Weil-Deligne group and Weil-Deligne group

(Yes, you read it correctly.)

Yesterday, I encountered two different definitions of Weil-Deligne groups. Although both definitions are extensions of a Weil group, one is a non-split extension by $\mathbb{C}$ and the other is just a direct product with $\mathrm{SL}_{2}(\mathbb{C})$.

Definition 1 (Weil-Deligne group) Let $F$ be a non-archimedean local field and $W_F$ be the Weil group of $F$. The Weil-Deligne group of $F$ is defined as a semi-direct product

\[W_{F}' := W_{F} \ltimes \mathbb{C}\]

with the action of $W_F$ on $\mathbb{C}$ given by

\[gxg^{-1} = ||g||x, \quad g \in W_F, x \in \mathbb{C}\]

where $g$ acts on the residue field of $F$ via $a \mapsto a^{||g||}$. The group structure of $W_F’$ is given by

\[(g_1, x_1)(g_2, x_2) := \left(g_1 g_2, x_1 + ||g_1||x_2 \right).\]

Definition 2 (Weil-Deligne group) Let $F$ be a non-archimedean local field and $W_F$ be the Weil group of $F$. The Weil-Deligne group of $F$ is defined as a direct product

\[W_F' := W_F \times \mathrm{SL}_{2}(\mathbb{C}).\]

These two definitions come from slightly different context. The first definition is more Galois-theoretic than the second definition, and it is also known that the category of $\ell$-adic representations of $\mathrm{Gal}(\overline{F}/F)$ embeds fully faithfully into the category of representations of Weil-Deligne group $W_F’$ (see Tate’s article [2]). The second definition is used to define $L$-parameters and their $L$-functions (see 12.2 of [3]). The goal of this post is to explain how the representations of Weil-Deligne groups with the above definitions become equivalent. Before we start, we’ll explain the representations of each group first.

Representations of $W_F’= W_F \ltimes \mathbb{C}$

Finite dimensional complex representations of $W_F’$ are continuous homomorphisms from $W_F’$ to $\mathrm{GL}_n(\mathbb{C})$. One can check that giving such representation $\sigma’ :W_F’ \to \mathrm{GL}_n(\mathbb{C})$ is equivalent to give a pair $(\sigma, N)$ where $\sigma :W_F \to \mathrm{GL}_n(\mathbb{C})$ is $n$-dimensional representation of $W_F$ and $N \in \mathrm{GL}_n(\mathbb{C})$ is a nilpotent element satisfying

\[\sigma(g) N \sigma(g)^{-1} = ||g||N\]

for $g \in W_F$. We can recover $\sigma’$ from $(\sigma, N)$ by¹

\[\sigma'((g, x)) = \exp(xN)\sigma(g),\]

which becomes a homomorphism from $W_F’$ to $\mathrm{GL}_n(\mathbb{C})$:

\[\begin{align*} \sigma'((g_1, x_1)(g_2, x_2)) &= \sigma'\left(g_1 g_2, x_1 + ||g_1||x_2\right) \\ &= \exp \left(\left(x_1 + ||g_1||x_2\right)N\right) \sigma(g_1 g_2) \\ &= \exp \left(x_1 N\right) \exp(||g_1||x_2 N) \sigma(g_1) \sigma(g_2)\\ &= \exp(x_1N) \sigma(g_1) \exp(x_2 N) \sigma(g_2) \\ &= \sigma'((g_1, x_1)) \sigma'((g_2, x_2)). \end{align*}\]

Here we use the equation $\sigma(g) \exp(xN) \sigma(g)^{-1} = \exp(||g|| x N)$ that follows from $\sigma(g) N \sigma(g)^{-1} = ||g||N$ by exponentiating both sides. Conversely, we can find $\sigma, N$ corresponding to $\sigma’ : W_F’ \to \mathrm{GL}_n(\mathbb{C})$ via

\[\sigma = \sigma'|_{W_F}, \quad N = \frac{\log \sigma'(x)}{x}.\]

(One need to check that the log function is well-defined and $N$ is independent of the choice of $x \in \mathbb{C}$ considered as an element $(1, x)$ of $W_F’$. See section 3 of [1] for the details.)

We define the direct sum and tensor product of such representations as

\[\begin{align*} (\sigma_1, N_1) \oplus (\sigma_2, N_2) &:= (\sigma_1 \oplus \sigma_2, N_1 \oplus N_2)\\ (\sigma_1, N_1) \otimes (\sigma_2, N_2) &:= (\sigma_1 \otimes \sigma_2, N_1 \otimes 1+ 1 \otimes N_2) \end{align*}\]

We call that $\sigma’ =(\sigma, N)$ is admissible if $\sigma$ is semisimple, and indecomposable if the space of $\sigma’$ cannot be written as a direct sum of proper subspaces invariant under $W_F’$. For each $n$, we can define a *special representation of dimension $n$, $\mathrm{sp}(n)$, as follows. Let $e_0, e_1, \dots, e_{n-1} \in \mathbb{C}^{n}$ be the standard basis, and define $\mathrm{sp}(n)$ via

\[\begin{align*} \sigma(g) e_j &= ||g||^{j} e_j, \quad 0 \leq j \leq n-1 \\ Ne_j &= e_{j+1}, \quad 0 \leq j \leq n-2 \\ Ne_{n-1} &= 0. \end{align*}\]

(You may found that $N$ corresponds to the standard nilpotent matrix with one’s on the off-diagonal. Also, it is not hard to check that this indeed a representation of Weil-Deligne group.) For $n> 1$, $\mathrm{sp}(n)$ is not irreducible since $\ker(N) = \mathbb{C}e_{n-1}$ is a 1-dimensional invariant subspace. However, it is indecomposable: if $\mathbb{C}^{n} = U \oplus W$ for some proper invariant subspaces $U$ and $W$, both $U \cap \ker N$ and $V \cap \ker N$ should be non-trivial, which is impossible since $\ker N$ has dimension 1. In fact, all the admissible indecomposable representations of $W_F’$ are tensor products of irreducible representations of $W_F$ and special representations.

