Pure and Applied Mathematics Quarterly

Volume 18 (2022)

Number 5

Special Issue in honor of Professor Benedict Gross’s 70th birthday

Guest Editors: Zhiwei Yun, Shouwu Zhang, and Wei Zhang

Twisted composition algebras and Arthur packets for triality $\operatorname{Spin}_8$

Pages: 1951 – 2130

DOI: https://dx.doi.org/10.4310/PAMQ.2022.v18.n5.a3


Wee Teck Gan (Department of Mathematics, National University of Singapore)

Gordan Savin (Department of Mathematics, University of Utah, Salt Lake City, Ut., U.S.A.)


The purpose of this paper is to construct and analyze certain square-integrable automorphic forms on the quasi-split simply-connected groups $\operatorname{Spin}_8$ of type $D_4$ over a number field $F$. Since the outer automorphism group of $\operatorname{Spin}_8$ is $S_3$, these quasi-split groups are parametrised by étale cubic $F$-algebras $E$ and we denote them by $\operatorname{Spin}^E_8$ (to indicate the dependence on $E$). We shall specialize to the case when $E$ is a cubic field: this gives the so-called triality $\operatorname{Spin}_8$.

W.T.G. is partially supported by a Singapore government MOE Tier 1 grant R-146-000-320-114.

G. Savin is partially supported by a National Science Foundation grant DMS-1901745.

Received 14 June 2021

Received revised 29 December 2021

Accepted 10 January 2022

Published 12 January 2023