Triality and Functoriality

TL;DR

The paper exploits the triality automorphism of the $D_4$ Dynkin diagram to produce new Langlands functorial liftings, notably a weak Spin lifting from ${ m PGSp}_6$ to ${ m GL}_8$ and a tensor (Rankin–Selberg) lifting from ${ m GL}_2 imes{ m GSp}_4$ to ${ m GL}_8$. It achieves these liftings by combining similitude theta correspondences with the triality automorphism and by analyzing how Satake parameters transform under dual maps, endoscopy, and isogenies. A major arithmetic application is the detailed description of the spinor $L$-function attached to Siegel cusp forms for ${ m Sp}_6(f Z)$, including meromorphic continuation, functional equation, and the precise shape of the spin lift’s automorphic parameters in both generic and non-generic cases. Overall, the work interlaces triality, theta correspondences, endoscopy, and automorphic transfer to produce concrete functorial liftings with explicit local-global behavior and L-function consequences.

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

Theorems & Definitions (52)

• Theorem 1.1
• Corollary 1.2
• Proposition 3.2
• Lemma 3.3
• proof
• proof
• Proposition 3.5
• Proposition 3.7
• Theorem 3.8
• Lemma 3.9