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The Langlands dual and unitary dual of quasi-split $PGSO_8^E$

Author: Caihua Luo
Journal: Represent. Theory 24 (2020), 292-322
MSC (2010): Primary 22E35
Published electronically: July 21, 2020
MathSciNet review: 4126655
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Abstract: This paper serves two purposes, by adopting the classical Casselman–Tadi$\acute {c}$’s Jacquet module machine and the profound Langlands–Shahidi theory, we first determine the explicit Langlands classification for quasi-split groups $PGSO^E_8$ 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 $PGSO^E_8$ 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 $PGSO_8^E$.

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Additional Information

Caihua Luo
Affiliation: Department of Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, Chalmers Tvärgata 3, SE-412 96 Göteborg, Sweden
MR Author ID: 1253923
ORCID: 0000-0001-9523-638X

Keywords: Langlands classification (dual), unitary dual, Jacquet module, principal series, adjoint $^{3}D_4$.
Received by editor(s): May 5, 2019
Received by editor(s) in revised form: December 25, 2019, and February 21, 2020
Published electronically: July 21, 2020
Article copyright: © Copyright 2020 American Mathematical Society