88 L. CLOZEL, SPECTRAL THEORY OF AUTOMORPHIC FORMSIf 7f is an irreducible, unitary representation of G(As) it can be written as an
infinite tensor product
7r=@ 7fp
p!f;S
(For the correct description see [31, p. 311] as well as [A2]). There is a new finite
set of primes T (depending on 7r) such that 1fp is unramified for p tf. T; it then has
a unique KP-invariant vector ep (up to complex numbers of absolute value 1). The
(essential) uniqueness of ep guarantees that 7f is uniquely defined - for a precise
discussion see Moore [A2]. (This is what fails for nilpotent, non-Abelian groups).
Let K = IIKp c G(As). The product topology forces:Lemma A.2. - Any irreducible representation T of K can be writtent T = @Tp,
p!f;S
where Tp is trivial for p tf. T = T( T) (a finite set) and each Tp has finite image.Recall now the notion of "large compact subgroup". We say that K C G
(K compact, G locally compact) is large if any irreducible representation 7f of G
restricts to K with finite multiplicities. If G contains a large compact subgroup, G
is tame [22, 15.5.2].
Bernstein h as shown that Gp was tame by showing that any compact-open
subgroup is large in Gp [Al]^7
We complete the proof by showing that Kc G(As) is large. Let T = ®Tp be a
representation of K , and 7f = ®1fp a representation of G(As). Write 7f = 1fT ® 1fT,
T = TT@TT where Tis finite, and 7fT unramified, and TT ~ :ii.. Then the multiplicity
decomposes as a product; TT h as multiplicity 1in7fT by definition of the (restricted)
tensor product; TT = @Tp and Bernstein's theorem implies: mult(Tp, 1fp) < oo
pET
whence mult(TT,7rT) < oo.References for the Appendix[Al] J. BERNSTEIN.- All reductive p-adic groups are tame, Funct. Anal. Appl.
8 (1974), 91 - 93.
[A2] C.C. MOORE.- Decomposition of unitary representations defined by discrete
subgroups of nilpotent groups, Ann. Math. 82 (1965), 146- 182.
[A3] M.H. SLIMAN.- Theorie de Mackey pour les groupes adeliques, Asterisque
115 , 1982.(^7) This was a lso Harish-Chandra's proof in the real case.