from the action of $W$ on $T$. We denote the invariants of this action by $\mathbb{C}[X_{\bullet}(T)]^{W}$.

Now we can state the Satake isomorphism as follows:

$\displaystyle \mathcal{H}(\mathrm{GL}_{n}(F),\mathrm{GL}_{n}(\mathcal{O}_{F}))\cong \mathbb{C}[X_{\bullet}(T)]^{W}$.

Let us now see how the Satake isomorphism helps us prove the unramified local Langlands correspondence. We define the dual torus $\widehat{T}$ to be $\mathrm{Spec}(\mathbb{C}[X_{\bullet}(T)])$. Then homomorphisms $\mathbb{C}[X_{\bullet}(T)]^{W}\to\mathbb{C}$ correspond to $W$-conjugacy classes of elements in $\widehat{T}(\mathbb{C})$. These conjugacy classes, in turn, are in bijection with the conjugacy classes of diagonalizable elements in $\mathrm{GL}_{n}(\mathbb{C})$. But these conjugacy classes are in bijection with the unramified Weil-Deligne representations, as mentioned earlier. At the same time, by the Satake isomorphism $\mathbb{C}[X_{\bullet}(T)]^{W}\to\mathbb{C}$ corresponds to maps $\mathcal{H}(\mathrm{GL}_{n}(F),\mathrm{GL}_{n}(\mathcal{O}_{F}))\to\mathbb{C}$, and therefore to irreducible $\mathcal{H}(\mathrm{GL}_{n}(F),\mathrm{GL}_{n}(\mathcal{O}_{F}))$-modules, and therefore finally to unramified irreducible admissible representations of $\mathrm{GL}_{n}(F)$. This gives us the unramified local Laglands correspondence for $\mathrm{GL}_{n}(F)$.

As mentioned earlier, all of this can also be applied to more general reductive groups, with appropriate generalizations of what it means for an irreducible admissible representation to be unramified. In this case, the conjugacy classes involved will be that of the Langlands dual group.

There is also a “geometric” version of the Satake isomorphism which relates the representations of the Langlands dual group to the category of perverse sheaves on a very special geometric object called the affine Grassmannian. We will discuss more of this in future posts.

References:

Satake isomorphism on Wikipedia

Unramified representations and the Satake isomorphism by James Newton

Topics in automorphic forms (notes by Chao Li from a course by Jack Thorne)

Langlands correspondence and Bezrukavnikov’s equivalence by Anna Romanov and Geordie Williamson

# The mod p local Langlands correspondence for GL_2(Q_p)

In The Local Langlands Correspondence for General Linear Groups, we stated the local Langlands correspondence for $\mathrm{GL}_n(F)$ where $F$ is a finite extension of $\mathbb{Q}_{p}$. We recall that it states that there is a one-to-one correspondence between irreducible admissible representations of $\mathrm{GL}_{n}(F)$ and F-semisimple Weil-Deligne representations of $\mathrm{Gal}(\overline{F}/F)$. Here all the relevant representations are over $\mathbb{C}$.

In this post, we will consider representations over a finite field $\mathbb{F}$ of characteristic $p$. While we may hope that some sort of “mod p local Langlands correspondence” will also hold in this case, at the moment all we know is the case of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$, which we will discuss in this post. It is a sort of stepping stone to the “p-adic local Langlands correspondence” (where the representations are over some extension of $\mathbb{Q}_{p}$), which, as in the mod p case, is only known for the case of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$. The p-adic local Langlands correspondence for $\mathrm{GL}_{2}(\mathbb{Q}_{p})$ plays an important role in the proofs of the known cases of the Fontaine-Mazur conjecture, which concerns when a Galois representation comes from the etale cohomology of some variety.

Let us start by discussing some representation theory of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$ over $\mathbb{F}$. This will be somewhat similar to our discussion in The Local Langlands Correspondence for General Linear Groups, as we will see later when we state the classification of the irreducible admissible smooth representations of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$, but we will also need some new ingredients.

A Serre weight is an absolutely irreducible representation (absolutely irreducible means it is irreducible over the algebraic closure of $\mathbb{F}$) of $\mathrm{GL}_{2}(\mathbb{F}_{p})$ over $\mathbb{F}$. This is the same as an absolutely irreducible smooth representation of $\mathrm{GL}_{2}(\mathbb{Z}_{p})$ over $\mathbb{F}$.

Serre weights are completely classified and can be explicitly described. Let $r\in\lbrace 0,\ldots,p-1\rbrace$ and let $s\in \lbrace 0,\ldots, p-2\rbrace$. Then a Serre weight is always of the form $\mathrm{Sym}^{r}\mathbb{F}^{2}\otimes\mathrm{det}^{s}$.

The name “Serre weight” originates from its relationship to Serre’s modularity conjecture, which is a conjecture about when a residual representation comes from a modular form, and what the level and the weight of the modular form should be. Avner Ash and Glenn Stevens made the observation that a residual representation $\overline{\rho}$ is modular of weight $k$ ($k\geq 2$) and level $\Gamma_{1}(N)$ if and only if $H^{1}(\Gamma_{1}(N),\mathrm{Sym}^{k-2}\mathbb{F}^{2})_{\mathfrak{m}_{\overline{\rho}}}$ (here $\mathfrak{m}_{\overline{\rho}}$ is a certain maximal ideal of the Hecke algebra associated to $\overline{\rho}$) is nonzero.

For convenience, from here on in this post we shall consider Serre weights not just as representations of $\mathrm{GL}_{2}(\mathbb{Z}_{p})$ but as representations of $\mathrm{GL}_{2}(\mathbb{Z}_{p})\mathbb{Q}_{p}^{\times}$, which sends the uniformizer of $\mathbb{Q}_{p}$ to $1$.

Serre weights are important because from them we can obtain induced representations of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$. Let $\mathrm{c-Ind}_{\mathrm{GL}_{2}(\mathbb{Z}_{p})\mathbb{Q}_{p}^{\times}}^{\mathrm{GL}_{2}(\mathbb{Q}_{p})}\sigma$ be the representation of $\mathrm{GL} _{2}(\mathbb{Q}_{p})$ coming from compactly supported functions from $\mathrm{GL}_{2}(\mathbb{Q}_{p})$ to $\sigma$ which satisfy $f(kg)=\sigma(k)f(g)$.

The endomorphisms of $\mathrm{c-Ind}_{\mathrm{GL}_{2}(\mathbb{Z}_{p})\mathbb{Q}_{p}^{\times}}^{\mathrm{GL}_{2}(\mathbb{Q}_{p})}\sigma$ form the Hecke algebra, which is isomorphic to $\mathbb{F}[T]$. In other words, we can consider $\mathrm{c-Ind}_{\mathrm{GL}_{2}(\mathbb{Z}_{p})\mathbb{Q}_{p}^{\times}}^{\mathrm{GL}_{2}(\mathbb{Q}_{p})}\sigma$ as a module over $\mathbb{F}[T]$, and we can take the quotient $(\mathrm{c-Ind}_{\mathrm{GL}_{2}(\mathbb{Z}_{p})\mathbb{Q}_{p}^{\times}}^{\mathrm{GL}_{2}(\mathbb{Q}_{p})}\sigma)/(T)$. This quotient is irreducible, and it is an important class of absolutely irreducible admissible smooth representations of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$ called the supersingular representations.

