# 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