Triality and Functoriality

TL;DR

This paper leverages triality automorphisms of type D4 algebraic groups to establish new instances of Langlands functoriality, including spin and tensor product liftings, with applications to properties of L-functions of Siegel modular forms.

Contribution

It introduces novel functorial liftings from GSp_6 and GSp_4 to GL_8 using triality automorphisms, advancing the Langlands program.

Findings

Proves weak spin lifting from GSp_6 to GL_8.

Establishes tensor product lifting from GL_2 x GSp_4 to GL_8.

Analyzes properties of spinor L-functions for Siegel modular forms.

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.