The rest of the absolutely irreducible admissible smooth representations of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$ are very similar to what we discussed in The Local Langlands Correspondence for General Linear Groups. Namely, they can be obtained from principal series representations, which are induced representations of characters from the Borel subgroup $B(\mathbb{Q}_{p})$ of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$ (i.e. the upper-triangular matrices in $\mathrm{GL}_{2}(\mathbb{Q}_{p})$).

To recall, the principal representations are $\mathrm{Ind}_{B(\mathbb{Q}_{p})}^{\mathrm{GL}_{2}(\mathbb{Q}_{p})}\chi_{1}\otimes\chi_{2}$, which means they come from functions $f:\mathrm{GL}_{2}(\mathbb{Q}_{p})\to\mathbb{F}$ such that $f(hg)=\chi_{1}\otimes\chi_{2}(h)f(g)$ for $h\in B(\mathbb{Q}_{p})$ and $g\in\mathrm{GL}_{2}(\mathbb{Q}_{p})$, where $\chi_{1}$ and $\chi_{2}$ are characters of $\mathbb{Q}_{p}^{\times}$, and $\chi_{1}\otimes\chi_{2}$ as a function on $B(\mathbb{Q}_{p})$ means it sends an element $\begin{pmatrix}a& b\\0& d\end{pmatrix}$ of $B(\mathbb{Q}_{p})$ to $\chi_{1}(a)\otimes \chi_{2}(d)$.

In the case that $\chi_{1}\neq\chi_{2}$, the principal series representations will be absolutely irreducible, in which case we obtain another class of absolutely irreducible admissible smooth representations. Otherwise, we may obtain absolutely irreducible representations as a quotient. These will be twists (this means a tensor product by a character) of the Steinberg representation, which is defined as the quotient $\mathrm{Ind}_{B(\mathbb{Q}_{p})}^{\mathrm{GL}_{2}(\mathbb{Q}_{p})}(1\otimes 1)/\mathbf{1}$ (here $\mathbf{1}$ is the trivial representation of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$). This gives a third class of absolutely irreducible admissible representations. Finally we have the characters, which give a fourth class.

In summary, the absolutely irreducible admissible smooth representations of $\mathrm{GL}_{2}(\mathbb{Q}_{p})$ over $\mathbb{F}$ can be classified into the following four kinds as follows:

• One-dimensional representations (characters) $\delta: \mathrm{GL}_{2}(\mathbb{Q}_{p})\to\mathbb{F}^{\times}$
• Principal series representations $\mathrm{Ind}_{B(\mathbb{Q}_{p})}^{\mathrm{GL}_{2}(\mathbb{Q}_{p})}\chi_{1}\otimes\chi_{2}$ for $\chi_{1},\chi_{2}: \mathbb{Q}_{p}^{\times}\to\mathbb{F^{\times}}, \chi_{1}\neq\chi_{2}$
• Twists of Steinberg representations $\delta\otimes\mathrm{Ind}_{B(\mathbb{Q}_{p})}^{\mathrm{GL}_{2}(\mathbb{Q}_{p})}(1\otimes 1)/\mathbf{1}$ for $\delta:\mathrm{GL}_{2}(\mathbb{Q}_{p})\to\mathbb{F}^{\times}$
• Supersingular representations $(\mathrm{c-Ind}_{\mathrm{GL}_{2}(\mathbb{Z}_{p})\mathbb{Q}_{p}^{\times}}^{\mathrm{GL}_{2}(\mathbb{Q}_{p})}\sigma)/(T)$ for $\sigma$ a Serre weight

Let us now discuss the other side of the correspondence, the “Galois side”. For ease of notation let us also denote $\mathrm{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p})$ by $G_{\mathbb{Q}_{p}}$.

Let $g$ be an element of the inertia subgroup of $G_{\mathbb{Q}_{p}}$. Serre’s level 2 fundamental character $\omega_{2}$ is given by composing the map

$\displaystyle g\mapsto \frac{g(\sqrt[p^{2-1}]{-p})}{\sqrt[p^{2-1}]{-p}}$

which takes values in $\mathbb{\mu}_{p^{2}-1}$ with the isomorphism $\mu_{p^{2}-1}\xrightarrow{\sim}\mathbb{F}_{p^{2}}^{\times}$.

Let $h$ be a natural number which is mutually prime to $p+1$. We have that $\mathrm{Ind}_{G_{\mathbb{Q}_{p^{2}}}}^{G_{\mathbb{Q}_{p}}}\omega_{2}^{h}$ is an irreducible 2-dimensional representation of $G_{\mathbb{Q}_{p}}$. In fact, any absolutely irreducible representation of $G_{\mathbb{Q}_{p}}$ over $\mathbb{F}$ is of this form, possibly tensored with the unramified character $\lambda_{a}$ which takes the inverse of the Frobenius to $a\in\mathbb{F}^{\times}$.

The mod p local Langlands correspondence is now the bijection described explicitly as follows:

To the supersingular representation $(\mathrm{c-Ind}_{\mathrm{GL}_{2}(\mathbb{Z}_{p})\mathbb{Q}_{p}^{\times}}^{\mathrm{GL}_{2}(\mathbb{Q}_{p})}\sigma)/(T)$ where $\sigma=\mathrm{Sym}^{r}\mathbb{F}^{2}$, we associate the Galois representation $\mathrm{Ind}_{G_{\mathbb{Q}_{p^{2}}}}^{G_{\mathbb{Q}_{p}}}\omega^{r+1}$.

To the representation $\pi$ which is obtained as the extension $0\to\mathrm{Ind}_{B(\mathbb{Q}_{p})}^{\mathrm{GL}_{2}(\mathbb{Q}_{p})}\chi_{1}\otimes\chi_{2}\varepsilon^{-1}\to\pi\to \mathrm{Ind}_{B(\mathbb{Q}_{p})}^{\mathrm{GL}_{2}(\mathbb{Q}_{p})}\chi_{2}\otimes\chi_{1}\varepsilon^{-1}\to 0$ we associate the Galois representation $\overline{\rho}$ which is obtained as an extension $0\to\chi_{1}\to\overline{\rho}\to\chi_{2}\to 0$. Here $\varepsilon$ is the reduction mod p of the p-adic cyclotomic character, and $\chi_{1}$ and $\chi_{2}$ are characters $\mathbb{Q}_{p}^{\times}\to\mathbb{F}^{\times}$ which are not equal to each other nor to the product of the other by the p-adic cyclotomic character or its inverse.

The p-adic local Langlands correspondence, which, as stated earlier concerns representations over some finite extension of $\mathbb{Q}_{p}$ and is important in the Fontaine-Mazur conjecture, needs to be compatible with the mod p local Langlands correspondence as well. Its statement is more involved than the mod p local Langlands correspondence, and its proof involves $(\varphi,\Gamma)$-modules. We reserve further discussion of the p-adic local Langlands correspondence to future posts.

References:

The emerging p-adic Langlands programme by Christophe Breuil

Representations of Galois and of GL_2 in characteristic p by Christophe Breuil

Towards a modulo p Langlands correspondence for GL_2 by Christophe Breuil and Vytautas Paskunas

# Trace Formulas

A trace formula is an equation that relates two kinds of data – “spectral” data related to representations (or eigenvalues of certain operators), and “geometric” data, related to integrals along “orbits” on some space.

The name “trace formula” comes from how this equation is obtained – by expanding the “trace” of a certain operator (let’s call it $R_{f}$. It will depend on a compactly supported “test function” $f(x)$ on a topological group $G$) on square-integrable functions on a compact quotient $\Gamma\backslash G$ of $G$ (which give a representation of $G$ by translation) by a discrete subgroup $\Gamma$.

