A Refined Gross-Prasad Conjecture for Unitary Groups

Let F be a number field, AF its ring of adeles, and let pin and pin+1 be irreducible, cuspidal, automorphic representations of SOn( AF ) and SOn+1( AF ), respectively. In 1991, Benedict Gross and Dipendra Prasad conjectured the non-vanishing of a certain period integral attached to pi n and pin+1 is equivalent to the non-vanishing of L(1/2, pin ⊠ pin+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-correspondence.