Phacopsrana

I’ve recently been studying the Local Langlands correspondence for $\mathrm{GL_n}$ between complex admissible representations of $\mathrm{GL_n(K)}$ for $\mathrm{K}$ an extension of $\mathbb{Q}_p$ and admissible representations of $\mathrm{W'_K}$ , the Weil-Deligne group of $\mathrm{K}$ . I’ve been found all the different conventions and notations very confusing so this blog post is my attempt to untangle things. As usual, I plan to be as concise as possible and so this means I will not include most proofs. To compensate, I will try to be as careful as possible to provide good references. The goal of this post is to state the facts clearly, rather than spend too much time on intuition. None of the below is original and comes primarily from the references listed below.

On the Galois side of Local Langlands, we have four groups whose representations we consider.

The Weil group $\mathrm{W_K}$ of $\mathrm{K}$ .
The absolute Galois group $\mathrm{G_K}$ of $\mathrm{K}$ .
The Weil Deligne group which is given by $\mathrm{W_K} \rtimes \mathrm{G_a}$ where $\mathrm{W_K}$ acts by $wxw^{-1}=||w||x$ .
The Langlands Group $\mathrm{W_K} \times \mathrm{SL_2}(\mathbb{C})$ .

We begin by drawing our focus to the first two entries of this list. When we consider representations of these two groups we consider finite dimensional representations of complex vector spaces as well as $l$ -adic ones ( $\overline{\mathbb{Q}_l}$ -vector spaces for $l \neq p$ ). Notice that $\mathbb{C}, \overline{\mathbb{Q}_l}$ are non-canonically isomorphic as fields but carry different topologies.

We typically impose one of two continuity conditions on representations.

A representation $\rho: \mathrm{G} \to \mathrm{GL(V)}$ is continuous if it is continuous as a function between topological groups.
A representation is smooth if the stabilizer of each vector is an open subgroup.

The second condition doesn’t depend on the topology of the target so smooth $l$ -adic representations are the same thing as smooth complex representations.

A continuous complex representation of $\mathrm{G_K}$ has finite image. This result comes from the fact that $\mathrm{GL_n}(\mathbb{C})$ has no small subgroups: there exists an open subset $\mathrm{U} \subset \mathrm{GL_n}(\mathbb{C})$ containing the identity but no nontrivial subgroups. Thus for $\rho: \mathrm{G} \to \mathrm{GL(V)}$ , $\rho^{-1}(\mathrm{U})$ is open so contains an open subgroup whose image is a subgroup of $\mathrm{U}$ and so is the identity. An open subgroup of $\mathrm{G_K}$ has finite index.

For the $\mathrm{G_K}$ case any complex representation with finite image is continuous. If a complex representation of $\mathrm{G_K}$ is smooth then we take a basis $\{e_1, ..., e_n\}$ and an open subgroup $\mathrm{U_i} \subset \mathrm{G_K}$ fixing $e_i$ . Then $\cap_i \mathrm{U_i}$ is open and fixes the entire representation.

Combining the above: smooth and continuous complex representations are the same thing.

We now consider complex representations of $\mathrm{W_K}$ . The same analysis we applied above to $\mathrm{G_K}$ holds for $\mathrm{I_K}$ . In particular smooth and continuous representations of $\mathrm{I_K}$ have finite image and are trivial on an open subgroup. The pre-image of any subset of the image is an open subset of $\mathrm{W_K}$ since it is a union of translates of the kernel of the representation on $\mathrm{I_K}$ . Thus smooth and continuous complex representations of $\mathrm{W_K}$ are the same.

In conclusion, the eight combinations of the words $\mathrm{W_K, G_K}, l$ -adic, complex, smooth, and continuous split into 4 distinct sets:

Smooth and continuous complex $\mathrm{W_K}$ -representations, smooth $l$ -adic $\mathrm{W_K}$ representations.
Smooth and continuous complex $\mathrm{G_K}$ -representations, smooth $l$ -adic $\mathrm{G_K}$ representations.
Continuous $l$ -adic representations of $\mathrm{W_K}$ .
Continuous $l$ -adic representations of $\mathrm{G_K}$ .

Notice that restricting an element of 2, 4 to $\mathrm{W_K}$ gives an element of 1, 3 respectively. Also notice that 1, 2 are examples of 3, 4 respectively. Thus 3 is the most general type of object and in fact a subcategory of $3$ is the object on the Galois side of the Local Langlands correspondence.

