20

J. TATE

where An is the ring of locally constant g-valued functions on 0nL

(4.1.1)

DEFINITION

[D3, (8.3.6)]. The Weil-Deligne group WF is the group scheme

over Q which is the semidirect product of WF by Gai on which WF acts by the rule

wxw~l

= \\w\\x.

Let E be a field of characteristic 0. The group WF(E) of points of W'F with

coordinates in Eh just E x WF with the law of composition^, H^X^, H2) =

(fl! + || w21| a2» Wxw2) for au a2e E and wu w2 e WF.

Let K be a finite-dimensional vector space over E. A homomorphism of group

schemes over E

p': WF xQE GL(K)

determines, and is determined by, a pair (p, N) as in (4.1.2) below, such that, on

points, p'((a w)) = exp(aN) -p(w). That is the explanation for the following defini-

tion:

(4.1.2)

DEFINITION

[D3, (8.4.1)]. Let E be a field of characteristic 0. A representa-

tion of W'F over 2J is a pair p' = (p, N) consisting of:

(a) A finite-dimensional vector space V over E and a homomorphism p: WF -

GL(K) whose kernel contains an open subgroup of/, i.e., which is continuous for

the discrete topology in GL(F).

(b) A nilpotent endomorphism N of V, such that p(w)Np(w)~l = \\w\\N, for

weW.

(4.1.3) 0-semisimplicity. Let p' = (p, N) be a representation of W'F over E. Define

v : WF - Z by \\w\\ = q-viw)t There is a unique unipotent automorphism uof K

such that u commutes with N and with p(WF) and such that

Qxp(aN)p(w)u~v(w)

is

a semisimple automorphism of V for all a e is and all w e WF—/ [D3, (8.5)]. Then

Pss — (pu~v N)ls called the 0-semisimplification of p', and p' is called 0-semisimple

if and only if p' = p^, i.e., u = 1, i.e., the Frobeniuses act semisimply. For this it

is necessary and sufficient that the representation p of WF be semisimple in the ordi-

nary sense, because p(0) generates a subgroup of finite index in p(WF), and in

characteristic 0 a representation of a group is semisimple if and only if its restriction

to a subgroup of finite index is semisimple. In his article in these

PROCEEDINGS,

Borel discusses admissible morphisms WF -*

LG;

when G = GLM, these are just our

0-semisimple (p, JV)'s.

(4.1.4)

EXAMPLE.

Sp(«) is the following representation (p, N) of WF over Q.

V=Q» = Qe0 + Qex + ••• + Qen.u

p(w)e{ = o),(w)^ (= H | ««?,•),

to, = e,+1 (0 £ i « - 1), JVV_! = 0.

(4.1.5) Given any (p, N), Ker N is stable under WF. Hence (p, N) is irreducible o

N = 0 and p is irreducible. It is not hard to show that the 0-semisimple indecom-

posable representations of WF are those of the form p ® Sp(rc) with p' irreducible.

(The ® is defined by (p, N) ® (Pl, N{) = (p ® pl5 W ® 1 4- 1 ® A^).)

(4.1.6) Let (p, N, K) be a representation of W^ over E. We put F^ = (Ker

N)1

and define a local L-factor, a conductor, and a local constant by