10

J. TATE

resf/F

induction

ind£/F

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

WFh.

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* «

WFh,

are of the form % =

z~NcoS9

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 =

lndF/F(z~Ncos)

with N 0. (For N = 0 this induced represen-

tation is reducible:

(2.2.2.1) 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

WF — — ^ GL(F) • PGL(F)

a

are equal; hence p = a ® x f°

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.