10 ARMAND BOREL, AUTOMORPHIC FORMS ON REDUCTIVE GROUPS
£ (resp. μ). Then a Casimir operator can be written Cg = -I: xr + I: yJ, and
Ce = L:i xr is a Casimir operator for t Let n = 2Ce + Cg. It is an analytic elliptic
operator. We claim that f is annihilated by some non-constant polynomial in n,
which will prove our assertion. The function f is annihilated by an ideal J of finite
codimension in Z(g) (by (A3) ) and, since f is K-finite on the right, it is annihilated
by an ideal (of finite codimension of U(£). Therefore f is annihilated by an idea
of finite codimension of the subalgebra Z(g).U(£) of U(g). But then there exists a
polynomial of strictly positive degree P(D) in n belonging to that ideal.
3.2. A function a on G is said to be K-invariant if a(k.x) = a(x.k) for all x E G,
k EK. We have the following theorem:
Theorem. Given a neighborhood U of 1 in G, there exists a K -invariant function
a E C;;"(U) such that f = f *a.
This follows from the fact that f is Z-finite and K-finite on one side, by a
theorem of Harish-Chandra. See ([13], theorem 1) or ([1], 3.1), and for SL 2 (JR.)
([6], 2.13). Also see Section 9.4 below.
3.3. A smooth function u on G is said to be of uniform moderate growth (bounded
by m) if there exists m E Z such that vm(D !) < oo for all DE U(g).
An elementary computation shows that if vm(u) < oo, then
vm(U* a)< oo
for any a E C'(;"(G). Since, D(f a)= fDa, the previous theorem implies the
Corollary. An automorphic form is of uniform moderate growth. More generally,if f has moderate growth, then f * a is of uniform moderate growth.
3.4. We intercalate some facts needed in the sequel. Since G is a closed subgroup
of SLN(JR.), it is clear that 11-11 has a strictly positive minimum, say to on G. Fix a
Haar measure on G. Fort 2 to, let
(7) Gt = {g E GI llgll :::; t}.
There there exists m E N such that
(8) vol (Gt)-< tm, (t 2 to).
(cf Lemma 37 of [14]). (We shall sketch the proof in Section 4.6, after having
recalled some facts about the structure of semisimple groups.) It implies
(9)To see this, fix a compact neighborhood C of 1 such that
r n c.c-^1 = {1}.
In view of ( nl), there exists a constant d > 0 such that
llxll :::; dlbll) ('y Er, x E C."f).
So
whence
LJ C'Y c Gdt, (disjoint union),
-yEG,nr