Certain Induced Complementary Series of the Universal Covering of the Symplectic Group

Let S̃p(q,R) be the universal covering of the symplectic group. In this paper, we study the unitarity problem for the representation induced from a one dimensional character ( , v) of G̃L(q − p) tensoring with a unitary representation σ0 of a smaller S̃p(p,R). We establish the unitarity when the real character v is in a certain interval depending on and σ0 satisfies a certain growth condition. We then apply our result inductively to construct complementary series for degenerate principal series with multiple G̃L-factors. In particular, in class , there are 2q principal complementary series of size at least (0, c ) q with c = min(|1−2 |, 1−|1−2 |). Various complementary series of the linear group Sp(q,R) have been constructed and studied by Kostant ( [18]), Knapp-Stein ( [16]), Speh-Vogan ( [26]). Their complementary series are close to the tempered dual, while our complementary series are often “far away ”from the tempered dual. More recently, Barbasch obtain all spherical unitary representations of Sp(q,R) ( [2]). Our approach is quite different and works well for the universal covering group. Essentially, we realize the underlying Ind functor as a Howe type duality with respect to a degenerate principal series representation I( , v) of S̃p(p + q,R) ( [12]). Then we construct an induced intertwining operator for the induced representation under consideration from the intertwining operator on I( , v). The positivity of the induced intertwining operator is established by a standard deformation argument based on the positivity of the intertwining operator on I( , v). ∗This research is supported in part by an NSF grant and LSU.