Triality and Functoriality

Abstract

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 ${\mathrm{GSp}}_6$ to ${\mathrm{GL}}_8$ and the tensor product lifting from ${\mathrm{GL}}_2\times {\mathrm{GSp}}_4$ to ${\mathrm{GL}}_8$. As an arithmetic application, we establish the expected properties of the spinor L-function attached to an arbitrary Siegel modular cusp form for ${\mathrm{Sp}}_6(\mathbb{Z})$ generating a holomorphic discrete series.

48 pages v2 : minor corrections, new subsection 7.7 discussing a few examples