The operator $R_{f}$ takes a function $\phi(x)$ on the group $G$, translates it by some element $y$ (recall for example that acting on functions by translation is how we defined the representation of the group $\mathbb{R}$ in Representation Theory and FourierÂ Analysis), multiplies it by the test function $f(x)$, then integrates over the group $G$ (the group $G$ must have a measure called “Haar measure” to do this) to obtain a new function $(R_{f}\phi)(y)$:

$\displaystyle (R_{f}\phi)(y)=\int_{G}\phi(xy)f(x)dx$

We can also express this as

$\displaystyle (R_{f}\phi)(y)=\int_{G}\phi(x)f(y^{-1}x)dx$

Let $\Gamma$ be a discrete subgroup of $G$, such that the quotient $\Gamma\backslash G$ is compact (this will turn out to be important later). Instead of integrating over all of $G$ we may instead integrate over the quotient $\Gamma\backslash G$ by re-expressing the integrand as follows:

$\displaystyle (R_{f}\phi)(y)=\int_{\Gamma\backslash G}\phi(x)\sum_{\gamma\in\Gamma}f(y^{-1}\gamma x)dx$

The sum $\sum_{\gamma\in\Gamma}f(y^{-1}\gamma x)$ is called the “kernel” of the operator $R_{f}$ and is denoted by $K(x,y)$. We have

$\displaystyle (R_{f}\phi)(y)=\int_{\Gamma\backslash G}K(x,y)\phi(x)dx$

So the operator $R_{f}$ looks like the integral of $K(x,y)\phi(x)dx$ over the quotient $\Gamma\backslash G$. Compare this with how a matrix with entries $A_{mn}$ acts on a finite dimensional vector $v_{n}$:

$\displaystyle v_{m}=\sum_{n}A_{mn}v_{n}$

Note that we think of integrals as analogous to sums for infinite dimensions, as functions are analogous to vectors in infinite dimensions. Now we can see that the kernel $K(x,y)$ is the analogue of the entries of some matrix!

The “trace” of a matrix is just the sum of its diagonal entries, i.e. the sum of $A_{nn}$ for all n. Therefore, the trace of the operator defined above is the integral of $K(x,x)$ (i.e. we set $x=y$) over $\Gamma\backslash G$.

$\displaystyle \mathrm{tr}(R_{f})=\int_{\Gamma\backslash G} K(x,x)dx$

Now recall that the kernel $K(x,y)$ is given by the sum $\sum_{\gamma\in\Gamma}f(y^{-1}\gamma x)$. Therefore the trace will be given by

$\displaystyle \mathrm{tr}(R_{f})=\int_{\Gamma\backslash G}\sum_{\gamma\in\Gamma}f(x^{-1}\gamma x)dx$.

Some analysis manipulations will allow us to re-express the trace as the sum

$\displaystyle \mathrm{tr}(R_{f})=\sum_{\gamma\in\lbrace \Gamma\rbrace}\mathrm{vol}(\Gamma_{\gamma}\backslash G_{\gamma})\int_{G_{\gamma}\backslash G} f(x^{-1}\gamma x)dx$

over representatives $\gamma$ of conjugacy classes in $\Gamma$ of the integrals of $f(x^{-1}\gamma x)$ over the quotient $G_{\gamma}\backslash G$ where $G_{\gamma}$ is the centralizer of $\gamma$ in $G$, multiplied by some factor called the “volume” of $\Gamma_{\gamma}\backslash G_{\gamma}$.

The integral of $f(x^{-1}\gamma x)$ over $G_{\gamma}\backslash G$ is called an “orbital integral“. This expansion of the trace is going to be the “geometric side” of the trace formula.

We consider another way to expand the trace. Recall that to define the operator $R_f$ we needed to act by translation. In this case that the quotient $\Gamma\backslash G$ is compact, as we stated earlier, this representation (let us call it $R$) by translation decomposes into a direct sum of irreducible representations $\pi$, with multiplicities $m(\pi,R)$. So we decompose first before getting the trace!

This other expansion is called the “spectral side“. Since we have now expanded the same thing, the trace, in two ways, we can equate the two expansions:

$\displaystyle \sum_{\gamma\in\lbrace \Gamma\rbrace}\mathrm{vol}(\Gamma_{\gamma}\backslash G_{\gamma})\int_{G_{\gamma}\backslash G}f(x^{-1}hx)dx=\sum_{\pi} m(\pi,R)\mathrm{tr}(\int_{G}f(x)\pi(x)dx)$

This equation is what is called the “trace formula”. Let us test it out for $G=\mathbb{R}$, $H=\mathbb{Z}$, like in Representation Theory and FourierÂ Analysis.

In the geometric side, $f(x^{-1}\gamma x)=f(\gamma)$, since $\mathbb{R}$ is abelian. $\mathbb{Z}$ is also abelian, so the conjugacy classes are just elements of $\mathbb{Z}$. We have $G_{\gamma}=G$ and $\Gamma_{\gamma}=\Gamma$. One can check that the volume is $1$ and the orbital integral is just $f(\gamma)$. Replacing $\gamma$ by $n$ for notational convenience, we see that the geometric side is just a sum of $f(n)$ over each integer $n$ in $\mathbb{Z}$.

Let us now look at the spectral side. Recall that the representation decomposes into irreducible representations, each with multiplicity $1$, which are given by multiplication by $e^{2 \pi i k x}$. We consider the operator $R_{f}$ now.

Recall that we let our representation act, then multiply it with the test function f, then integrate. We broke it up into irreducible representations, which act by multiplication by $e^{2 \pi i k x}$. What is multiplication of a function of the form $e^{2 \pi i k x}$ and integrating over $x$?

This is just the Fourier transform of the test function $f$! Since we have an irreducible representation for every integer $k$, we sum over those. So we have an equality between the sum of $f(n)$ where $n$ is an integer, and the corresponding sum of its Fourier transforms!

This is actually a classical result in Fourier analysis known as Poisson summation:

$\displaystyle \sum_{n\in\mathbb{Z}}f(n)=\sum_{k\in\mathbb{Z}}\int_{\mathbb{R}} e^{2\pi i k x}f(x)dx$

Atle Selberg famously applied the trace formula to the representation of $G=\mathrm{SL}_{2}(\mathbb{R})$ on functions on a double quotient $H\backslash\mathrm{SL}_2(\mathbb{R})/\mathrm{SO}(2)$. Note that the quotient $\mathrm{SL}_2(\mathbb{R})/\mathrm{SO}(2)$ is the upper half-plane. H is chosen by Selberg so that the double quotient is a Riemann surface of genus $g\geq 2$.

Selberg used the trace formula to relate lengths of geodesics (given by orbital integrals) to eigenvalues of the 2D Laplacian. Note that the Laplacian already appears in our example of Poisson summation, because $e^{2 \pi i k x}$ is also an eigenfunction of the 1D Laplacian.

This may be why the spectral side is called “spectral”. The trace formula is fascinating on its own, but very commonly used with it is to study representations of certain groups via more familiar representations of other groups.

To do this, note that the spectral side contains information related to representations. If we could only somehow find a way to relate the geometric sides of trace formulas of two different representations, then we can relate their spectral sides!

This is an approach to the part of representation theory known as Langlands functoriality, which studies how representations are related given that the respective groups have “Langlands duals” that are related. Relating the geometric sides involves proving difficult theorems such as “smooth transfer” and the “fundamental lemma”.

Finally, it is worth noting that the spectral side is also used to study special values of L-functions. This is inspired by the work of Hecke expressing completed L-functions as Mellin transforms of modular forms. But that is for another time!

