- ARITHMETIC SUBGROUPS. REDUCTION THEORY 23
5.6. Proposition. Let f be a smooth Z(g)-finite function on f\G. Then there
exist exponential polynomials Qi on A and polynomials
Pi E Z(g) such that
f(x.a) = L Qi(a)Pd(x) (a E AG, x E^0 G).More generally, this is valid with respect to a decomposition G = G' x A, with A c
AG and^0 G C G', and the proof reduces to the case where A is one-dimensional,
in which case it is an exercise in ODE. A proof is given in ([2], II, §3, lemma 3) or
([14], p. 20).
5.7. We shall also have to consider moderate growth when AG =J=. {l}. It
is possible to express it as in Section 2.1 by means of Hilbert-Schmidt norms if
G C GLN provided G is closed not only in GLN(IR) but a lso in the space MN(IR)
of N x N real matrices. One way to guarantee this is to arrange that the weights
of AG in JRN are invariant under >. ,___, ->.. However, we shall also need to express
moderate growth in terms of the functions a>-.
Lemma. Let f = a>-.P(loga) be an exponential polynomial on AG. Then there
exist finitely many μi E X(AG) {l :::; i :::; l) such that
lf(a)I -<Laμ; (a E AG)·
Choose a basis (.Ai) of X(A) (1 :::; i :::; d = dimAG) and let ai = a>.; be the
corresponding coordinates. Set>.= I:; liAi andμ= I:; mi.Ai where mi > llil· Then
lf(a)I -< aμ in the subspace lail 2 1 (1 :::; i :::; d) of X(A) but it may not be so if
some a/s tend to 0. For each d-tuple t = (t1, ... , td), ti = ±1, let μ, = I:;i tiμi.
Then it is clear that lf(a)I -<aμ• in the region ai 2 1 if ti = 1, ai :::; 1 if ti = -1.
Therefore
(57)
which proves the lemma. Clearly, given finitely many exponential polynomials fi
(i EI), one can findμ such that (57) holds for each k
5.8. Proposition. Let f be an automorphic form on f\G. Then there exists
. E X(A n °G) and finitely many μi E X(AG) (i EI) such that
(58) lf(g)I-< a(g)>. L a(g)μ;
It suffices to prove this for g in a standard fundamental set
D = Do x AG, where Do is a standard fundamental set for r in °G. There exists
. E X(A n °G) such that
(59) lf(x)I-< a(x)>. (x E Do).
Since f is of uniform moderate growth, this a lso holds for all derivatives D f, (D E
Z(a)). The proposition now follows from the above lemma and Proposition 5.6.
5.9. For later reference, we end this section with an elementary fact about the
function a(x)>-. Namely, if C is compact in G, then
(60) a(x.c)>-:::::: a(x)>. (x E G, c EC)