Formal degrees and local theta correspondence

The Gross–Prasad conjecture and local theta correspondence

“…We shall also denote by (weak P1) the part of the conjecture (P1) concerning only the correspondence of L-parameters φ → φ ′ ; likewise we have (weak P2). Then we recall that in our earlier paper [17], we have shown: Proposition 1.1. The statements (Θ), (weak P1) and (weak P2) hold.…”

“…We first consider the case dim V = dim W = n. We shall consider the theta correspondence for U(V ǫ n ) × U(W ǫ ′ n ). The following summarises some results of [17]: Theorem 4.1. Let φ be an L-parameter for U(W ± n ).…”

“…• First, by our results in [17], the nontempered case can be reduced to the tempered case on smaller unitary groups.…”

“…By the Howe duality, which was proved by Waldspurger [59] for p = 2 and by the first author and Takeda [20], [21] for any p (so that the assumption p = 2 can be removed from the results of [17] stated below), the maximal semisimple …”

“…We may assume that Θ ψ,V,W (π) = 0. If φ is tempered, then Θ ψ,V,W (π) is irreducible and tempered by [17,Proposition C.4(i)]. In general, by Proposition 9.1, π is a standard module of the form Ind(( …”