 
Summary: BULLETIN OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 83, Number 4, July 1977
A TRUNCATION PROCESS FOR REDUCTIVE GROUPS
BY JAMES ARTHUR1
Communicated by J. A. Wolf, January 26, 1977
Let G be a reductive group defined over Q. Index the parabolic subgroups
defined over Q, which are standard with respect to a minimal (O)P, by a partially
ordered set 4. Let 0 and 1 denote the least and greatest elements of 9 respec
tively, so that (l)P is G itself. Given u E 9, we let (")N be the unipotent radical
of (u)p, (u)M a fixed Levi component, and (U)A the split component of the cen
ter of (u)M. Following [1, p. 328], we define a map (u)H from (")M(A) to
(u)a =
Hom(X((U)M)Q, R) by
ex,(u)H(m)> = Ix(m)l, x E X((U)M)Q, m E ()M(A).
If K is a maximal compact subgroup of G(A), defined as in [1, p. 328], we ex
tend the definition of (")H to G(A) by setting
(u)H(nmk) = ()H(m), n e (u)N(A), m E ()M(A), k e K.
Identify (°)a with its dual space via a fixed positive definite form ( ,) on (°)a
which is invariant under the restricted Weyl group Q2. This embeds any (u)a into
