Fargues-Scholze spectral action

In Fargues–Scholze theory, the spectral action is a monoidal action

\operatorname{Perf}\!\left( [Z^1(W_E,\widehat G)/\widehat G] \right) \curvearrowright D_{\mathrm{lis}}(\operatorname{Bun}_G)

of perfect complexes on the stack of local $L$ -parameters on the derived category of lisse $\ell$ -adic sheaves on the stack of $G$ -bundles on the Fargues–Fontaine curve.

Precise coefficient rings and compactness conditions are part of the theorem.

Construction principle

Geometric Satake supplies Hecke functors indexed by representations of the dual group $\widehat G^I$ , and the leg divisors carry independent Weil-group actions. Their compatibility for all finite sets $I$ satisfies the relations of perfect complexes on the cocycle stack. This categorical upgrade packages the local excursion operators.

Consequences

Taking endomorphisms of the tensor unit gives the map from the spectral Bernstein center to the ordinary Bernstein center. The action also decomposes the sheaf category according to connected components of the parameter stack.

For a point $\phi$ with centralizer $S_\phi$ , the fiber category remembers representations of $S_\phi$ , which is the structure expected to encode the members of an $L$ -packet across the groups $G_b(E)$ .

Theorem versus categorical conjecture

The spectral action itself is constructed. A much stronger conjecture says that acting on a Whittaker sheaf gives a fully faithful functor from perfect complexes on the parameter stack and extends to an equivalence between an appropriate coherent spectral category and the automorphic sheaf category. That categorical local Langlands equivalence is not proved in full generality.

References

Laurent Fargues and Peter Scholze, “Geometrization of the local Langlands correspondence,” Chapters VIII–X. arXiv.