- ARITHMETIC SUBGROUPS. REDUCTION THEORY 21
5.2.3. Lemma. Let I C J be subsets of .0. and 61 = 6 p 1 ,t,w a Siegel set with respect
to P1. Let c > t be a constant and setU1,J = {x E 61la(x)°' :Sc for a E J - I}
Then U 1 ,J belongs to a Siegel set with respect to PJ.Proof. We have N1 = NJ(MJ n N1) and M 1 c MJ. Therefore w belongs to a
compact subset of NJMJ. There remains to examine U1,J n A1,t (we write A1 and
AJ for Ap 1 and ApJ). The Lie algebra a 1 of A 1 is the orthogonal direct sum (with
respect to the Killing form) of aJ and af = mJna1, therefore A1 = Af x AJ, where
Af = expaf. Let a E U1,J, write accordingly a= af.aJ (af E Af,aJ E AJ). The
elements of J are equal to 1 on AJ, therefore, if a E J - I, then a""= (af)"", hence
t :S a"" :S c. The restrictions of the elements of J - I form a coordinate system on
Af, therefore {af,a E U1,J} is contained in a compact subset C of Af.
.0.(P 1 ,A 1 ) (respectively .0.(PJ,AJ)) is the restriction of .0. - I (respectively
.0. - J) to A 1 (respectively AJ) and .0. - I= (J - I) U (.0. - J). Let now a E .0. - J
and a E U1,J· We have a""= (af)""a'J and, since af belongs to the compact set C,
we have a"" ::=:: a'J (a E U1 ,J, a E .0. - J) , whence the existence of a constant t' > 0
such that U1,J C CAJ,t'· Since CC MJ, the lemma is proved.
5.3. Theorem. We keep the previous notation. Let P be a minimal parabolic
IQ-subgroup and 6 = 6P,t,w a Siegel set with respect to P and C a finite subset of
G(IQ). Then the set(51) {r Er, 'Y(C.6) n C.6-/= 0}
is finite (Siegel property). There exists such a set C and 6 such that G = r.C.6.
A set C.6 satisfying the last condition will be called a standard fundamental
set. c is minimal with that property if the groups cp represent the r-conjugacy
classes of minimal parabolic IQ-subgroups.
In view of 5.2.2, the theorem can also be expressed by stating that if P 1 , ... , P s
are representatives of the r-conjugacy classes of minimal parabolic IQ-subgroups,
then there exist Siegel sets 6P,,t,,w,, the union of which covers r\G. It cannot be
covered by less than s such subsets. Each is said to represent a cusp. The one-cusp
case occurs for some classical arithmetic subgroups such as SLn(Z) or Sp 2 n(Z),
more generally for arithmetic subgroups of IQ-split groups associated to admissible
lattices in the sense of Chevalley. In the adelic case, where the role of r is played
by G(IQ), there is only one cusp.
5.4. Moderate growth condition. Assume here that prk <QIG = 0. Fix a
maximal (not necessarily split) torus T:) A. There is a corresponding root system
<P(Tc, gc); the restriction of any a E <P(Tc, gc) to any Ap is either zero or an
element of <P(G,Ap). We choose a positive system in <P(Tc,gc) so that for each
a E <P(Tc, gc)+, the restriction of a to Ap is either zero or an element of <P(P, Ap ).
Identity G with a subgroup of some GLN via a IQ-embedding p. The represen-
tation pis the a finite sum of irreducible ones, with highest weights >- 1 , ... , Ak, say;
these are characters of T c that are real-valued and positive on Ap.
Let 6 be a Siegel set with respect to the parabolic subgroup P. We claim that
(52)