- CONSTANT TERMS. THE BASIC ESTIMATE 25
(b) We may take for H.S. norm on Lp the restriction of the H .S. norm on G
for a given embedding. Then, clearly, if f is of moderate growth (resp.
fast decrease) on G, then so is f p on Lp.
(c) It remains to see that if f is Z(g)-finite, then fp is Z(lp)-finite. This
relies essentially on a theorem of Harish-Chandra. There is a natural
homomorphism v : Z(g) ----> Z(Cp) such that z E v(z) + U(g).np. Now let
J be an ideal of Z(g) annihilating f. Let z E J, and write z = v(z)+w(z),
where w(z) E U(g).np. Then(zf)p = z.fp = v(z)fp + w(z)f p.
But since f p is left-Np-invariant, we have w(z).f p = 0, whence
(zf)p = v(z).fp
which shows that if the ideal J annihil ates f, then v(J) annihilates f p.
By a theorem of Harish-Chandra, Z(lp) is a finitely generated module
over Z(g), hence if J has finite codimension in Z(g), then v(J) has finite
codimension in Z(lp) and condition (A3) is fulfilled by f p.6.5. Transitivity of the constant term. Let Q be a proper parabolic Q-
subgroup contained in P , and let Q = Q n Lp. Then its image Q in Lp is a
parabolic Q-subgroup, so that (! p) • Q makes sense. We have
(64)
We leave this as an exercise. In particular, f p = 0 implies f Q = 0.
As a consequence of this and Section 6.3(a), f is cuspidal if and only if fp = 0
when P runs through a set of representatives modulo r of proper maximal parabolic
Q-subgroups.
6.6. Fix P E PQ· We shall write P = N.A.M the Langlands decomposition of P.
Let f E C^00 (rN\G). We are interested in the asymptotic behavior off - fp on a
Siegel set 6 = 6 P ,t,w with respect to P.
We can find a decreasing sequence of Q-subgroups of N, normal in N.A,
N = N 1 ::> N2 ::> · · · ::> Nq ::> Nq+l = {1}
with dimensions decreasing by one. Thus, q = dimN. Let rj = r n Nj. It is
cocompact in Nj and rj /rJ+ 1 is infinite cyclic. Let /3j be the weight of A on
nj/nj+l· (Thus /3j runs through the elements of <P(P,A), where f3 occurs dimg13
times.) Fix Xj E nj such that expXj generates rj/rJ+l· Let fj be the "partial
constant term"
(65) fj(x) = r f(nx) dn
lr;\N;where dn gives volume 1 to rj\Nj. Thus Ji= fp and fq = f, and
(66)Choose a basis {Yi} of g.
f - fp = L:)h+1 - fj)
j6.7. Proposition (the basic estimate). Let f E C^1 (rN\G) be of moderate
growth, bounded by A E X(A) on Siegel sets w .r.t to P. L et 6 be one and 61 be