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Non-vanishing theorems for the cohomology of certain arithmetic quotients.

This paper is concerned with constructions of automorphic forms on classical groups. Our main tool is a very old one, namely that of theta series. Suppose (G, G') is a real reductive dual pair [14]. Let Γ, Γ' be arithmetic lattices in G and G' respectively. Starting from a square integrable irreducible representation of G' occurring in L(F'\G'\ we wish to obtain (via theta lifting) interesting automorphic forms on G, especially those related to cohomology. This approach was suggested to us by Sarnak, and depends on the fact that one has now a relatively good understanding of the occurrence of discrete series in L [9], [10], [30]. Note that Kazhdan [15], Borel-Wallach [7] and Anderson [3] have previously used the same approach in the case where G' is compact; in that case the representations of G thus obtained are all holomorphic.