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
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