On a Rankin–Selberg integral of three Hermitian cusp forms

Let K = Q(i) K = Q ( i ) . We study the Petersson inner product of a Hermitian Eisenstein series of Siegel type on the unitary group U₅(K) U 5 ( K ) , diagonally-restricted on U₂(K)× U₂(K)× U₁(K) U 2 ( K ) × U 2 ( K ) × U 1 ( K ) , against two Hermitian cuspidal eigenforms F , G of degree 2 and an elliptic cuspidal eigenform h (seen as a Hermitian modular form of degree 1), all having weight k ≡ 0 4 k ≡ 0 ( mod 4 ) . We obtain, through this consideration, an integral representation of a certain Dirichlet series, together with an additional residue term. By taking F to belong in the Maass space, we are able to show that the Dirichlet series possesses an Euler product. Moreover, its p -factor for an inert prime p can be essentially identified with the twist by h of a degree six Euler factor attached to G by Gritsenko. The question of whether the same holds for the primes that split remains unanswered here, even though we make considerable steps in that direction too. Our paper is inspired by a work of Heim, who considered a similar question in the case of Siegel modular forms.