Triality and Functoriality

We use the triality automorphism of simple algebraic groups of type $D_4$ to prove some new instances of global Langlands functorial lifting. In particular, we prove the (weak) spin lifting from ${\mathrm{G}\mathrm{S}\mathrm{p}}_6$ to ${\mathrm{G}\mathrm{L}}_8$ and the tensor product lifting from ${\mathrm{G}\mathrm{L}}_2\times {\mathrm{G}\mathrm{S}\mathrm{p}}_4$ to ${\mathrm{G}\mathrm{L}}_8$. As an arithmetic application, we establish the expected properties of the spinor L-function attached to an arbitrary Siegel modular cusp form for ${\mathrm{S}\mathrm{p}}_6(\mathbb{Z})$ generating a holomorphic discrete series.