Categorical local Langlands correspondence

Establish an equivalence between the derived category of compact lisse-étale sheaves on the moduli stack of $G$-bundles over the Fargues–Fontaine curve and the derived category of bounded coherent complexes with quasi-compact support on the moduli stack of Langlands parameters, together with the stated $mathcal Coh^{b,\mathrm{qc}(\mathrm{Par}_G)$-linear spectral action.

Background

The paper presents the Fargues–Scholze categorical local Langlands correspondence as a conjectural derived categorical upgrade of the classical local Langlands correspondence. The two sides are the lisse-étale sheaf theory of BunG\mathrm{Bun}_G and the coherent sheaf theory of the parameter stack ParG\mathrm{Par}_G.

The conjecture also requires compatibility with the spectral action, which is intended to recover representation-theoretic information from coherent sheaves on the parameter stack. The paper treats this equivalence as foundational for its proposed construction of Arthur packets.

References

There is an equivalence of categories

— A conjectural construction of Arthur packets in Fargues-Scholze's categorical local Langlands correspondence  (2608.24341 - Tam, 25 Aug 2026) in Conjecture 1, Section 2 (Categorical LLC), labeled FS-main-statement

According to Conj 1.7.3, the equivalence (\ref{FS-main-statement-equivalence}) is conjectured to be restricted from the functor

— A conjectural construction of Arthur packets in Fargues-Scholze's categorical local Langlands correspondence  (2608.24341 - Tam, 25 Aug 2026) in Remark immediately following Conjecture FS-main-statement, Section 2

One may wonder whether the theory of CLLC is functorial with respect to this curve input.

— A conjectural construction of Arthur packets in Fargues-Scholze's categorical local Langlands correspondence  (2608.24341 - Tam, 25 Aug 2026) in End of Section 1 (Introduction)

$\mathcal Perf( C_\varphi) \simeq \mathcal Db([*/ A_\varphi]) \simeq \mathcal Rep_{\mathrm{fin}(A_\varphi)$.

— A conjectural construction of Arthur packets in Fargues-Scholze's categorical local Langlands correspondence  (2608.24341 - Tam, 25 Aug 2026) in Conjecture immediately after the discussion of equivariant fundamental groups, Section 7.1 (Regular conormal bundles)