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On the constant term of a square integrable automorphic form. (English) Zbl 0554.22004
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:
 2.2e+47 Semisimple Lie groups and their representations