Theta Correspondence and Quantum Induction

Theta correspondence θ over R was established by Howe in ( [5]). In [10], we proved that θ preserves unitarity under certain restrictions, generalizing the result of Jian-Shu Li ( [16]). The goal of this paper is to elucidate the idea of constructing unitary representation through the propagation of theta correspondences. We show that under a natural condition on the sizes of the related dual pairs which can be predicted by the orbit method ( [23], [26], [18]), one can compose theta correspondences to obtain unitary representations. We call this process quantum induction. Some of our results were announced at ICM-2002.