References:

Arthur-Selberg trace formula on Wikipedia

Poisson summation formula on Wikipedia

An introduction to the trace formula by James Arthur

Selberg’s trace formula: an introduction by Jens Marklof

# Automorphic Forms

An automorphic form is a kind of function of the adelic points (see also Adeles and Ideles) of a reductive group (see also Reductive Groups Part I: Over Algebraically Closed Fields), that can be used to investigate its representation theory. Choosing an important kind of function of a group that will be helpful in investigating its representation theory was also discussed in Representation Theory and Fourier Analysis, where we found the square-integrable functions on the circle to be useful in studying its representations (or that of the real line) since it decomposed into a direct sum of irreducible representations. In fact, the cuspidal automorphic forms we will introduce later on in this post will also have this property (called semisimple) of decomposing into a direct sum of irreducible representations.

Remark: We have briefly mentioned, in the unramified case, cuspidal automorphic forms as certain functions on $\mathrm{Bun}_{G}$ in The Global Langlands Correspondence for Function Fields over a Finite Field. The function field (over a finite field) version of the cuspidal automorphic forms we define here are actually obtained as linear combinations of translates of such functions hence why it is enough to study them. In this post, we will discuss automorphic forms in more detail, beginning with the version over the field of rational numbers before generalizing to more general global fields.

### Defining modular forms as functions on $\mathrm{GL}_{2}(\mathbb{A})$

In a way, automorphic forms can also generalize modular forms (see also Modular Forms), and this will give us a way to connect the two theories. We shall take this route first and recast modular forms in a new language – instead of functions on the upper half-plane, we shall now look at them as functions on the group $\mathrm{GL}_{2}(\mathbb{A})$ (here $\mathbb{A}$ denotes the adeles of $\mathbb{Q}$).

Let $K_{f}$ be a compact open subgroup of $\mathrm{GL}_{2}(\mathbb{A}_{f})$ whose elements all have determinants in $\widehat{\mathbb{Z}}^{\times}$. Here $\mathbb{A}_{f}$ stands for the finite adeles, which are defined in the same way as the adeles, except we don’t include the infinite primes in the restricted product. We have that

$\displaystyle \mathrm{GL_{2}}(\mathbb{A})=\mathrm{GL_{2}}(\mathbb{Q})\mathrm{GL}_{2}(\mathbb{R})^{+}K_{f}$

where $\mathrm{GL}_{2}(\mathbb{R})^{+}$ is the subgroup of $\mathrm{GL}_{2}(\mathbb{R})$ consisting of elements that have positive determinant. Now let us take the double quotient $\mathrm{GL_{2}}(\mathbb{Q})\backslash\mathrm{GL}_{2}(\mathbb{A})/K_{f}$. By the above expression for $\mathrm{GL}_{2}(\mathbb{A})$ as a product, we have

$\displaystyle \mathrm{GL_{2}}(\mathbb{Q})\backslash\mathrm{GL}_{2}(\mathbb{A})/K_{f}\simeq \Gamma\backslash\mathrm{GL_{2}}(\mathbb{R})$

where $\Gamma$ is the subgroup of $\mathrm{GL}_{2}(\mathbb{R})$ given by projecting $\mathrm{GL}_{2}(\mathbb{Q})\cap \mathrm{GL}_{2}(\mathbb{R})^{+}K_{f}$ into its archimedean component. Now suppose we are in the special case that $K_{f}$ is given by $\displaystyle \prod_{p} \mathrm{GL}_{2}(\mathbb{Z}_{p})$. Then it turns out that $\Gamma$ is just $\mathrm{SL}_{2}(\mathbb{Z})$! Using appropriate choices of $K_{f}$, we can also obtain congruence subgroups such as $\Gamma_{0}(N)$ (see also Modular Forms).

The group $\mathrm{GL}_{2}(\mathbb{R})^{+}$ acts on the upper half-plane by fractional linear transformations, i.e. if we have $\displaystyle g_{\infty}=\begin{pmatrix} a & b \\ c & d \end{pmatrix}\in\mathrm{GL}_{2}(\mathbb{R})^{+}$, then $g_{\infty}$ sends $\tau$ in the upper half-plane to $\displaystyle g_{\infty}(\tau)=\frac{az+b}{cz+d}$. Let

$\displaystyle j(g_{\infty},\tau)=\mathrm{det}(g_{\infty})^{-\frac{1}{2}}(c\tau+d)$.

Now given a modular form $f$ of weight $m$ and level $\Gamma_{0}(N)$, we may associate to it a function $\phi_{\infty}(g_{\infty})$ on $\mathrm{GL}_{2}(\mathbb{R})^{+}$ as follows:

$\displaystyle f\mapsto \phi_{\infty}(g_{\infty})=f(g_{\infty}(i))j(g_{\infty},i)^{-m}$

We can also go the other way, recovering $f$ from such a $\phi_{\infty}$:

$\displaystyle \phi_{\infty}\mapsto f(g_{\infty}(i))=\phi_{\infty}(g_{\infty})j(g_{\infty},i)^{m}$

for any $g_{\infty}$ such that $g_{\infty}(i)=\tau$. Ultimately we want a function $\phi$ on $\mathrm{GL}_{2}(\mathbb{A})$, and we achieve this by setting $\phi(g)=\phi(\gamma g_{\infty} k_{f})$ to just have the same value as $\phi_{\infty}(g_{\infty})$.

### Translating properties of modular forms into properties of functions on $\mathrm{GL}_{2}(\mathbb{A})$

#### Invariance under $\mathrm{GL}_{2}(\mathbb{Q}$ and $K_{f}$

Now we want to know what properties $\phi$ must have, so that we can determine which functions on $\mathrm{GL}_{2}(\mathbb{A})$ come from modular forms. We have just seen that we must have

$\displaystyle \phi(g)=\phi(\gamma g_{\infty} k_{f})=\phi_{\infty}(g)$.

#### The action of $Z_{\infty}^{+}$ and $K_{\infty}^{+}$

Let us now consider the action of the center of $\mathrm{GL}_{2}(\mathbb{R})^{+}$ (which we denote by $Z_{\infty}$) and the action of $\mathrm{SO}(2)$, which is a maximal compact subgroup of $\mathrm{GL}_{2}(\mathbb{R})^{+}$ (and therefore we shall also denote it by $K_{\infty})^{+}$. The center $Z_{\infty}$ is composed of the matrices of the form $z_{\infty}$ times the identity matrix, and it acts trivially on the upper half-plane. Therefore we will have

$\displaystyle j(z_{\infty}g_{\infty},\tau)=\mathrm{sgn}(z_{\infty})\mathrm{det}(g_{\infty})^{-\frac{1}{2}}(c\tau+d)$

Now for the maximal compact subgroup $K_{\infty}^{+}$. As previously mentioned, this is the group $\mathrm{SO}(2)$, and may be expressed as matrices of the form

$\displaystyle k_{\theta}=\begin{pmatrix}\mathrm{cos}(\theta) & \mathrm{sin}(\theta)\\-\mathrm{sin}(\theta) & \mathrm{cos}(\theta)\end{pmatrix}$.

Then in the action of $\mathrm{GL}_{2}(\mathbb{R})^{+}$ on the upper half-plane, $Z_{\infty}K_{\infty}^{+}$ is the stabilizer of $i$. We will also have

$\displaystyle j(z_{\infty}k_{\theta},i)=\mathrm{sgn}(z_{\infty})e^{i\theta}$

This leads us to the second property our function $\phi$ must satisfy. First we consider $\phi_{\infty}$. For $z_{\infty}k_{\theta}\in Z_{\infty}K_{\infty}^{+}$, we must have

$\displaystyle \phi_{\infty}(g_{\infty}z_{\infty}k_{\theta})=\phi_{\infty}(g_{\infty})\mathrm{sgn}(z)^{m}(e^{i\theta})^{m}$.

