Theta Correspondence and Quantum Induction

Hongyu He

Published 2002-10-23, updated 2005-10-24Version 3

In his article "Transcending Classical Invariant Theory" (J.A.M.S., 1989, Vol 2), Roger Howe established a correspondence between representations of a dual pair of reductive groups. This correspondence is known as Howe's correspondence or local theta correspondence. In my paper "Unitary Representations and Theta Correspondence for Classical Group of Type I"(to appear in J. Func. Anal.), we proved that theta correspondence over the real field preserves unitarity under certain restrictions, generalizing some results of Jian-Shu Li (Invent. Math., Vol 97, 1989). In this paper, we study the compositions of theta correspondences. Under certain favorable conditions, we prove that the composition of theta correspodence yields unitary representations. We call this process quantum induction. Quantum induction resembles parabolic induction in various ways. Some results in this paper were announced at ICM-2002.