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

Abstract

We $\text{unconditionally}$ prove a $\text{generic}$ 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 $\infty$-category of $\text{generic}$ Bernstein blocks on the automorphic side to the stable $\infty$-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 ${\mathrm{GL}}_n$. Moreover, for an $\text{arbitrary}$ quasi-split reductive $p$-adic group $G$, we formulate a classical local Langlands framework under which a classical correspondence can be lifted to an $\infty$-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 $G$.

44 pages; comments welcome