32 ARMAND BOREL, AUTOMORPHIC FORMS ON REDUCTIVE GROUPSLemma, Z(g) operates by homotheties. Therefore, every element of VJ( is Z(g)-
finite.9.6. Discrete spectrum and L^2 -automorphic forms. Assume now that V C
L^2 (f\G) is irreducible. Then VJ( consists of elements which are K-finite, Z(g)-
finite. By Proposition 3.6, they are automorphic forms.
Let L^2 (f\G)dis be the smallest closed subspace of L^2 (r\G) containing all ir-
reducible G-invariant subspaces. It is stable under G and is a Hilbert direct sum
of irreducible subspaces. It follows from Theorem 7.4 that it has finite multiplic-
ities, in the sense of Corollary 8.5. Indeed, let W be an irreducible summand of
L^2 (f\G)dis and let .A Ek be such that WA i-0. Then WA consists of automorphic
forms of a prescribed type (J, .A), for some ideal J of Z(g). Since A(f, J, .A) is finite
dimensional, L^2 (f\G)dis can have only finite many summands isomorphic to Was
G-modules.
Let f E L^2 (f\G) be Z(g)-finite and K-finite. Then, the smallest invariant
closed subspace of L^2 (r\ G) containing f is a finite sum of closed irreducible sub-
spaces ([4], 5 .26), therefore f belongs to L^2 (f\G)dis· As a consequence, the L^2 -
automorphic forms are the elements of L^2 (f\G)dis which are Z(g)-and K-finite.
Remark: We have referred to statements in [4] in which G is assumed to be
connected. Our G is not necessarily so, but it has at most finitely many connected
components. It is standard and elementary that the restriction to a normal sub-
group of finite index of an irreducible representation is a finite sum of irreducible
representations, so the extension to our case is immediate.
9.7. If f\G is compact, then °L^2 (f\G) = L^2 (f\G)dis = L^2 (f\G). In general
(^0) L (^2) (f\G) c L (^2) (f\G)dis, by Corollary 8.5. Its complement there is called the
residual spectrum L^2 (f\G)res· The complement of L^2 (f\G)dis in L^2 (f\G) is the
continuous spectrum L^2 (f\G)ct· It is analyzed by means of the analytic continua-
tion of Eisenstein series. The elements of L^2 (f\G)res occur as residues or iterated
residues at poles of analytically continued Eisenstein series.
As a generalization of the statements made in the proof of Theorem 8.4, it is also
true that the restriction of *a (a E C;;"'(G)) to L^2 (f\G)dis is of trace class. This
was proved by W. Muller first for a K-finite a, and then in general, independently,
by W. Muller and L. Ji.
- A decomposition of the space of automorphic forms.
The main purpose of this section is to define a projector of the space A(f\G) of
automorphic forms on r\ G onto the space^0 A(r\ G) of cusp forms.
10.1. The parabolic subgroups P , Q E P<Q are said to be associate if a G(Q)-
conjugate of one has a common Levi Q-subgroup with the other. Let Ass(G) be
the set of Q-conjugacy classes of associated parabolic Q-subgroups. It is finite,
since f\P<Q is finite.
Before giving the following definition, let us remark that if
f E Cm 9 (f\G) and g E CJd(f\G), then f.g is fast decreasing and a fortiori in
L1 (f\ G), so that the scalar product
(f, g) = r f(x)g(x)dx
lr\G