The Langlands dual and unitary dual of quasi-split 𝑃𝐺𝑆𝑂₈^{𝐸}

Luo, Caihua

doi:10.1090/ert/545

The Langlands dual and unitary dual of quasi-split $PGSO_8^E$
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Represent. Theory 24 (2020), 292-322 Request permission

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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