Howe quotients of unitary characters and unitary lowest weight modules

Abstract

In this paper, let ( G , G ′ ) (G,G’) be the dual pair ( S p ~ ( p , R ) , O ~ ( n , m ) ) (\widetilde {\mathrm {Sp}}(p,\mathbb {R}), \tilde {\mathrm O}(n,m)) . We will determine the composition series of the Howe quotients of G ′ G’ which are lifts from one-dimensional unitary representations of G G and unitary lowest weight modules of G G . We will also determine the unitarizability of the subquotients. Our method also works for the dual pairs ( U ~ ( p , q ) , U ~ ( n , m ) ) (\widetilde {\mathrm U}(p,q), \widetilde {\mathrm U}(n,m)) and ( O ~ ∗ ( 2 p ) , S p ~ ( n , m ) ) (\tilde {\mathrm O}^*(2p), \widetilde {\mathrm {Sp}}(n,m)) .