LECTURE 3. KNOWN BOUNDS FOR THE CUSPIDAL SPECTRUM 69This follows from theorems of Harish-Chandra (in the real case) and J. Bern-
stein (in the p-adic case) asserting that G is of type l. For the meaning of this
property see [22, § 8.6, 9, 18]. For the fact that it implies Theorem 3.3 see [22,
Thm. 8.6.5 and 8.6.6]^4.
If p has a decomposition (3.6), the support ofμ - a closed subset of G -is called
the support of p, notation: Supp(p). If 7f E G, we say that 7f is weakly contained
in p if 7f E Supp(p). More generally, if p 1 is a unitary representation of G, we say
that p 1 is weakly contained in p if Supp(p 1 ) C Supp(p).
An important fact is that weak inclusion can be characterized intrinsically. If
( 7f, 7-i) is a representation of G, a coefficient of 7f is a function on G of the form
c(g) = cv(g) = (7r(g)v, v) where v E 7-i.
Theorem 3.4. -The following properties are equivalent for p, p 1 unitary:
(i) P1 is weakly contained in p.
(ii) Any coefficient c of p 1 is limit, uniformly on each compact subset of G
of a sequence Um) of positive linear combinations of coefficients of p.
Proof: Dixmier [22, Theorem 3.4.4 and Prop. 8.6.8].
We insert a few remarks. The dual G is generally not separated (this is French
for "Hausdorff"). On the other hand it is separable: then are countable bases of
neigbourhoods, so it is enough to test convergence on sequences.
Note that Theorem 3.4 says in particular that coefficients c associated to ir-
reducibles 7f E Supp(p) can be approximated by positive linear combinations of
coefficients of p.
Finally, I could not find the following useful lemma in the literature (but com-
pare [11,§ 3 ]):
Lemma 3.5 (Notations as in Theorem 3.4). - Assume that Vi c 1i 1 is a dense
vector subspace and that any coefficient C associated to v E Vi is a uniform limit on
compacta of positive combinations of coefficients of p. Then P1 is weakly contained
in p.
Proof: Assume u E 1i1 is a vector of norm 1 and assume llvn -ull < 2 ~,
Vn E Vi and llvnll = 1. Moreover let (wk) be an increasing and exhaustive sequence
of compact subsets of G. Then we have:
1
I Cvn (g) - Cu (g) I :::; - (g E C) ·
n
Assume, for given n, f n ,m verifies the condition (ii) of the Theorem, relative to
Cv". For any k we can find N = N ( n, k) such that
1
lcvn (g) - fn,N(g)I:::; - (g E Wk)·
nSet f(n) = fn,N with N = N(n, n). Then
2
lcu(g) - f(n)(g)I:::; - (g E Wn),q.e.d.
n(^4) Dixmier's formulation is slightly different. He writes
P = EBn=l, oo k EEr 'Hrr dμn(1f),
where the μn are mutually singular ( "etrangeres" ). The point is that two mutually singular
measures μ, μ' are supported on two disjoint Borel subsets E , E' (see Bourbaki, Integration
Ch. V - for Radon measures, alack!). From this our formulation follows.