1549380232-Automorphic_Forms_and_Applications__Sarnak_

5. ARITHMETIC SUBGROUPS. REDUCTION THEORY 19

(where A+ is the positive Weyl chamber), also called a Cartan decomposition. Let

p E X(A) be defined, as usual, by

(44) 2p = L dim(g/3),8 = L m 0 a

/3E~+ oE6

We claim now that we have the more precise version of Section 3.4 (8):

(45)

Let

(46)

vol( Gt) --< tm where m = L m 0

oE6

D(a) = IT (a/3 - a-/3r{J (a EA, n/3 = dimg/3)

/3E~+

On G there is a Haar measure adapted to ( 43)

(47) dg = D(a).da.dk.dk'

where dk, dk' and da are Haar measures on K, K and A respectively, meaning that

if f is say, a compactly supported integrable function on G, then:

1

f(g)dg = ( D(a).da ( f(k.a.k')dk.dk'

G } A+ JKxK

([15], X, Prop. 1.17, p. 381 - 382). Since llk.a.k'll::::: llall (k,k' EK, and a EA) we

see, by applying ( 4 7) to the characteristic function of Gt, that:

(48) vol(Gt)--< ( D(a).da--< ( a^2 Pda

}A+ }A+

since D(a) is clearly majorized by a positive multiple of a^2 P on A+. On A we take

the a^0 , a E .0., as coordinates. We have

therefore

(49)

and (45) now follows from (48) and (49).

5. Arithmetic subgroups. Reduction theory

From now on F = Q, IR (and F = C as before).

5.1. Let G be a Q-group and G C GLn a Q-embedding. A subgroup r C G(Q)
is arithmetic if it is commensurable with G n GLn (Z) (i.e. r n ( G n GLn (Z)) is
of finite index in both groups). This notion is independent of the embedding and
compatible with surjective Q-morphisms.
Assume G to be connected. Then the group^0 G can also be defined as
nxEX(G) kerx^2 , and hence can be viewed as a Q-group (or rather the group of
real points in one).

A rational character x maps r into {±1} (since r, being arithmetic in GL 1 ,

leaves invariant a lattice in Q, i.e. an infinite cyclic group), hence r c^0 G. The
quotient r\^0 G has finite invariant volume. It is compact if and only if LG is
anisotropic over Q, or if and only if prk Q(LG) = 0 and the image of r in LG