Transfer of Siegel cusp forms of degree 2

Let π be the automorphic representation of GSp₄(A) generated by a full level cuspidal Siegel eigenform that is not a Saito-Kurokawa lift, and τ be an arbitrary cuspidal, automorphic representation of GL₂(A). Using Furusawa’s integral representation for GSp₄×GL₂ combined with a pullback formula involving the unitary group GU(₃,₃), we prove that the L-functions L(s,π×τ) are “nice”. The converse theorem of Cogdell and Piatetski-Shapiro then implies that such representations π have a functorial lifting to a cuspidal representation of GL₄(A). Combined with the exterior-square lifting of Kim, this also leads to a functorial lifting of π to a cuspidal representation of GL₅(A). As an application, we obtain analytic properties of various L-functions related to full level Siegel cusp forms. We also obtain special value results fforms. We also obtain special value results for GSp₄×GL₁ and GSp₄×GL₂.