Fargues construction of elliptic parameter sheaves

Construct, for every elliptic Langlands parameter $\varphi$, a complex of sheaves $\mathcal F_\varphi$ on $\mathrm{Bun}_G$ equipped with an action of $S_\varphi$ that satisfies the Hecke-eigensheaf, cuspidality, L-packet, and spectral-action construction properties stated in Conjecture 4.4.

Background

The conjecture seeks to realize the representation-theoretic local Langlands correspondence geometrically. For an elliptic parameter, the desired sheaf should be a Hecke eigensheaf with eigenvalue determined by the parameter.

Its restrictions to the basic strata of BunG\mathrm{Bun}_G are expected to recover the representations in the corresponding L-packet, while the spectral action of the skyscraper sheaf at the parameter should produce the entire object.

References

Given an elliptic parameter $\varphi$, there exists a complex of sheaves $\mathcal F_\varphi\in \mathcal D_{\mathrm{lis}(\mathrm{Bun}G)$, equipped with an action of $S\varphi$, satisfying the following properties.

— A conjectural construction of Arthur packets in Fargues-Scholze's categorical local Langlands correspondence  (2608.24341 - Tam, 25 Aug 2026) in Conjecture 4.4, Section 5 (Fargues' conjecture), labeled Fargues-conjecture-detailed

Regarding each $\rho$ as a skyscraper sheaf supported at the point $x_\varphi \in \mathrm{Par}_G$, we further conjecture that

— A conjectural construction of Arthur packets in Fargues-Scholze's categorical local Langlands correspondence  (2608.24341 - Tam, 25 Aug 2026) in Equation (5.2), immediately after Conjecture Fargues-conjecture-detailed, Section 5

One may wonder whether an analogue of (\ref{Fargues-conjecture-individual}) also holds for non-elliptic parameters (if we use $x_{\varphi ,!}\Lambda$ instead of $\mathcal E_\varphi = x_{\varphi, *}\Lambda$).

— A conjectural construction of Arthur packets in Fargues-Scholze's categorical local Langlands correspondence  (2608.24341 - Tam, 25 Aug 2026) in Paragraph following Equation (5.2), Section 5

\item $\mathcal A_C$ is a perfect complex (see Cor VIII.2.10).

— A conjectural construction of Arthur packets in Fargues-Scholze's categorical local Langlands correspondence  (2608.24341 - Tam, 25 Aug 2026) in Conjecture labeled main conjecture, Section 7.3 (Conjectural Arthur packets in CLLC)