Note the appearance of the weight $m$. Now when we extend this function $\phi_{\infty}$ on $\mathrm{GL}_{2}(\mathbb{R})^{+}$ to a function $\phi$ on $\mathrm{GL}_{2}(\mathbb{A})$, we must replace $Z_{\infty}$ by its connected component $Z_{\infty}^{+}$.

#### Holomorphicity and the action of the Lie algebra

Next we must translate the property that the modular form $f$ is holomorphic into a property of $\phi$. For this we shall introduce certain “raising” and “lowering” operators.

Let $\mathfrak{g}_{0}$ be the (real) Lie algebra of $\mathrm{GL}_{2}(\mathbb{R})^{+}$. An element $X\in\mathfrak{g}_{0}$ acts on the space of smooth functions on $\mathrm{GL}_{2}(\mathbb{R})^{+}$ as follows:

$\displaystyle X\phi(g_{\infty})=\frac{d}{dt}\phi(g_{\infty}\mathrm{exp}(tX))\bigg\vert_{t=0}$

We can extend this to an action of the complexified Lie algebra $\mathfrak{g}$, defined to be $\mathfrak{g}\otimes_{\mathbb{R}} \mathbb{C}$, by setting

$\displaystyle (X+iY)\phi=X\phi+iY\phi$

We now look at two special elements of $\mathfrak{g}$. They are

$\displaystyle X_{+}=\frac{1}{2}\begin{pmatrix}1 & i\\i & -1\end{pmatrix}$

and

$\displaystyle X_{-}=\frac{1}{2}\begin{pmatrix}1 & -i\\-i & -1\end{pmatrix}$.

Let us now look at how these special elements act on the smooth functions on $\mathrm{GL}_{2}(\mathbb{R})^{+}$. We have

$\displaystyle X_{+}\phi(g_{\infty}k_{\theta})=\phi(g_{\infty})(e^{i\theta})^{m+2}$

and

$\displaystyle X_{-}\phi(g_{\infty}k_{\theta})=\phi(g_{\infty})(e^{i\theta})^{m-2}$

In other words, the action of $X_{+}$ raises the weight by $2$, while the action of $X_{-}$ lowers the weight by $2$. Now it turns out that the condition that the function $f$ on the upper half-plane is holomorphic is the same condition as the function $\phi$ on $\mathrm{GL}_{2}(\mathbb{R})^{+}$ satisfying $X_{-}\phi=0$!

#### Holomorphicity at the cusps

Now we have expressed the holomorphicity of our modular form $f$ as a condition on our function $\phi$ on $\mathrm{GL}_{2}(\mathbb{A})$. However not only do we want our modular forms to be holomorphic on the upper half-plane, we also want them to be “holomorphic at the cusps”, i.e. they do not go to infinity at the cusps. This is going to be accomplished by requiring the function $g_{\infty}\mapsto \phi(g_{\infty}g_{f})$ to be “slowly increasing” for all $g_{f}\in\mathrm{GL}_{2}(\mathbb{A}_{f})$. This means that for all $g_{f}\in\mathrm{GL}_{2}(\mathbb{A}_{f})$, we have

$\displaystyle \vert \phi(g_{\infty}g_{f})\geq C\Vert g_{\infty}\Vert^{N}$

where $C$ and $N$ are some positive constants and the norm on the right-hand side is given by, for $g_{\infty}=\begin{pmatrix}a & b\\c & d\end{pmatrix}$,

$\displaystyle \Vert \phi(g_{\infty}g_{f})\Vert=(a^{2}+b^{2}+c^{2}+d^{2})(1+\mathrm{det}(g_{\infty}^{-2}))=\mathrm{Tr}(g_{\infty}^{T}g_{\infty})+\mathrm{Tr}((g_{\infty}^{-1})^{T}g_{\infty}^{-1})$.

#### Summary of the properties

Let us summarize now the properties we want our function $\phi$ to have in order that it come from a modular form $f$:

• For all $\gamma\in\mathrm{GL}_{2}(\mathbb{Q})$, we have $\phi(\gamma g)=\phi(g)$.
• For all $k_{f}\in K_{f}$, we have $\phi(gk_{f})=\phi(g)$.
• For all $g_{f}\in\mathrm{GL}_{2}(\mathbb{A}_{f})$, the function $g_{\infty}\mapsto \phi(g_{\infty}g_{f})$ is smooth.
• For all $k_{\theta}\in K_{\infty}$ we have $\phi(gk_{\theta})=\phi(g)e^{i\theta}$.
• The function $\phi$ is invariant under $Z_{\infty}^{+}$.
• We have $\displaystyle X_{-}\phi=0$.
• The function given by $g_{\infty}\mapsto\phi(g_{\infty}g_{f})$ is slowly increasing.

#### Cuspidality

Now let us consider the case where $f$ is a cusp form. We want to translate the cuspidality condition to a condition on $\phi$, and we do this by noting that this means that the Fourier expansion of $f$ has no constant term. Given that Fourier coefficients can be expressed using Fourier transforms, we make use of the measure theory on the adeles to express this cuspidality condition as

$\displaystyle \int_{\mathbb{Q}\setminus\mathbb{A}}\phi\left(\begin{pmatrix}1 & x\\0&1\end{pmatrix}\right)dx=0$.

### Automorphic forms

We have now defined modular forms as functions on $\mathrm{GL}_{2}(\mathbb{A})$, and enumerated some of their important properties. Modular forms, as functions on $\mathrm{GL}_{2}(\mathbb{A})$, turn out to be merely be specific examples of more general functions on $\mathrm{GL}_{2}(\mathbb{A})$ that satisfy similar, but more relaxed, properties. These are the automorphic forms.

The first few properties are the same, for instance for all $\gamma\in\mathrm{GL}_{2}(\mathbb{Q})$, we want $\phi(\gamma g)=\phi(g)$, and for all $k_{f}\in K_{f}$, where $K_{f}$ is a compact subgroup of $\mathrm{GL}_{2}(\mathbb{A}_{f})$, we want $\phi(g k)=\phi(g)$. We will also want the function given by $g_{\infty}\mapsto \phi(g_{\infty}g_{f})$ to be smooth for all $g_{f}\in\mathrm{GL}_{2}(\mathbb{A}_{f})$.

What we want to relax a little bit is the conditions on the actions of $K_{\infty}$, $Z_{\infty}^{+}$, and the Lie algebra $\mathfrak{g}$, in that we want the space we get by having them act on some function $\phi$ to be finite-dimensional. Instead of looking at the action of the Lie algebra $\mathfrak{g}$, it is often convenient to instead look at the action of its universal enveloping algebra $U(\mathfrak{g})$. The universal enveloping algebra is an honest to goodness associative algebra that contains the Lie algebra (and is in fact generated by its elements) such that the commutator of the universal enveloping algebra gives the Lie bracket of the Lie algebra. We shall denote the center of $U(\mathfrak{g})$ by $Z(\mathfrak{g})$. Now it turns out that $Z(\mathfrak{g})$ is generated by the Lie algebra of $Z_{\infty}^{+}$ and the Casimir operator $\Delta$, defined to be

$\displaystyle \Delta=H^{2}+2X_{+}X_{-}+2X_{-}X_{+}$

where $H$ is the element given by $\begin{pmatrix}0&-i\\i &0\end{pmatrix}$. Therefore, the action of the center of the universal enveloping algebra encodes the action of $Z_{\infty}^{+}$ and the Lie algebra $\mathfrak{g}$ at the same time.

