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

Abstract

We prove a generic categorical (arithmetic) 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 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 [BZCHN24] for $\mathrm{GL}_n$. 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 $\infty$-categorical correspondence. Furthermore, combined with the recent work of Hansen-Mann [HM26] and assuming the expected compatibility of Fargues-Scholze construction with spectral Eisenstein series, our results give the full Fargues--Scholze categorical local Langlands equivalence [FS24], without the genericity condition, for a large class of quasi-split reductive $p$-adic groups $G$.