The Langlands dual and unitary dual of quasi-split PGSO(8)(E)

Journal article, 2020

This paper serves two purposes, by adopting the classical Casselman-Tadic's Jacquet module machine and the profound Langlands-Shahidi theory, we first determine the explicit Langlands classification for quasi-split groups PGSO(8)(E) which provides a concrete example to guess the internal structures of parabolic inductions. Based on the classification, we further sort out the unitary dual of PGSO(8)(E) and compute the Aubert duality which could shed light on the final answer of Arthur's conjecture for PGSO(8)(E). As an essential input to obtain a complete unitary dual, we also need to determine the local poles of triple product L-functions which is done in the appendix. As a byproduct of the explicit unitary dual, we verified Clozel's finiteness conjecture of special exponents and Bernstein's unitarity conjecture concerning AZSS duality for PGSO(8)(E).