Let us now define automorphic forms in general. Even though the focus on this post is on $\mathrm{GL}_{2}$ and over the rational numbers $\mathbb{Q}$, we can just give the most general definition of automorphic forms now, even for more general reductive groups and more general global fields. So let $G$ be a reductive group and let $F$ be a global field. The space of automorphic forms on $G$, denoted $\mathcal{A}$, is the space of functions $\phi:G(\mathbb{A}_{F})\to\mathbb{C}$ satisfying the following properties:

• For all $\gamma\in G(F)$, we have $\phi(\gamma g)=\phi(g)$.
• For all $k_{f}\in K_{f}$, $K_{f}$ a compact open subgroup of $G(\mathbb{A}_{f})$, we have $\phi(gk_{f})=\phi(g)$.
• For all $g_{f}\in G(\mathbb{A}_{F,f})$, the function $g_{\infty}\mapsto \phi(g_{\infty}g_{f})$ is smooth.
• The function $\phi$ is $K_{\infty}$-finite, i.e. the space $\mathbb{C}[K_{\infty}]\cdot\phi$ is finite dimensional.
• The function $\phi$ is $Z(\mathfrak{g})$-finite, i.e. the space $Z(\mathfrak{g})\cdot\phi$ is finite dimensional.
• The function $g_{\infty}\mapsto \phi(g_{\infty}g_{f})$ is slowly increasing.

Here slowly increasing means that for all embeddings $\iota:G_{\infty}\to\mathrm{GL}_{n}(\mathbb{R})$ of the infinite part of $G(\mathbb{A}_{F})$, we have

$\displaystyle \Vert \phi(g_{\infty}g_{f})\Vert=\mathrm{Tr}(\iota(g_{\infty})^{T}\iota(g_{\infty}))+\mathrm{Tr}((\iota(g_{\infty}^{-1})^{T}\iota(g_{\infty})^{-1})$.

Furthermore, we say that the automorphic form $\phi$ is cuspidal if, for all parabolic subgroups $P\subseteq G$, $\phi$ satisfies the following additional condition:

$\displaystyle \int_{N(\mathbb{F})\setminus N(\mathbb{A}_{F})}\phi(ng)dn=0$

where $N$ is the unipotent radical (the unipotent part of the maximal connected normal solvable subgroup) of the parabolic subgroup $P$.

These cuspidal automorphic forms, which we denote by $\mathcal{A}_{0}$, form a subspace of the automorphic forms $\mathcal{A}$.

### Automorphic forms and representation theory

As stated earlier, automorphic forms give us a way of understanding the representation theory of $G(\mathbb{A}_{F})$ where $G$ is a reductive group. Let us now discuss these representation-theoretic aspects.

We will actually look at automorphic forms not as representations of $G(\mathbb{A})$, but as $(\mathfrak{g},K_{\infty})\times G(\mathbb{A}_{F,f})$-modules. This means they have actions of $\mathfrak{g}$, $K_{\infty}$, and $G(\mathbb{A}_{F,f})$ all satisfying certain compatibility conditions. A $(\mathfrak{g},K_{\infty})\times G(\mathbb{A}_{F,f})$-module is called admissible if any irreducible representation $K_{\infty}\times K_{f}$ shows up inside it with finite multiplicity, and irreducible if it has no proper subspaces fixed by $\mathfrak{g}$, $K_{\infty}$, and $G(\mathbb{A}_{F,f})$. Recall from The Local Langlands Correspondence for General Linear Groups, irreducible admissible representations of $G(F_{v})$, where $F_{v}$ is some local field, are precisely the kinds of representations that show up in the automorphic side of the local Langlands correspondence.

In fact the global and the local picture are related by Flath’s theorem, which says that, for an irreducible admissible $(\mathfrak{g},K_{\infty})\times G(\mathbb{A}_{F,f})$-module $\pi$, we have the following factorization

$\displaystyle \pi=\bigotimes'_{v\not\vert\infty}\pi_{v}\otimes \pi_{\infty}$

into a restricted tensor product (explained in the next paragraph) of irreducible admissible representations $\pi_{v}$ of $G(F_{v})$, running over all places $v$ of $F$. At the infinite place, $\pi_{v}$ is an irreducible admissible $(\mathfrak{g}, K_{\infty})$-module.

The restricted tensor product is a direct limit over $S$ of $V_{S}=\bigotimes_{s\in S} \pi_{s}$ where for $S\subset T$ we have the inclusion $V_{S}\hookrightarrow V_{T}$ given by $x_{S}\mapsto x_{S}\otimes\bigotimes_{v\in T\setminus S}\xi_{v}^{0}$, where $\xi_{v}$ is a vector fixed by a certain maximal compact open subgroup (called hyperspecial) $K_{v}$ of $G(F_{v})$ (a representation of $G(F_{v})$ containing such a fixed vector is called unramified).

We have that $\mathcal{A}$ and $A_{0}$ are $(\mathfrak{g},K_{\infty})\times G(\mathbb{A}_{f})$-modules. An automorphic representation of a reductive group $G$ is an indecomposable $(\mathfrak{g},K_{\infty})\times G(\mathbb{A}_{F,f})$-module that is isomorphic to a subquotient of $\mathcal{A}$. A cuspidal automorphic representation is an automorphic representation that is isomorphic to a subquotient of $\mathcal{A}_{0}$. It is a property of the space of cuspidal automorphic forms that it is semisimple, i.e. it decomposes into a direct product of cuspidal automorphic representations (a property that is not necessarily shared by the bigger space of automorphic forms!).

An automorphic form generates such an automorphic representation, and a theorem of Harish-Chandra states that such a representation is admissible. In fact, automorphic representations are always admissible. Again recalling that irreducible admissible representations of $G(F_{v})$ make up the automorphic side of the local Langlands correspondence, we therefore expect that automorphic representations of $G$ will make up the automorphic side of the global Langlands correspondence.

However, to state the original global Langlands correspondence in general is still quite complicated, as it involves an as of-yet hypothetical object called the Langlands group, which plays a somewhat analogous role as the Weil group in the local Langlands correspondence. Instead there are certain variants of it that are considered easier to approach, for instance by imposing conditions on the representations such as being “algebraic at infinity“. These variants of the global Langlands correspondence will hopefully be discussed in future posts.

References:

Automorphic form on Wikipedia

Topics in automorphic forms (notes by Chao Li from a course by Jack Thorne)

MSRI Summer School on Automorphic Forms and the Langlands ProgramÂ by Kevin Buzzard

The Automorphic Project

An Introduction to the Langlands Program by Daniel Bump, James W. Cogdell, Ehud de Shalit, Dennis Gaitsgory, Emmanuel Kowalski, and Stephen S. Kudla (edited by Joseph Bernstein and Stephen Gelbart)

# The Global Langlands Correspondence for Function Fields over a Finite Field

In The Local Langlands Correspondence for General Linear Groups, we introduced some ideas related to what is known as the Langlands program, and discussed in a little more detail the local Langlands correspondence, at least for general linear groups.

