An approach to the characterization of the local Langlands correspondence

Bertoloni Meli, Alexander; Youcis, Alex

doi:10.1090/ert/643

An approach to the characterization of the local Langlands correspondence
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by Alexander Bertoloni Meli and Alex Youcis;

Represent. Theory 27 (2023), 415-430

DOI: https://doi.org/10.1090/ert/643

Published electronically: July 7, 2023

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Abstract:

Scholze and Shin [J. Amer. Math. Soc. 26 (2013), pp. 261–294] gave a conjectural formula relating the traces on the automorphic and Galois sides of a local Langlands correspondence. Their work generalized an earlier formula of Scholze, which he used to give a new proof of the local Langlands conjecture for $\mathrm {GL}_n$. Unlike the case for $\mathrm {GL}_n$, the existence of non-singleton $L$-packets for more general reductive groups constitutes a serious representation-theoretic obstruction to proving that such a formula uniquely characterizes such a correspondence. We show how to overcome this problem, and demonstrate that the Scholze–Shin equation is enough, together with other standard desiderata, to uniquely characterize the local Langlands correspondence for discrete parameters.

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Representation Theory

An approach to the characterization of the local Langlands correspondence
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Abstract: