The Refined Gross–Prasad Conjecture for Unitary Groups

Let F be a number field and
$\mathbb A_F$
be its ring of adèles, and let π
n
and π
n+1
be irreducible, cuspidal, automorphic representations of
${\rm SO}_n(\mathbb A_F)$
and
${\rm SO}_{n+1}(\mathbb A_F)$
, respectively. In 1991, Gross and Prasad [8] conjectured the nonvanishing of a certain period integral attached to π
n
and π
n+1
is equivalent to the nonvanishing of
$L(\frac {1}{2},\pi_n\boxtimes\pi_{n+1})$
. More recently, Ichino and Ikeda [18] 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 [34], while the second case will use the machinery of the Θ-correspondence.