In this post, we will discuss the global Langlands correspondence, but we will focus on the case of function fields over a finite field. This will be somewhat easier to state than the case of number fields, and at the same time perhaps give us a bit more geometric intuition. Let us fix a smooth, projective, and irreducible curve $X$, defined over a finite field $\mathbb{F}_{q}$. We let $F$ be its function field. For instance, if $X$ is the projective line $\mathbb{P}^{1}$ over $\mathbb{F}_{q}$, then $F=\mathbb{F}(t)$.

### The case of $\mathrm{GL}_{1}$: Global class field theory for function fields over a finite field

To motivate the global Langlands correspondence for function fields, let us first think of the $\mathrm{GL}_{1}$ case, which is a restatement of (unramified) global class field theory for function fields. Recall that in Some Basics of Class Field Theory global class field theory tells us that for global field $F$, its maximal unramified abelian extension $H$, also called the Hilbert class field of $F$, has the property that $\mathrm{Gal}(H/F)$ is isomorphic to the ideal class group.

We recall that there is an analogy between the absolute Galois group and the etale fundamental group in the case when there is no ramification. Therefore, in the case of function fields, the corresponding statement of unramified global class field theory may be stated as

$\displaystyle \pi_{1}(X,\overline{\eta})^{\mathrm{ab}}\times_{\widehat{\mathbb{Z}}}\mathbb{Z}\xrightarrow{\sim} \mathrm{Pic}(\mathbb{F}_{q})$

where $\pi_{1}(X,\overline{\eta})$ is the etale fundamental group of $X$, a profinite quotient of $\mathrm{Gal}(\overline{F}/F)$ through which its action factors ($\overline{\eta}$ here serves as the basepoint, which is needed to define the etale fundamental group). The Picard scheme $\mathrm{Pic}$ is the scheme such that for any scheme $S$ its $S$ points $\mathrm{Pic}(S)$ correspond to the isomorphism classes of line bundles on $X\times S$. This is analogous to the ideal class group. Taking the fiber product with $\mathbb{Z}$ is analogous to taking the Weil group (see also Weil-Deligne Representations and The Local Langlands Correspondence for General Linear Groups).

The global Langlands correspondence, in the case of $\mathrm{GL}_{1}$, is a restatement of this in terms of maps from each side to some field (we will take this field to be $\overline{\mathbb{Q}}_{\ell}$). It states that there is a bijection between characters $\sigma:\pi_{1}(X,\overline{\eta})\to \overline{\mathbb{Q}}_{\ell}^{\times}$, and $\chi:\mathrm{Pic}(\mathbb{F}_{q})/a^{\mathbb{Z}}\to \overline{\mathbb{Q}}_{\ell}^{\times}$ where $a$ is any element of $\mathrm{Pic}(\mathbb{F}_{q})$ of nonzero degree. Again this is merely a restatement of unramified global class field theory, and nothing has changed in its content. However, this restatement points to us the way in which it may be generalized.

### Generalizing to $\mathrm{GL}_{n}$, and then to more general reductive groups

To generalize this, we may take maps $\sigma:\pi_{1}(X,\overline{\eta})\to \mathrm{GL}_{n}(\overline{\mathbb{Q}}_{\ell})$ instead of maps $\sigma:\pi_{1}(X,\overline{\eta})\to \overline{\mathbb{Q}}_{\ell}^{\times}$, since $\overline{\mathbb{Q}}_{\ell}^{\times}$ is just $\mathrm{GL}_{1}(\overline{\mathbb{Q}}_{\ell})$. To make it look more like the case of number fields, we may also define this same map as a map $\sigma:\mathrm{Gal}(\overline{F}/F)\to \mathrm{GL}_{n}(\overline{\mathbb{Q}}_{\ell})$ which factors through $\pi_{1}(U,\overline{\eta})$ for some open dense subset $U$ of $X$. This side we call the “Galois side” (as it involves the Galois group).

What about the other side (the “automorphic side”)? First we recall that $\mathrm{Pic}(\mathbb{F}_{q})$ classifies line bundles on $X$. We shall replace this by $\mathrm{Bun}_{n}(\mathbb{F}_{q})$, which classifies rank $n$ vector bundles on $X$. It was figured out by Andre Weil a long time ago that $\mathrm{Bun}_{n}(\mathbb{F}_{q})$ may also be expressed as the double quotient $\mathrm{GL}_{n}(F)\backslash\mathrm{GL}_{n}(\mathbb{A}_{F})/\mathrm{GL}_{n}(\prod_{v}\mathcal{O}_{F_{v}})$ (this is known as the Weil parametrization). Now functions on this space will give representations of $\mathrm{GL}_{n}(\mathbb{A}_{F})$. We will be interested not in all functions on this space, but in particular certain kinds of functions called cuspidal automorphic forms, which gives a representation that decomposes into pieces that we then want to match up with the Galois representations.

In fact we can generalize even further and consider reductive groups (see also Reductive Groups Part I: Over Algebraically Closed Fields and Reductive Groups Part II: Over More General Fields) other than $\mathrm{GL}_{n}$! Let $G$ be such a reductive group over $F$. Instead of $\mathrm{Bun}_{n}(\mathbb{F}_{q})$ we now consider $\mathrm{Bun}_{G}(\mathbb{F}_{q})$, the moduli stack (see also Algebraic Spaces and Stacks) of $G$-bundles on $X$. As above, we consider the space of cuspidal automorphic forms on $\mathrm{Bun}_{G}(\mathbb{F}_{q})$, which we shall denote by $C_{c}^{\mathrm{cusp}}(\mathrm{Bun}_{G}(\mathbb{F}_{q})/\Xi,\overline{\mathbb{Q}}_{\ell})$. Here $\Xi$ is a subgroup of finite index in $\mathrm{Bun}_{Z}(\mathbb{F}_{q})$, where $Z$ is the center of $G$.

As we are generalizing to more general reductive groups than just $\mathrm{GL}_{n}$, we need to modify the other side (the Galois side) as well. Instead of considering Galois representations, which are group homomorphisms $\sigma: \mathrm{Gal}(\overline{F}/F)\to \mathrm{GL}_{n}(\overline{\mathbb{Q}}_{\ell})$, we must now consider L-parameters, which in this context are group homorphisms $\sigma: \mathrm{Gal}(\overline{F}/F)\to \widehat{G}(\overline{\mathbb{Q}}_{\ell})$, where $\widehat{G}$ is the dual group of $G$ (which as one may recall from Reductive Groups Part II: Over More General Fields, has the roots and coroots of $G$ interchanged).

We may now state the “automorphic to Galois” direction of the global Langlands correspondence for function fields over a finite field $\mathbb{F}_{q}$, which has been proven by Vincent Lafforgue. It says that we have a decomposition

$\displaystyle C_{c}^{\mathrm{cusp}}(\mathrm{Bun}_{G}(\mathbb{F}_{q})/\Xi,\mathbb{Q}_{\ell})=\bigoplus_{\sigma} \mathfrak{H}_{\sigma}$

of the space $C_{c}^{\mathrm{cusp}}(\mathrm{Bun}_{G}(\mathbb{F}_{q})/\Xi,\mathbb{Q}_{\ell})$ into subspaces $\mathfrak{H}_{\sigma}$ indexed by L-parameters $\sigma$. It is perhaps instructive to compare this with the local Langlands correspondence as stated in Reductive Groups Part II: Over More General Fields, to which it should be related by what is known as local-global compatibility.

(The “Galois to automorphic direction” concerns whether an L-parameter is “cuspidal automorphic”, and we will briefly discuss some partial progress by Gebhard BĂ¶ckle , Michael Harris, Chandrasekhar Khare, and Jack Thorne later at the end of this post.)

