Some results in the theory of genuine representations of the metaplectic double cover of GSp2n(F)GSp2n(F) over p-adic fields

Let F   be a p-adic field and let G(n)¯ and G0(n)¯ be the metaplectic double covers of the general symplectic group and symplectic group attached to a 2n dimensional symplectic space over F. We show here that if n   is odd then all the genuine irreducible representations of G(n)¯ are induced from a normal subgroup of finite index closely related to G0(n)¯. Thus, we reduce, in this case, the theory of genuine admissible representations of G(n)¯ to the better understood corresponding theory of G0(n)¯. For odd n   we also prove the uniqueness of certain Whittaker functionals along with Rodier type of Heredity. Our results apply also to all parabolic subgroups of G(n)¯ if n   is odd and to some of the parabolic subgroups of G(n)¯ if n   is even. We prove some irreducibility criteria for parabolic induction on G(n)¯ for both even and odd n. As a corollary we show, among other results, that while for odd n  , all genuine principal series representations of G(n)¯ induced from unitary representations are irreducible, there exist reducibility points on the unitary axis if n   is even. We also list all the reducible genuine principal series representations of G(2)¯ provided that the F is not 2-adic.