Tensor product of degenerate principal series and local theta correspondence

Abstract

We construct an intertwining map from the oscillator representation to the tensor product of two degenerate principal series of G and G1 which form a reductive dual pair in the sense of Howe. We derive results on local theta correspondence for the dual pairs (O(p, q), Sp(2q, ℝ)). (U(p, q), U(q, q)) and (Sp(p, q), O*(4q)). where p ≥ q. We also describe the correspondence completely for the pairs (O(p, q), SL(2, ℝ)), (U(p, q) U(1,1)), and (Sp(p, q), O*(4)). © 2001 Academic Press.