Furthermore the decomposition above must respect the action of Hecke operators (analogous to those discussed in Hecke Operators). Let us now discuss these Hecke operators.

### Hecke operators

Let $\mathcal{E},\mathcal{E}'$ be two $G$-bundles on $X$. Let $x$ be a point of $X$, and let $\phi:\mathcal{E}\to\mathcal{E}'$ be an isomorphism of $G$ bundles over $X\setminus x$. We say that $(\mathcal{E}',\phi)$ is a modification of $\mathcal{E}$ at $x$. A modification can be bounded by a cocharacter, i.e. a homomorphism $\lambda:\mathbb{G}_{m}\to G$. This keeps track and bounds the vector bundles’ relative position.

To get an idea of this, we consider the case $G=\mathrm{GL}_{n}$. Consider the completion $\mathcal{E}_{x}^{\wedge}$ of stalk of the vector bundle $\mathcal{E}$ at $x$. It is a free module over the completion $\mathcal{O}_{X,x}^{\wedge}$ of the structure sheaf at $x$, which happens to be isomorphic to $\mathbb{F}_{q}[[t]]$. Let $(\mathcal{E}',\phi)$ be a modification of $\mathcal{E}$ at $x$. There is a basis $e_{1},\ldots,e_{n}$ of $\mathcal{E}_{x}^{\wedge}$ such that $t^{k_{1}}e_{1},\ldots,t^{k_{n}}e_{n}$ is a basis of $\mathcal{E}_{x}^{'\wedge}$, where $k_{1}\geq\ldots\geq k_{n}$. But the numbers $k_{1},\ldots,k_{n}$ is the same as a cocharacter $\lambda:\mathbb{G}_{m}\to\mathrm{GL}_{n}$, given by $\mu(t)=\mathrm{diag}(t^{k_{1}},\ldots,t^{k_{n}})$.

The Hecke stack $\mathrm{Hck}_{v,\lambda}$ is the stack whose points $\mathrm{Hck}_{v,\lambda}(\mathbb{F}_{q})$ correspond to modifications $(\mathcal{E},\mathcal{E}',\phi)$ at $v$ whose relative position is bounded by the cocharacter $\lambda$. It has two maps $h^{\leftarrow}$ and $h^{\rightarrow}$ to $\mathrm{Bun}_{G}(\mathbb{F}_{q})$, which send the modification $(\mathcal{E},\mathcal{E}',\phi)$ to $\mathcal{E}$ and $\mathcal{E}'$ respectively. The Hecke operator $T_{\lambda,v}$ is the composition $h_{*}^{\rightarrow}\circ h^{\leftarrow *}$. In essence what it does is it sends a function $f$ in $C_{c}^{\mathrm{cusp}}(\mathrm{Bun}_{G}(\mathbb{F}_{q})/\Xi,\mathbb{Q}_{\ell})$ to the function which sends a point in $\mathrm{Bun}_{G}(\mathbb{F}_{q})$ corresponding to the $G$-bundle $\mathcal{E}$ to the sum of the values of $f(\mathcal{E}')$ over all modifications of $G$-bundles $\phi:\mathcal{E}'\to\mathcal{E}$ at $v$ bounded by $\lambda$. In this last description one can see that it is in fact analogous to the description of Hecke operators for modular forms discussed in Hecke Operators.

More generally given a representation $V$ of $\widehat{G}$, we can obtain a Hecke operator $T_{V}$, and these Hecke operators have the property that if $V=V'\oplus V''$, we must have $T_{V,v}=T_{V',v}+T_{V'',v}$, and if $V=V'\otimes V''$ , we must have $T_{V,v}=T_{V',v}T_{V'',v}$. If $V$ is irreducible, then we can build $T_{V,v}$ as a combination of $T_{\lambda,v}$, where the $\lambda$‘s are the weights of $V$.

Now let us go back to the decomposition

$\displaystyle C_{c}^{\mathrm{cusp}}(\mathrm{Bun}_{G}(\mathbb{F}_{q})/\Xi,\mathbb{Q}_{\ell})=\bigoplus_{\sigma} \mathfrak{H}_{\sigma}.$

The statement of the global Langlands correspondence for function fields over a finite field $\mathbb{F}_{q}$ additionally requires that the Hecke operators preserve the subspaces $\mathfrak{H}_{\sigma}$; that is, they act on each of these subspaces, and do not send an element of such a subspace to another outside of it. Additionally, we require that the action of the Hecke operators are “compatible with the Satake isomorphism”. This means that the action of a Hecke operator $T_{V,v}$ is given by multiplication by the scalar $\mathrm{Tr}_{V}(\sigma(\mathrm{Frob}_{v}))$. This is somewhat analogous to the Eichler-Shimura relation relating the Hecke operators and the Frobenius briefly mentioned in Galois Representations Coming From Weight 2 Eigenforms.

### Ideas related to the proof of the automorphic to Galois direction: Excursion operators and the cohomology of moduli stacks of shtukas

Let us now discuss some ideas related to Vincent Lafforgue’s proof of “automorphic to Galois direction” of the global Langlands correspondence for function fields over a finite field. An important part of these concerns the algebra of excursion operators, denoted by $\mathcal{B}$. These are certain endomorphisms of $C_{c}^{\mathrm{cusp}}(\mathrm{Bun}_{G}(\mathbb{F}_{q})/\Xi,\mathbb{Q}_{\ell})$ which include the Hecke operators. The idea of the automorphic to Galois direction is that characters $\nu:\mathcal{B}\to \overline{\mathbb{Q}}_{\ell}^{\times}$ correspond uniquely to some L-parameter $\sigma$.

To understand these excursion operators better, we will look at how they are constructed. The construction of the excursion operators involves the cohomology of moduli stacks of shtukas.

A shtuka is a very special kind of a modification of a vector bundle. Given an indexing set $I$, a shtuka over a scheme $S$ over $\mathbb{F}_{q}$ consists of the following data:

• A set of points $(x_{i})_{i\in I}:S\to X^{I}$ (the $x_{i}$ are called the “legs” of the shtuka)
• A $G$-bundle $\mathcal{E}$ over $X\times S$
• An isomorphism

$\displaystyle \phi: \mathcal{E}\vert_{(X\times S)\setminus (\bigcup_{i\in I}\Gamma_{x_{i}}}\xrightarrow{\sim}(\mathrm{Id}\times \mathrm{Frob}_{S})^{*}\mathcal{E}\vert _{(X\times S)\setminus (\bigcup_{i\in I}\Gamma_{x_{i}})}$

where $\Gamma_{x_{i}}$ is the graph of the $x_{i}$‘s. Let us denote the moduli stack of such shtukas by $\mathrm{Sht}_{I}$. We take note of the important fact that the moduli stack of shtukas with no legs, $\mathrm{Sht}_{\emptyset}$, is a discrete set of points and is in fact the same as $\mathrm{Bun}_{G}(\mathbb{F}_{q})$!

We now want to define sheaves on $\mathrm{Sht}_{I}$ which will serve as coefficients when we take its etale cohomology, and we want these sheaves to depend on representations $W$ of $\widehat{G}^{I}$, for the eventual goal of having the cohomology (or appropriate subspaces of it that we want to consider) be functorial in $W$. This is to be accomplished by considering another moduli stack, the moduli stack of modifications over the formal neighborhood of the legs $x_{i}$. This parametrizes the following data:

• The set of points $(x_{i})_{i\in I}:S\to X^{I}$
• A pair of $G$-bundles $\mathcal{E}$ and $\mathcal{E}'$ on the formal completion