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
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
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
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
« 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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