Local Langlands Correspondence for Unitary Groups via Theta Lifts

Abstract

Using the theta correspondence, we extend the classification of irreducible representations of quasi-split unitary groups (the so-called local Langlands correspondence) due to [Mok15] to non quasi-split unitary groups. We also prove that our classification satisfies some good properties, which characterize it uniquely. In particular, this provides an alternative approach to the works of [KMSW14] and [MR18].

Arthur's multiplicity formula for even orthogonal and unitary groups

“…If F is non-Archimedean, this is proved in [AG17b, Proposition 6.7] for Case O and [GI16, Proposition 9.1 & Applendix B] for Case U . Note that the proof in [AG17b] and [GI16] rely on the LLC for the orthogonal and unitary groups, these have been established in [Art13] and [Mok15] for the quasi-split group cases and by [CZ20a] and [CZ20b] for the non quasi-split group cases.…”

“…For Case U and F is non-Archimedean, this is first proved by Gan-Ichino [GI16] using the AMF, but one can easily extend Atobe's method to this case to avoid using any global method (for example, see our previous paper [CZ20b, Lemma 4.3.1& Corollary 6.3.2]). Note that the proof in [Ato18] and [GI16] rely on the LLC for the orthogonal and unitary groups, these have been established in [Art13] and [Mok15] for the quasi-split group cases and by [CZ20a] and [CZ20b] for the non quasi-split group cases. When F is Archimedean, this is due to Paul for Case O in [Pau05], and Case U in [Pau00].…”