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):


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 Anschütz and Arthur-César 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.


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 Anschütz and Arthur-César Le Bras

Cohomologue p-adique de la Tour de Drinfeld: le Cas de la Dimension 1 by Pierre Colmez, Gabriel Dospinescu, and Wiesława Nizioł

A p-adic 6-Functor Formalism in Rigid-Analytic Geometry by Lucas Mann

Lectures on Condensed Mathematics by Peter Scholze

Report from Oberwolfach from totallydisconnected


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.


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