Notice that the sets 3, 4 defined above depend apriori on a choice of $l$ . But we will see their stucture doesn’t in any particularly important way. Thus it woud be useful to to try to formulate these sets in a way that doesn’t explicitely use the topology of the target. This is accomplished in the literature by the notion of a Weil-Deligne representation. These can be defined as representations of a Weil-Deligne group as mentioned at the start of this post. However, there is an equivalent definition using less machinery and appearing much more frequently in the literature, which we now give.

Definition. A Weil-Deligne representation is an ordered triple $(\rho, N, V)$ consisting of

An $l$ -adic or complex vector space $V$ (as remarked earlier, smooth representations don’t depend on the topology of the target though one would need to choose an isomorphism between $\overline{\mathbb{Q}_l}$ and $\mathbb{C}$ to show this).
A smooth $\mathrm{W_K}$ -representation $\rho$ with image in $\mathrm{GL(V)}$ .
A nilpotent element $N \in \mathrm{End}(V)$ such that for all $w \in \mathrm{W_K}$ , $\rho(w)N\rho(w)^{-1}=||w||N$ .

We define a morphism of Weil-Deligne representations to be a morphism $f: (\rho_1, V_1) \to (\rho_2, V_2)$ of smooth representations so that $f \circ N_1=N_2 \circ f$ .

Key Theorem. The category of Weil-Deligne representations is equivalent ot the category of continuous $l$ -adic representations of $\mathrm{W_K}$ .

We will not prove this fact here though it comes as a consequence of Grothendieck’s $l$ -adic monodromy theorem which states that for $\rho$ a continuous $l$ -adic representation of $\mathrm{W_K}$ , there exists an open subgroup $U$ of $\mathrm{I_K}$ so that $\rho|_U$ is unipotent. In particular, $\rho|_U$ is given by $\rho(x)=exp(t_l(x) N)$ for some unique nilpotent operator $N$ where $t_l$ is the $l$ -adic cyclotomic character. This theorem then gives one a a way of “converting” a continuous $l$ -adic representation into a smooth one. Simply make a choice $\phi$ Frobenius and define $\rho'$ by $\rho'(\phi^ax)=\rho(\phi^a x) exp( -t_l(x) N)$ . So the equivalence of categories is given by $\mathrm{WD: Rep(W_K)} \to \mathrm{Rep(WD_K)}$ , $(\rho, V) \mapsto (\rho', N ,V)$ .

Theorem. The category of continuous $l$ -adic representations of $\mathrm{G_K}$ is equivalent to the category of Weil-Deligne representations such that $\rho(\phi)$ has eigenvalues with absolute value $1$ (where $\phi$ is a choice of Frobenius).

Semi-Simplifications

I find the various notions of semisimple Galois representations to be quite confusing. Let me try to explain what I think is the correct state of affairs.

The Galois side of the Local Langlands correspondence deals with Frobenius semisimple Weil-Deligne representations. These are tuples $( \rho, N, V)$ satisfying the Weil-Deligne representation conditions as well as the further condition that if $\phi$ is some choice of Frobenius in $\mathrm{W_K}$ , then $\rho(\phi)$ is semisimple. Since $\mathrm{I_K}$ is compact, this is in fact equivalent to the condition that $\rho$ is semisimple. However, (and confusingly!) this is not the same thing as a semisimple object in the category of Weil-Deligne representations. We have an equivalence of categories between Weil-Deligne representations and continuous $l$ -adic representations of $\mathrm{W_K}$ and this equivalence preserves the categorical notion of semisimplicity. Thus we are free to consider it in the category of continuous $l$ -adic representations. In particular, we want to show that if $\rho$ is a continuous $l$ -adic representation, then the associated $N$ must be $0$ . We can assume without loss of generality that $N$ is a matrix with $e_{ij}=1$ for $j=i+1$ and $0$ otherwise and it suffices to show that $N$ is a $1 \times 1$ matrix. Observe that the kernel of $N$ is one-dimensional in this case. But if $v \in ker(N)$ , then $0=\rho'(w)Nv=||w|| N \rho'(w)v$ and so $\rho(w)'v$ is a scalar multiple of $v$ for each $w \in \mathrm{W_K}$ . Thus $\rho'$ and therefore $\rho$ stabilize the one dimensional subspace spanned by $v$ . But this subspace has no complement when we restrict $\rho$ to $U \subset I_K$ and so $\rho$ is not semisimple.

