Endoscopy and the automorphic tensor product on the unitary group

We study some tempered endoscopic cases of Langlands functoriality on the $n$-variable unitary groups via the simple stable trace formula. This extends previous work of Rogawski and Clozel-Harris-Labesse. Ramakrishnan and Kim-Shahidi have proved the existence of the automorphic tensor product for $GL_2 \times GL_2$ and $GL_2 \times GL_3$. We apply our endoscopic results to, in a sense, "lift" these theorems and deduce the existence of the automorphic tensor product for the quasi-split unitary groups $U^*_2 \times U^*_2$ and $U^*_2 \times U^*_3$. We then formulate the notion of overconvergent Langlands functoriality on the definite unitary group. We apply some results of Chenevier to obtain morphisms of eigenvarieties that extend classical endoscopic Langlands functoriality. This enables us to obtain, in the \emph{small slope} setting, some classical cases of non-tempered endoscopic Langlands functoriality. To conclude, we apply the recent potential automorphy results of Barnet-Lamb-Gee-Geraghty-Taylor to obtain some cases of the potential automorphic tensor product for higher rank unitary groups.