1549380232-Automorphic_Forms_and_Applications__Sarnak_

20 ARMAND BOREL, AUTOMORPHIC FORMS ON REDUCTIVE GROUPS

consists of semisimple elements. The proofs of these statements reduces easily to

the case of reductive groups since, as is well-known, an arithmetic subgroup of a

connected unipotent IQ-group is co-compact (see e.g. [3]).

In the non-compact case, the purpose of reduction theory is to construct fun-

damental sets with respect to r. It suffices to do this when G is reductive, which

we assume. This construction relies on the notion of Siegel sets.

5.2. Siegel sets. We let PQ be the set of parabolic IQ-subgroups of G. It is

operated upon by conjugation by G(IQ), in particular by r. It is known that f \PQ

is finite. Let P E PQ· To remind us that we are dealing with the Langlands

decomposition of P r elative to IQ we shall write in bold face the subscript P. Fix

t > 0. We let

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(it is equivalent to require aa ;:::: t for all a E Q<i>+).

The Langlands decomposition of P can also be written as

P = Np.Mp.Ap. A Siegel set 6P,t ,w, where w C Np.Mp is relatively compact, is

that set

w .AP,t·K·

A simple computation shows that if AG= {1}, then 6P,t,w has finite volume with

respect to Haar measure.

Note that 6P,t ,w is the product of AG by a Siegel set in °G with respect to

Pn °G and Ap n °G.

5.2.l. We collect here some simple remarks on Siegel sets. The Siegel sets are

defined here as subsets of G. Traditionally, they were introduced as subsets of X,

namely, the orbits of the origin by our Siegel sets, and that mostly with respect

to a minimal parabolic subgroup, in the split case. For instance, if G = GLn(lR),

K = On, X is the space of positive quadratic forms and P is the group of upper

triangular matrices, then a Siegel set in X is the set of quadratic forms of t he form

na. t(na), where a is diagonal with entries ai satisfying ai ;:::: tai+l (1 ::=; i ::=; n) and

n is upper triangular, with coefficients n ij (i < j) bounded in absolute value by

some constant. For SLn(lR), one requires moreover IT ai = l. If n = 2 and X is

the upper half plane X = { z = x + iy E C , y > 0}, then a Siegel set is given by the

conditions lxl :::; c and y > t.

5.2.2. Lemma. Let 6 = 6t,w be a Siegel set with respect to P. Let c E G(IQ) and

P' = c P. Then c. 6 is contained in a Siegel set with respect to P'.

Proof. Note first that if p E P, then p.6 belongs to a Siegel set with respect to
P. (Write p = x.a (x E Np.Mp, a EA). Then p.6 = x. aw.a.Ap,tK.)
We have G = K.P and can write c = kp (p E P , k EK). By the remark just
made, we are reduced to showing that k .6 is contained in a Siegel set with respect
to P'. We have P' = k P and conjugation by k brings t he Langlands decomposition
with respect to P into the Langlands decomposition with respect to P'. Therefore

k.6 = kw. kAP,t·K

where kw is compact in Np' MP', and, obviously, k AP,t = AP',t·
The following lemma will be used in Section 6.8.

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