12 ARMAND BOREL, AUTOMORPHIC FORMS ON REDUCTIVE GROUPS
Assuming (A3) expressed in this way, we let A(r, J, ~) be the space of auto-morphic forms such that J.f = 0 and f * ~ = f, where, as before, J is an ideal of
finite codimension of Z(9).
We shall see that if r is arithmetic, then A(r, J, O is finite dimensional. This
theorem is due to Harish-Chandra.
- Reductive groups (review)
We review here what is needed in this course. The catch-words are split tori, roots
and parabolic subgroups. We first start with an example.
4.1. GLn(IR) and SLn(IR). Let A be the subgroup of diagonal matrices with strictly
positive entries and a its Lie algebra. The exponential is an isomorphism of a onto
A, with inverse the logarithm. Let X(A) be the smooth homomorphisms of A into
IR+. If A E X(A), we denote by a>. the value of A on a. Let), be the differential of A
at 1. It is a linear form on a and we can also write a>.= exp(5,(log(a))), a notation
which is often used in representation theory. Note that A i-; ), is an isomorphism
of X(A) onto a.
Let Ai be the character which associates to a its ith coordinate. The Ai span
a lattice in X(A) to be denoted X(A). Its elements are therefore the characters
a i-; a;7'^1 a;'^2 • .. a~n, for mi E Z.
[Interpretation: Let T = (Cr, and let X(T) be the group of rational mor-
phisms of T into C. It is a free abelian group of rank n and X (A) is the restriction
of X(T) to A. In particular, an element of X(A) extends canonically to T.]
4.1.1. Roots. For (3 E X(A), let
(14) 9 ,a = { x E 9 : Ad (a) .x = a.Bx}(3 is called a root if it is nonzero and 9 ,a # 0. If so, it is immediately checked that
there exist i , j ::::; n, i # j, such that g,a is one-dimensional, spanned by the matrix
eij with entry 1 at the ( i, j) spot and zero elsewhere, and that a.B = ad ay, i.e.
(3 =Ai - Aj·
We let = (A, G) = (a, 9) be the set of roots of G with respect to A (their
differentials are the roots of 9 with respect to a). They span a lattice of finite index
in X*(A) and form a root system of type An-l· We have
(15) 9 = a EB EB,aeI>9,B
Fix the ordering on X(A) defined by A 1 > A 2 > ... and let
/::;,, = { ai} 1 < i < n-1, where ai = Ai - Ai+ 1. The elements of /::;,, are the simple roots
(for the gi;e;;:-ordering). Any root is a linear combination of the ai with integral
coefficients of the same sign. The Weyl group W() of the root system may be
identified to
(16) W() = N(A)/Z(A)
Note that:(17) Z(A) = M x A, M = (Z/2z)n