NUMBER THEORETIC BACKGROUND

11

(2.2.5) In the local nonarchimedean case one can say much more about the

structure of primitive irreducible representations (see [K]). A first result of this sort

is

(2.2.5.1)

PROPOSITION.

Let F be local nonarchimedean and let V be a primitive

irreducible representation of WF. Then the restriction of V to the wild ramification

group P is irreducible.

This result is proved in [K] and [B]. The proof depends on the supersolvability

ofGF/P.

(2.2.5.2)

COROLLARY.

The dimension of V is a power of the residue characteristic p.

Indeed, P is a pro-/?-group.

(2.2.5.3) COROLLARY. If U is an irreducible (not necessarily primitive) representa-

tion of WF of degree prime top, then U is monomial.

Because U is induced from a primitive irreducible V whose dimension is prime to

p and a /?-power, hence 1.

(2.3) Inductive functions of representations. Let F be a local or global field. For a

representation V of WF, let [V]e R(WF) denote the virtual representation de-

termined by V. Let R°(WF) denote the group of virtual representations of degree

0 of WF, i.e., those of the form [V] - [V], with dim V = dim V.

(2.3.1)

PROPOSITION.

The group R(WF) is generated by the elements of the form

lndE/F[xlfor E/F finite and % a quasi-character of WE. Similarly, R°( WF) is generated

by the elements of the form IndE/F([%] — [#']).

It suffices to prove the second statement, because R(WF) = RP(WF) + Z • [1].

Let R%(WF) denote the subgroup of R°(WF) generated by the elements

IndE/F([x] — [/']). By the degree 0 variant of Brauer's theorem [D3, Proposition

1.5] we have R°(GF) c R%(WF). The formula Ind(p ® Res %) = (Ind p) ® % shows

that R% - x

c

R* for each quasi-character % of WF.

To prove the proposition we must show for each irreducible representation p of WF

that [p] — (dim p) [1] e R%( WF). For each p there is a finite extension E of F and & prim-

itive irreducible representation pE of WE such that p = lndE/F pE. Then [p] — (dimp) •

[1] is the sum of Ind£/F([pE]-(dim pE)[lE]) and (dim pE) (IndE/F[lE]-[E:F] [1F]).

The latter is of Galois type, so by the transitivity of induction we are reduced

to the case in which p is primitive and irreducible. But then p — a ®% with

a € R(GF) and % a quasi-character (2.2.4). If n = dim p = dim a

[p]-n[l] = ([a]~-n[l])[X] + n([X] - [I])

and this is in R%( WF) by the remarks above, since [a] — n [1] e R°(GF).

(2.3.2)

DEFINITION.

Let F be a local or global field. Let A be a function which

assigns to each finite separable extension E/F and each Ve M(WE) an element

X{V) in an abelian group X. We say X is additive over F is for each E and each

exact sequence 0 - V -• V -• V" - 0 of representations of WE we have X(V) =

X(V')1(V"). When that is so we can defined on virtual representations so that A;

R(E) - X is a homomorphism for each E. We say X is inductive over F if it is

additive over Fand the diagram