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Automorphic forms constructed from Whittaker vectors. (English) Zbl 0692.10029
Let G be a semisimple Lie group of split rank 1 and $$\Gamma$$ a discrete subgroup of finite covolume. Forming a generalized Poincaré series of weak Whittaker functions attached to the nonunitary principal series a meromorphic family M(s) of functions on $$\Gamma$$ $$\setminus G$$ is obtained that satisfy all axioms of automorphic forms except the condition of moderate growth. It is shown that the principal part at a pole $$s_ 0$$, Re $$s_ 0\geq 0$$ is square integrable and that for a fixed K-type all automorphic forms up to a finite dimensional subspace are obtained. For the trivial K-type all automorphic forms are derived this way. The Fourier coefficients of the M-series are computed and a functional equation is derived.
Trying to generalize this to higher rank leads to the observation that in certain cases (proved here for $$G=SO(n,1)^ d)$$ the moderate growth condition for automorphic forms is redundant. This may be called a Koecher principle. The conjecture is stated that a Koecher principle holds for all G of rank$$\geq 2$$ and $$\Gamma$$ irreducible.
Reviewer: A.Deitmar

##### MSC:
 11F70 Representation-theoretic methods; automorphic representations over local and global fields 22E40 Discrete subgroups of Lie groups 22E46 Semisimple Lie groups and their representations
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