Howe Correspondence for Real Unitary Groups

Roger Howe proved that for any reductive dual pair (G, G′) in the symplectic groupSp(2n, R), there is a one-to-one correspondence of irreducible admissible representations of some two-fold covers ofGandG′. We determine this correspondence explicitly for dual pairs of the form (U(p, q),U(r, s)) withr+s=p+q, and describe it in terms of Langlands parameters. In this case, the correspondence may be understood in a natural way as a correspondence of representations of the linear groups, rather than the appropriate covers. We show that every irreducible admissible representation ofU(p, q) occurs in the correspondence with precisely one unitary group of equal rank. This result verifies a conjecture of Harris, Kudla, and Sweet, who investigated the correspondence for unitary groups of equal size overp-adic fields. The correspondence of discrete series representations was determined by J.-S. Li. For induced representations, the correspondence is obtained in a natural way from the corresponding discrete series on the respective Levi factors of the parabolic subgroups ofU(p, q) andU(r, s). Generalizing a result of Li, we show that under the correspondence representations with nonzero cohomology are matched in an interesting way, with unitarity not necessarily preserved. The proof uses the induction principle which is due to Kudla, and an argument involvingK-types and the space of joint harmonics (Howe).