Transfer of Siegel cusp forms of degree 2

Let $\pi$ be the automorphic representation of $\GSp_4(\A)$ generated by a full level cuspidal Siegel eigenform that is not a Saito-Kurokawa lift, and $\tau$ be an arbitrary cuspidal, automorphic representation of $\GL_2(\A)$. Using Furusawa's integral representation for $\GSp_4\times\GL_2$ combined with a pullback formula involving the unitary group $\GU(3,3)$, we prove that the $L$-functions $L(s,\pi\times\tau)$ are "nice". The converse theorem of Cogdell and Piatetski-Shapiro then implies that such representations $\pi$ have a functorial lifting to a cuspidal representation of $\GL_4(\A)$. Combined with the exterior-square lifting of Kim, this also leads to a functorial lifting of $\pi$ to a cuspidal representation of $\GL_5(\A)$. As an application, we obtain analytic properties of various $L$-functions related to full level Siegel cusp forms. We also obtain special value results for $\GSp_4\times\GL_1$ and $\GSp_4\times\GL_2$.