satisfying the usual Frobenius reciprocity, for we can identify WE with a closed
subgroup of finite index in WF.
(2.2) Quasi-characters and representations of Galois type. Using the isomorphism
CF « Wfi we can identify quasi-characters of CF with quasi-characters of
For example, we will denote by cos, for seC, the quasi-character of WF associated
with the quasi-character c «-* ||c||F, where ||c||F is the norm of c e CF. Thus cos(w) =
||w||5 in the notation of (1.4).
On the other hand, since p: WF -• GF has dense image, we can identify the set
M(GF) of isomorphism classes of representations of GF with a subset of M(WF). We
will call the representations in this subset "of Galois type". Thus, by (1.4.5), a re-
presentation p of WF is of Galois type if and only if p{WF) is finite.
With these identifications, a character % of GF is identified with the character %
of CF to which i corresponds by the reciprocity law homomorphism.
(2.2.1) In the Z-cases, i.e., if Fis a global function field, or a nonarchimedean
local field, then every irreducible representation p of WF is of the form p = a ® os,
where a is of Galois type. This is a general fact about irreducible representations of
a group which is an extension of Zby a profinite group; some twist of p by a quasi-
character trivial on the profinite subgroup has a finite image; see [D3, §4.10].
(2.2.2) If Fis an archimedean local field, the quasi-characters of WF, i.e., of F* «
are of the form % =
where z: F -+ C is an embedding and N an integer
^ 0, restricted to be 0 or 1 if Fis real. If Fis complex, these are the only irreducible
representations of F* = WF. If Fis real, WF has an abelian subgroup WF = F* of
index 2, and the irreducible representations of WF which are not quasi-characters
are of the form p =
with N 0. (For N = 0 this induced represen-
tation is reducible:
( Ind/?/Fcos = cos © x_1cws+i
where x: F - C is the embedding of F in C.)
(2.2.3) Suppose Fis a global number field. A primitive (i.e., not induced from a
proper subgroup) irreducible representation p of WF is of the form p = a ® %
where a is of Galois type and % a quasi-character.
Choose a finite Galois extension E of F big enough so that p factors through
WE/F = WF/WE. Since p is primitive and irreducible, p(WEh) must be in the center
of GL(K), because WEh is an abelian normal subgroup of WE/F. In other words,
the composed map WF JL GL(V) - PGL(F) kills WE and therefore gives a pro-
jective representation of Gal(F/F). This projective representation of Gal(F/F)can
be lifted to a linear representation a0: GF - GL(F) (see [S3, Corollary of Theorem
4]). Let a = (To-cp. The two compositions
are equal; hence p = a ® x
r s o m e
quasi-character %.
(2.2.4) Note that, in all cases, global and local, the primitive irreducible represen-
tations of WF are twists of Galois representations by quasi-characters.
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