A refined Gross-Prasad conjecture for unitary groups

Let F be a number field, AF its ring of adeles, and let [pi]n and [pi]n+1 be irreducible, cuspidal, automorphic representations of SOn(AF) and SOn+1AF), respectively. In 1991, Benedict Gross and Dipendra Prasad conjectured the non-vanishing of a certain period integral attached to [pi]n and [pi]n+1 is equivalent to the non-vanishing of L(1/2, [pi]n [X] [pi]n+1). More recently, Atsushi Ichino and Tamotsu Ikeda gave a refinement of this conjecture, as well as a proof of the first few cases (n = 2,3). Their conjecture gives an explicit relationship between the aforementioned L-value and period integral. We make a similar conjecture for unitary groups, and prove the first few cases. The first case of the conjecture will be proved using a theorem of Waldspurger, while the second case will use the machinery of the [theta] correspondenc