Proposition 1 Every admissible indecomposable representation of $W_F’$ is isomorphic to $\pi \otimes \mathrm{sp}(n)$ for some irreducible representation of $W_F$ and $n \geq 1$.

Proof. See Proposition 3.1.3 of [4]. $\square$

Also, any admissible (finite-dimensional) representation has a unique decomposition with factors of the above form.

Proposition 2 Every admissible representation $\sigma’$ of $W_F’$ has a decomposition of the form

\[\sigma' = \bigoplus_{i=1}^{s} \pi_i \otimes \mathrm{sp}(n_i)\]

which is unique up to permuting the factors.

Proof. Decomposability follows from the induction on the dimension of $\sigma’$. For uniqueness, see Corollary 2 of [1]. $\square$

Representations of $W_F’= W_F \times \mathrm{SL}_2(\mathbb{C})$

Representation of $W_F \times \mathrm{SL}_2(\mathbb{C})$ is more simpler than that of above: it is just a combination of representation of $W_F$ and $\mathrm{SL}_2(\mathbb{C})$. Finite dimensional (irreducible) representations are given by the symmetric product representations $\mathrm{Sym}^{n-1}$ for $n \geq 2$, which are $n$-dimensional representations of $\mathrm{SL}_2(\mathbb{C})$ where it acts on the space of homogeneous polynomials of degree $n-1$ in two variables $x$ and $y$, via

\[(\mathrm{Sym}^{n-1}(g)f)(x, y) = f(ax + cy, bx + dy).\]

for $g = (\begin{smallmatrix} a & b \\ c & d \end{smallmatrix})$. With the standard basis $x^{n-1}, x^{n-2}y, \dots, y^{n-1}$, we can also write it as a matrix form. For example, we have

\[\mathrm{Sym}^{2}\left(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\right) = \begin{pmatrix} a^2 & ab & b^2 \\ 2ac & ad + bc & 2bd \\ c^2 & cd & d^2 \end{pmatrix}.\]

It is known that symmetric power representations exhaust the irreducible representations of $\mathrm{SL}_2(\mathbb{C})$.

Proposition 3 Irreducible representations of $\mathrm{SL}_2(\mathbb{C})$ are $\mathrm{Sym}^{k}$ for $k \geq 0$, where $\mathrm{Sym}^{0}$ is the trivial representation and $\mathrm{Sym}^{1}$ is the standard representation given by the inclusion $\mathrm{SL}_2(\mathbb{C}) \hookrightarrow \mathrm{GL}_2(\mathbb{C})$.

Proof. This is a pretty standard fact in Lie group & Lie algebra representation theory, and you may find the proof in any Lie group & Lie algebra representation textbook. For example, see this note from Ivan Losev [5]. $\square$

Hence any irreducible representation of $W_F \times \mathrm{SL}_2(\mathbb{C})$ has a form of $\sigma \boxtimes \mathrm{Sym}^{k}$, where $\sigma$ is an irreducible representation of $W_F$.

Equivalence between representations of $W_F’$ and $W_F’$

The following proposition in [1] explains the equivalence between representations of two different groups. More precisely, we have a bijection between (indecomposable) admissible representations of $W_F’$ and (irreducible) semisimple representations of $W_F \times \mathrm{SL}_2(\mathbb{C})$.

Proposition 4 There is a bijection $\sigma’ \leftrightarrow \eta$ between isomorphism classes of indecomposable admissible representations $\sigma’ = (\sigma, N)$ of $W_F’$² and isomorphism classes of irreducible semisimple representations of $W_F \times \mathrm{SL}_{2}(\mathbb{C})$ given as follows:

\[\sigma' = \sigma \otimes \mathrm{sp}(n) \leftrightarrow \eta = \sigma \boxtimes \mathrm{Sym}^{n-1}\]

In general case, a representation $\sigma’= \sigma_1’ \oplus \cdots \oplus \sigma_k’$ with each $\sigma_i’$ indecomposable corresponds to $\eta = \eta_1 \oplus \cdots \oplus \eta_k$ where each $\eta_i$ corresponds to $\sigma_i$ as above.

Proof. Here we reproduce the proof in [1]. It is enough to check for indecomposable $\sigma’$’s and irreducible $\eta$’s. For given $\eta = \pi \boxtimes \mathrm{Sym}^{n-1}$, we can recover $\pi$ and $n$ from $\eta|_{W_F}$: $\pi$ is the only irreducible representation occuring in $\eta$ and it occurs with multiplicty $n$, and this proves injectivity. For surjectivity, any representations of $W_F \times \mathrm{SL}_2(\mathbb{C})$ are external tensor products of irreducible representations of $W_F$ and $\mathrm{SL}_2(\mathbb{C})$, and now we apply the Proposition 3. $\square$

References:

[1] D. Rolich, Elliptic Curves and the Weil-Deligne Group, Centre de Recherches Mathèmatiques (1994)

[2] J. Tate, Number theoretic background. Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977)

[3] J. Getz and H. Hahn, An Introduction to Automorphic Representations, 2022

[4] P. Deligne, Formes modulaires et representations de $\mathrm{GL}(2)$, Modular Functions of One Variable, Springer (1973)

[5] I. Losev, Lecture 3: Representation Theory of $\mathrm{SL}_2(\mathbb{C})$ and $\mathfrak{sl}_2(\mathbb{C})$, online note

Tags: math