Appendix: All reductive adelic groups a re tame
This appendix^6 is a complement to our § 3.2. Assume G is a reductive group
over Q (or more generally over a number field). In § 3.2 we recalled the "type
I" property of G(Qv) for a lo cal completion «:l!v· We will use the terminology of
Kirillov and Bernstein and call G "tame" if it is of type I and "wild" otherwise.
Since a product of tame groups is tame [22, 13.11.7], this extends to G(As) for S
finite.
In fact G(Qv) is tame of G is a connected linear algebraic group over «:l!v ([22,
13 .11.12], [A3]). However when we consider G(A) the situation is quite different. If
N is a nilpotent, non-Abelian group over Q, N(A) is generally wild as discovered
by Moore [A2].For a wild group decompositions such as (1.17) containing an integral on G
may make sense but there is no assurance that they are unique. Since this kind
of continuous decomposition often occurs for G(A), it is comforting to know the
following fact, true despite a contrary assertion in the literature:
Theorem A.1. - If G is a connected reductive group over Q , G(A) is tame.
Proof: Since this is intended as a reference we work in the most general case.
(If G is reductive over a number field, reduce to Q by Weil's restriction of scalars).
It is known that Gp = G x Qp is unramified [Sa] for almost all primes. This
means that Gp is quasi-split (i.e., has a Borel subgroup over Qp) and splits over
an unramified extension. Better, it means that there is a smooth reductive group
scheme G over Zp yielding Gp by extension of scalars -just as for Chevalley groups.
See Tits's lecture in [CJ.
The usual theory of spherical representations applies to (Gp, Kp)
(G(Qp), G(Zp)). In particular, if 7rp is an irreducible (Hilbert unitary, or admissi-
ble) representation of Gp on 1{ then 1{K v is zero - or one - dimensional [8a].
Let's throw out the finite set S of places where G may be ramified, as well as
oo. It suffices to check that the restricted product G(A.^8 ) = IJ' Gp is tame.
prf:.S
(^6) Written with the kind assistance of J .-B. Bost and R. Godement.
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