Certain Complementary Series of the (universal Covering of) the Symplectic Group

Complementary series of semisimple Lie groups near the tempered dual are studied by Kostant, Knapp-Stein, Speh-Vogan and many others. There are also complementary series arising from degenerate principal series that are “far away” from the tempered dual, for example the Stein representations. In this talk, I will focus on sympletic groups. I will construct certain complementary series induced from a one dimensional character on the $GL$ factors tensoring with certain unitary representations on the symplectic factor. One main ingredient is to study the restriction of the degenerate complementary series, classified completely by Sahi and others, to a pair of symplectic subgroups, in a way that resembles Howe’s duality.