- CONVOLUTION OPERATORS ON CUSPIDAL FUNCTIONS 29
7.3. Lemma. Let Z be a locally compact space with a positive measure μ such
that μ(Z) is finite. Let V be a closed subspace of L^2 (Z, μ) consisting of essentially
bounded functions. Then V is finite dimensional.
This lemma is due to R. Godement. For Hi::irmander's proof of it, see [14] p.
17-18 or [6], 8.3.7.4. Theorem (Harish-Chandra). Let J be an ideal of finite codimension
in Z(g) and ~ a standard idempotent for K (Proposition 3.6). Then the space
A(r, J, ~) of automorphic forms on r \ G of type ( J, ~) is finite dimensional.
See [14], Theorem 1 and, for SL 2 (1R), [6], §8. The proof is by induction on
rk <Q(G). Assume that prk<Q(G) > 0. We use Proposition 5.6 to reduce the proof
to f \^0 G. The functions Pd have K-types defined by~· Moreover, they are anni-
hilated by J' =Jn Z(^0 g). Therefore, the Pd belong to the space of automorphic
forms on r\^0 G of type (J', ~),which is finite dimensional by induction. Proposition
5.6 then shows that A(r, J, ~) is finite dimensional.
From now on, prk<Q(G) = 0. If rk<Q(G) = 0, then r\G is compact (Section 5.1),
all automorphic forms are L^2 and bounded. The theorem follows in this case from
Godement's lemma.
Let now rk<Q(G) > 0. Let Q be a set of representatives of r-conjugacy classes
of proper maximal parabolic Q-subgroups. It is finite ([3], 15 .6). Let μ be the
collection of the maps f f-) fp (P E Q). We have seen in Section 6.4( c), for any
P in fact, that f p belongs to a space of a utomorphic forms on Lp of a fixed type,
determined by J and~· By induction and our original argument, they form a finite
dimensional space, hence the image of μ is finite dimensional. This reduces us to
prove that kerμ is finite dimensional. In view of Section 6.4, ker μ =^0 A(r, J, ~).
By Theorem 6.9, it consists of bounded functions. To deduce from Lemma 7.3 that
it is finite dimensional, it suffices to show that it is a closed subspace of L^2 (f\G).
The argument is the same as the one given in [14], lemma 18, and in [6], proof of
8.3 for SL2(1R).- Convolution operators on cuspidal functions
The proofs of the main results here rely mostly on Section 3.5, Proposition 5.6 and
on the basic estimates and its consequences. To formulate them, it will be useful
to introduce some function spaces.
8.1. We let Cmg(f\G) (respectively C~g(f\G)) be the space of functions of mod-
erate growth (respectively smooth functions of uniform moderate growth). They
are endowed with the seminorms
vn(f) = sup lf(x)JJJxJl-n (respectively vo,nU) = sup JDf(x)JIJx-nJI, for D E
U(g), n E Z .)
For n E Z, let Cmg(f\G,n) (respectively C~g(r\G,n)) be the subspace of
Cmg(f\G) (respectively c~g(r\G)) on which the Vn (respectively all VD,n, D E
U(g)) are finite.
Then Section 5.5 says that:
() Let a E C':° ( G). Then there exists n E N such that the map f f-) f a
induces a continuous map of L^1 (r\G) into C~g(r\G, n).8.2. Let Cfd(r\G) (respectively C.tl<l(r\G)) be the space of continuous functions
on r \ G which are fast decreasing (respectively, of smooth functions such that D f