To summarize, the following are equivalent:

$\rho$ is a semisimple continuous $l$ -adic representation of $\mathrm{W_K}$ .
$\rho$ is a semisimple smooth $l$ -adic representation of $\mathrm{W_K}$ .
$\rho$ is a smooth $l$ -adic representation of $\mathrm{W_K}$ such that $\rho(\phi)$ is semisimple (where $\phi$ is a choice of Frobenius)

We complete this section by describing what the representations in question look like.

By our discussions above, a continuous complex representation $(\rho, V)$ of $\mathrm{G_K}$ has finite image and so in particular factors through some finite extension. This means that all such representations are given by an embedding of some finite Galois group $\mathrm{Gal(L/K)} \hookrightarrow \mathrm{GL(V)}$ .

We introduce the character $\omega: \mathrm{W_K} \to \mathbb{C}^{\times}$ such that $\omega(\mathrm{I_K})=1$ and if $\phi$ is a lift of Frobenius then $\omega(\phi)=q^{-1}$ where $q$ is the cardinality of the residue field of $K$ . More generally we have characters of the form $\omega^s$ so that $\omega^s(\phi)=q^{-s}$ . Then any irreducible complex representation of $\mathrm{W_K}$ is isomorphic to $\rho \otimes \omega^s$ where $\rho$ is the restriction of an irreducible complex representation of $\mathrm{G_K}$ .

We now consider Weil-Deligne representations. We define the special representation $\mathrm{Sp_n}$ by considering vectors $e_0, ..., e_{n-1}$ and defining $\rho(g)e_j=\omega(g)^je_j$ for $g \in \mathrm{W_K}$ and $Ne_j=e_{j+1}$ and $Ne_{n-1}=0$ . Then any indecomposable Weil-Deligne representation is isomorphic to $\rho \otimes \mathrm{Sp_n}$ where $\rho$ is an irreducible complex representation of $\mathrm{W_K}$ .

Langlands Group

The above theory enables us to adequately understand the Galois side of the Local Langlands correspondence (at least for the $\mathrm{GL_n}$ case!). However the situation is slightly unsatisfactoy in that we are forced to deal with Frobenius semisimple representations which form an unusual subcategory of Weil-Deligne representations. The above terminology is the most common in the literature but we can work with a slightly more familiar class of representations by considering the Langlands group $\mathrm{L_K}=\mathrm{W_K} \times \mathrm{SL_2}(\mathbb{C})$ .

A representation of $\mathrm{L_K}$ is a continuous complex finite dimensional representation of $\mathrm{L_K}$ so that the restriction to $\mathrm{SL_2}(\mathbb{C})$ is complex analytic. Such a representation is semisimple if and only if its restriction to $\mathrm{W_K}$ is semisimple.

Theorem. There is a bijection between Frobenius semisimple Weil-Deligne representations and semisimple representations of $\mathrm{L_K}$ given by $\pi \otimes \mathrm{Sp_n} \mapsto \pi \boxtimes \mathrm{sym}(n)$ (where $\mathrm{sym}(n)$ is the natural representation of $\mathrm{SL_2}(\mathbb{C})$ on homogeneous polynomials of degree $n-1$ .

References

Rohrlich’s exposition as posted on Dylan Yott’s website Rohrlich

Toby Gee’s notes from Arizona Winter School Gee

Commelin’s exposition Commelin

Taylor’s paper (page 5) Taylor

Last week we showed how to associate a linear algebraic object, the Dieudonne module, to a $p$ -divisible group. Furthermore, we saw how in the algebraically closed case the Dieudonne module is determined by its slopes. Today’s talk generalizes the previous setting in two ways. Firstly we will broaden our perspective to that of isocrystals with additional structure. This is analogous to how when we study Abelian varieties we often keep track of extra data such as the endomorphism ring and polarizations. Secondly, we move to the more general language of Tannakian categories. This will make the key results more straightforward to prove and understand and is generally very useful to know in many situations, such as when one is studying Shimura varieties or related constructions.

Continue reading “P-Divisible Groups Mini Course: Talk 3” →