Tensor Product of Degenerate Principal Series and Local Theta Correspondence

We construct an intertwining map from the oscillator representation to the tensor product of two degenerate principal series of G and G′ 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)).