28 ARMAND BOREL, AUTOMORPHIC FORMS ON REDUCTIVE GROUPS
proper parabolic Q-subgroups containing P are zero. Let nqf = fq. This is an
idempotent. The nq (Q :J P) commute with one another, and f,F(Yp) is the
kernel of
(80) prcusp = IJ (1 - nq).
G:JQ:JPIf F(Yp) = C~g(Yp) then, as we saw, the elements of f,F(Yp) are uniformly fast
decreasing on any Siegel set with respect to P. If P is minimal, and there is only
one cusp, these are cuspidal functions on I'\G, but not so otherwise. Section 10
will give a projection of C~g(r\G) onto °Cumg(I'\G) in the general case.
There is a converse to Theorem 6.9, due to L. Clozel [10]:
6.11. Theorem. Assume AG = l. Then any fast decreasing automorphic form
on I'\ G is a cusp form.
(cf. 5.3 in [10]. It is proved there adelically. We transcribe the argument in
the present framework.) Fast decrease is meant with respect to a given H.S. norm
on G.
Let P be a proper parabolic Q-subgroup of G. We have to prove that fp = 0.
There is a compact neighborhood C of 1 in Np such that
fp(x) = fc f(n.x)dn
therefore
lf?(x)I-< llxllm fc llnllmdn
which shows that fp is fast decreasing, too. As usual, we view f P as an automorphic
form on the Levi quotient Lp of P (Section 6.4). Identify Lp to a Levi subgroup
L of P, under the inverse of the restriction to L of the projection P--> Lp. This
provides a H.S. norm on Lp with respect to which fp is fast decreasing. On
Lp = Mp.Ap, it is described as a function of x .a (x E Mp, a E Ap) by Proposition
5.6. For a fixed x E Mp, the function a r-t f(m.a) is a finite sum of exponential
polynomials. Being fast decreasing, it must be equal to zero. Since this is true for
every x E Mp, the theorem is proved.
- Finite dimensionality of A(I', J, ~)
The proof that A(r, J, ~) is finite dimensional is essentially the same as that given
in [14] for semisimple groups and in [6] for SL 2 (IR). We shall be brief and refer to
both for details.
7.1. Definition. A locally L^1 -function ofr\G is cuspidal if its constant term with
respect to any proper parabolic subgroup is zero.
If V is the space of locally L^1 functions, then ° V denote the subspace of cuspidal
elements in V.
We recall two known lemmas.
7.2. Lemma, Assume pr~G = 0 and let p E [1, oo]. Then ° LP(I'\G) is closed in
LP(I'\G).
cf. 8.2 in [6] or, for p = 2, the proof of lemma 18 in [14].