NUMBER THEORETIC BACKGROUND
21
Z(V, t) = det (1 - 0t | Vff)-\ and L(V, s) = Z(F, ?-*), when E c C;
a{V) = a(p) + dim V* - dim Vf,9
e(V)-e(p)det(-®\V'/V&),
and
e(K,/) = e ( F ) ^ .
Here, for e, the usual (]) and dx are understood, but omitted from the notation.
These quantities do not change if we replace Fby its 0-semisimplification; but
note that they are not additive as functions of V, because VN is not. If N = 0, they
are the same as before.
One of the main reasons for introducing the Weil-Deligne group is the fantastic
generalization of local class field theory embodied in:
(4.1.7) Conjecture. Let F be a nonarchimedean local field and n an integer ^ 1.
There is a (in fact more than one) natural bijection between isomorphism classes of
0-semisimple representations of WF of degree n, and of irreducible admissible re-
presentations of GL(n9 F).
For n = 1 this is local class field theory. For n = 2, it is discussed at length in
[D2, (3.2)]. In this conjecture, for any n, the irreducible representations of WF
(which are just irreducible representations of WF) should correspond to the cuspidal
representations of GL(«, F). I understand that Bernitein and Zelevinsky have
shown that the way in which arbitrary admissible representations of GL(/i, F) are
built out of cuspidal ones follows the same pattern as the way in which arbitrary
$~semisimple representations of WF are built up out of irreducible ones. Thus the
main problem is now the correspondence between irreducibles and cuspidals.
A more general conjecture, involving an arbitrary reductive group G rather than
just GL(n% relates admissible representations of G(F) to homomorphisms of WF
into the "Langlands dual" of G (see Borel's article in these
PROCEEDINGS).
This
more general conjecture is the nonarchimedean local case of "Langlands' philo-
sophy".
(4.2) A-adic representations. Now suppose / is a prime different from the residue
characteristic p of F and let tt: IF Qt be a nonzero homomorphism. (Such a tt
exists and is unique up to a constant multiple, because the wild ramification group
P is a pro-/?-group, and the quotient I/P is isomorphic to the product IT*/*^/) We
have
ti(waw~l)
= ||w|j tt{a)y for a e Z, w e W,
because conjugation by w induces raising to the
||H||
power in I/P. Let 0 be an in-
verse Frobenius element (4.1.8). Suppose Ex is a finite extension of Qt. A ^-adic
representation of WF is a finite-dimensional vector space Vx over Ex and a homo-
morphism of topological groups px: WF GLEx(Vx) where GL^(F^) has the ^-adic
topology (i.e., the topology given by the valuation).
(4.2.1)
THEOREM (DELIGNE
[D3, §8]). The relationship Vx = V and
px(®na)
= p(^V)exp(^(^) N), oeI,ne Z,
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