NUMBER THEORETIC BACKGROUND 9
ard relationship between global and local canonical classes, and the vanishing of
W(GaKEJFv), CE).
Combining (1.6.1) and (1.5.2) we obtain
COROLLARY. The diagram
WFv
:
GFv
iv
WF GF
is unique up to isomorphism, and the automorphisms of it are the inner automorphisms
of W, defined by elements w eW°, which induce an automorphism of WFv {i.e., for v
nonarchimedean, w fixed by GFv;for v archimedean, w a product of an element of
by an element
of(W°)GFp).
2. Representations. Let G be a topological group. By a representation of G we
shall mean, in this section, a continuous homomorphism p: G - GL(F) where Fis
a finite-dimensional complex vector space. By a quasi-character of G we mean a
continuous homomorphism %: G -• C*. If (p, V) is any representation of G, then
det p is a quasi-character which we may sometimes denote also by det V. The map
V H* det V sets up a bijection between the isomorphism classes of representations
V of dimension 1 and quasi-characters. Of course we can identify quasi-characters
of G with quasi-characters of
Gab.
We let M{G) denote the set of isomorphism classes of representations of G, and
R(G) the group of virtual representations. A function A on M(G) with values in an
abelian group Zcan be "extended" to a homomorphism R(G) JTif and only if it
is additive, i.e., satisfies X(V) = X{V) X{V") whenever 0 - V - V -* V" Oisan
exact sequence of representations of G.
(2.1) Let F be a local or global field, F an algebraic closure of F, and WF a Weil
group for F/F. Let (p, V)be a representation of WF. Since WF = proj lim {WE/F}
and GL(V) has no nontrivial small subgroups, p must factor through WE/F, for
some finite Galois extension E of F in F. It follows that if a is an essentially inner
automorphism of WF in the sense of (1.5), then
Va
« F. Thus essentially inner
automorphisms act as identity on M(WF) and R{WF). By (1.5.1) we can therefore
safely think of M{WF) as a set depending only on F, not on a particular choice of F
or of Weil group WF for F/F, and the same for R(WF). In this sense, if v is a place
of a global F, the "restriction" map M{WF) - M{WF) induced by the map dv of
Proposition (1.6.1) depends only on v, not on a particular choice of the maps iv and
6V in that proposition, and the same for R{WF) - R{WF). We shall indicate this
map by p ^ pv or K •- Kp. (The independence from ^y results from (1.6.1), and the
independence from iv, from (1.5.1).)
If EjF is any finite separable extension, we have canonical maps
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