22 ARMAND BOREL, AUTOMORPHIC FORMS ON REDUCTIVE GROUPS
We can assume Ap to be in diagonal form. The entries are then weights of p. Given
such a weight μ, there exists i such that μ = Ai - 2: ma.a:, where o: E Q/J.(P, Ap)
and mi EN. Whence 119112 -< l:i ap(g)^2 >.'. The reverse inequality is obvious.
Let f be a function on 6. As a consequence, we see that f is of moderate
growth on 6 (in the sense of Section 2), if and only if there exists A E X(Ap) such
that
(53) lf(g)I -< a(g)>- (g E 6).
Consider the following three conditions on a continuous function f on r\G:
(i) f has moderate growth.
(ii) f has moderate growth in the sense of (53) on each Siegel set of G.
(iii) f has moderate growth in the sense of (53) on a standard fundamental
set n (as defined in Section 5. 3).Claim. These conditions are equivalent.
Clearly, (i) implies (ii), by the above, and (ii) implies (iii) by 5.2.2. Assume now
(iii). Then f has moderate growth on a fundamental set n. However, f(rg) = f(g)
for all 1 E r, g E G, whereas llfgll depends of course also on/. But we have
(54) 11911 :;::: inf 11!.c.gll (r E r, c E C, g E 6),
1Er
a result of Harish-Chandra which clearly shows that (iii) implies (i). Harish-
Chandra's proof of (54) is given in ([2], II, §1, Prop. 5).
More explicitly, it is obvious that 1191 I >--inf, I l1cgl I, so that the main point in
(54) is the reverse inequality, i.e. the existence of d > 0 such that
(55) (g E 6, c EC,/ Er)
Note also that if (i),(ii), (iii) are expressed in terms of Hilbert-Schmidt norms, we
can also add "bounded by m" with the same m in all three.
A continuous function f on a subset D C G is fast decreasing if it is of moderate
growth, bounded by m , for all m E Z, i.e. iflf(g)I-< 11911 = (g ED) for any m E Z
If D = 6 is a Siegel set, this can also be expressed by requiring that lf(g)I -< a(g)>-
(g E 6) for any A E X(A).
The previous argument shows that if f is left r-invariant, (i),(ii), (iii) remain
equivalent if "has moderate growth" or "has moderate growth in the sense of (53)"
is replaced by "is fast decreasing."
5.5. In Section 2, we assumed, to avoid preliminaries, that G was semi-simple.
However, the definition is of course valid if G is reductive. In the present case, we
shall have to know more precisely the dependence on AG. To express this, we need
the following definition, in which A is a closed connected subgroup of AG.
A function f on A is an exponential polynomial if there exist elements Ai E
X (A) and polynomials Pi on a ( 1 ~ i ~ s) such that
(56)Then one has the following result: