On the constant term of a square integrable automorphic form. (English) Zbl 0554.22004

Wallach, N. R.

Operator algebras and group representations, Proc. int. Conf., Neptun/Rom. 1980, Vol. II, monogr. Stud. Math. 18, 227-237 (1984).

[For the entire collection see Zbl 0515.00017.]
Let G be the set of real points of an affine reductive complex group defined over \({\mathbb{Q}}\), and let \(\Omega\) be a subgroup of \(G_{{\mathbb{Z}}}\) of finite index. Denote by \({\mathcal A}(\Omega,G)\) the (\({\mathfrak g},K)\)- module of automorphic functions of G with respect to \(\Gamma\). For a parabolic subgroup \(P=MN\) of G defined over \({\mathbb{Q}}\), and \(f\in {\mathcal A}(\Omega,G)\) set \(\lambda_ p(f)(m)=\int f(nm)d\dot n,\) where \(\dot n\) ranges over \(N\cap \Gamma \setminus N,\) and \(m\in M\). Then \(f\in {\mathcal A}(\Gamma,G)\) is called a cusp form if \(\lambda_ p(f)\equiv 0\) for all proper P as above. In general \(\lambda_ p(f)\in {\mathcal A}(\Gamma_ M,M).\) Let \(\circ {\mathcal A}(\Gamma,G)\) denote the space of cusp forms. The main result of this paper is the following (cf. Theorem (4.3)): Let V be a tempered (\({\mathfrak g},K)\)-module and let \(T: V\to {\mathcal A}(\Gamma,G)\cap L^ 2(\Gamma \setminus G)\) be a (\({\mathfrak g},K)\)-homomorphism. Then \(T(V)\subset \circ {\mathcal A}(\Gamma,G).\) The proof uses two ingredients: (1) Global estimate of the growth of \(\lambda_ p(f)\), for \(f\in {\mathcal A}(\Gamma,G)\cap L^ 2(\Gamma \setminus G)\) (2) Let \(T: V\to {\mathcal A}(\Gamma,G)\) be a (\({\mathfrak g},K)\)-homomorphism. Then \(\lambda_ p\circ T\) factors through \({\mathfrak n}V\). If V is tempered, weights of V/\({\mathfrak n}V\) satisfy certain growth condition, which, coupled with (1), implies the theorem. The author also obtains another estimate for growth of automorphic functions in \(L^ 2(\Gamma \setminus G)\) (cf. Theorem (4.4)).

Reviewer: H.Hecht

MSC:

22E46

Semisimple Lie groups and their representations

Keywords:

tempered representation; affine reductive complex group; automorphic functions; parabolic subgroup; cusp form

Citations:

Zbl 0515.00017