A generic categorical local Langlands correspondence for quasi-split reductive groups

2026/06/08 by David Helm, Maarten Solleveld, Yujie Xu
#math.NT #math.AG #math.RT

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Abstract

We unconditionally prove a generic categorical local Langlands conjecture for a large class of quasi-split reductive p-adic groups G, including all quasi-split classical groups and some non-classical groups. More precisely, we construct a natural fully faithful functor from the stable ∞-category of generic Bernstein blocks on the automorphic side to the stable ∞-category of ind-coherent sheaves on the moduli stack of (arithmetic) L-parameters, generalizing earlier work of the first author with Ben-Zvi, Chen and Nadler [BZCHN24] for GL_n. Moreover, for an arbitrary quasi-split reductive p-adic group G, we formulate a classical local Langlands framework under which a classical correspondence can be lifted to an ∞-categorical correspondence. Our result builds upon the phenomenal recent work of Zhu [Zhu25] on the unipotent block, as well as structure results such as [Sol22] on the automorphic side and [DHKM25] on the spectral side. In particular, our work establishes [HM26,Conjecture 8.2.1], which implies that the conditional proof of [HM26] for the Fargues-Scholze categorical local Langlands equivalence [FS24] (conditional on the conjectured compatibility of the Fargues-Scholze construction with spectral Eisenstein series) applies as well to a large class of quasi-split reductive p-adic groups G.

Citations

• Hecke algebras and local Langlands correspondence for non-singular depth-zero representations
• Hecke algebras for p-adic reductive groups and Local Langlands Correspondences for Bernstein blocks

Related

• Local Langlands in families: The banal case
• Genericity and contragredience in the local Langlands correspondence
• Tame categorical local Langlands correspondence
• The Explicit Local Langlands Correspondence for G₂
• Centre de Bernstein dual pour les groupes classiques