I.4. - Upper bounds as functions of the constant term

Summary

Lemma

Let ϕ be an automorphic form on G(k)\G. For every standard parabolic subgroup P = MU of G, let us take a set

of cuspidal data for (see 1.3.3). Then there exists c > 0 such that for all g є S, we have the upper bound

see 1.3.3 for the definition of Reл; deg(Q) is the total degree of Q. More generally for all, there exists such that for all g є S, we have the upper bound

where µ^M is the projection of onto (see 1.1.6 (9)).

Proof (a) We proceed by induction on the semi-simple rank of G. Suppose the lemma is proved for every proper standard Levi subgroup M of G. We immediately deduce a similar lemma concerning automorphic forms on M(k)U(A)\G for every proper standard parabolic P = MU of G. Note that if are two such subgroups, we have the equality of cuspidal components

We deduce from this that for all and all, we have an upper bound

A fortiori, we can replace the sum over P′ by the sum over all P′ G and restrict ourselves to g ε S.