Weil-Deligne Representations

Let F be a finite extension of the p-adic numbers \mathbb{Q}_{p}. In Galois Representations we described some continuous Galois representations of \mathrm{Gal}(\overline{F}/F), but all of them were p-adic (or rather \ell-adic, see the discussion in that post for the explanation behind the terminology). What about complex Galois representations? For instance, since the complex \ell-adic numbers (the completion of the algebraic closure of the \ell-adic numbers) are isomorphic to the complex numbers, if we fix such an isomorphism we could just base change to the complex numbers to get a complex Galois representation.

Complex Galois representations, also known as Artin representations, are in fact an interesting object of study in number theory. However, the issue is that if we require these Galois representations to be continuous, like we have required for our \ell-adic representations, we will find that they always have finite image, which also means in essence that we might as well just have been studying representations of finite Galois groups, not the absolute one as we intend to do.

To get a complex representation that will be as interesting as the p-adic ones, we have to make certain modifications. We will look at certain representations of a certain subgroup of the Galois group instead, called the Weil group, and together with some additional information in the form of a “monodromy operator“, we will have a complex representation that will in a way carry the same information as a \ell-adic representation.

Let us first define this Weil group. F be a local field and let \kappa be its residue field. The absolute Galois groups of F and \kappa fit into the following exact sequence

\displaystyle 0\to I\to \mathrm{Gal}(\overline{F}/F)\to\mathrm{Gal}(\overline{\kappa}/\kappa)\to 0

where I is the kernel of the surjective map \mathrm{Gal}(\overline{F}/F)\to\mathrm{Gal}(\overline{\kappa}/\kappa) and is called the inertia subgroup (this can be considered the “local” and also “absolute” version of the exact sequence discussed near the end of Splitting of Primes in Extensions).

The residue field \kappa is a finite field, say of some cardinality q. Finite fields have the property that they have a unique extension of degree n for every n, and the Galois groups of these extensions are cyclic of order n. As a result, the absolute Galois group \mathrm{Gal}(\overline{\kappa}/\kappa) of the residue field \kappa is isomorphic to the inverse limit \varprojlim_{n} \mathbb{Z}/n\mathbb{Z}, also known as the profinite integers and denoted \widehat{\mathbb{Z}}.

There is a special element of \mathrm{Gal}(\overline{\kappa}/\kappa) called the Frobenius, which corresponds to raising to the power of q. The powers of Frobenius give us a subgroup isomorphic to the integers \mathbb{Z} inside \mathrm{Gal}(\overline{\kappa}/\kappa) (which again is isomorphic to \widehat{\mathbb{Z}}). The inverse image of this subgroup under the surjective morphism \mathrm{Gal}(\overline{F}/F)\to\mathrm{Gal}(\overline{\kappa}/\kappa) is what is known as the Weil group of F (denoted W_{F}). Since \widehat{\mathbb{Z}} is the completion of \mathbb{Z}, the Weil group may be thought of as a kind of “decompletion” of the Galois group \mathrm{Gal}(\overline{F}/F).

It follows from local class field theory (see also Some Basics of Class Field Theory) that we have an isomorphism between the abelianization W_{F}^{\mathrm{ab}} of the Weil group and F^{\times}.

A Weil-Deligne representation is a pair (\rho_{0},N) consisting of a representation \rho_{0} of the Weil group W_{F}, together with a nilpotent operator N called the monodromy operator, which has to satisfy the property

\displaystyle \rho_{0}(\sigma)N\rho_{0}(\sigma)^{-1}=\Vert\sigma\Vert N

for all \sigma in W_{F}, where \Vert\sigma\Vert is the valuation of the element of F^{\times} corresponding to \sigma under the isomorphism given by local class field theory as mentioned above.

Grothendieck’s monodromy theorem them says that given a continuous p-adic representation \rho we can always associate to it a unique Weil-Deligne representation (\rho_{0},N) satisfying the property that, if we express an element of the absolute Galois group as \phi^{m}\sigma where \phi is a lift of Frobenius and \sigma belongs to the inertia group, then \rho(\phi^{m}\sigma)=\rho_{0}(\phi^{m}(\sigma))\mathrm{exp}(Nt(\sigma)), where t:\mathrm{Gal}(F^{\mathrm{tame}}/F^{\mathrm{ur}})\to\mathbb{Z}_{\ell}, F^{\mathrm{tame}} being the “tamely ramified” extension of F and F^{\mathrm{ur}} the unramified extension of F. The point is that, we can now associated to a p-adic Galois representation a complex representation in the form of the Weil-Deligne representation, which is the goal we stated in the beginning of this post.

It turns out that certain Weil-Deligne representations (those which are called F-semisimple) are in bijection with irreducible admissible representations of the \mathrm{GL}_{n}(F), thus linking two kinds of representations – those of Galois groups like we have discussed here, and those of reductive groups, similar to what was hinted at in Representation Theory and Fourier Analysis. This will be discussed in a future post.

References:

Weil group on Wikipedia

MSRI Summer School: Automorphic Forms and the Langlands Program (Lecture Notes) by Kevin Buzzard

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