Hecke eigensheaf - Knowlpedia

Definition

Let $E$ be a $\widehat G$ -local system on $X$ . A Hecke eigensheaf with eigenvalue $E$ is an automorphic $D$ -module $\mathcal F$ on $\operatorname{Bun}_G(X)$ , together with compatible isomorphisms

H_V(\mathcal F)\simeq \mathcal F\boxtimes E_V

for every finite-dimensional representation $V$ of $\widehat G$ . Here $H_V$ is the geometric Hecke functor and $E_V$ is the vector local system associated to $E$ through $V$ .

The isomorphisms must be tensor-compatible in $V$ and compatible with fusion at several points. A single eigen-isomorphism for one representation is generally not enough.

Pointwise and categorical forms

Constructing $\mathcal F_E$ for an individual $E$ is the eigensheaf form of geometric Langlands. The categorical form organizes all spectral parameters at once as an equivalence of sheaf categories.

Scope

For reducible or otherwise singular spectral parameters, eigensheaves can have derived multiplicities and automorphisms. The modern formulation is therefore not a naive bijection between isomorphism classes.

References

Alexander Beilinson and Vladimir Drinfeld, Quantization of Hitchin’s Integrable System and Hecke Eigensheaves, preprint. author manuscript.
Edward Frenkel, Dennis Gaitsgory, and Kari Vilonen, “On the geometric Langlands conjecture,” Journal of the American Mathematical Society 15 (2002), 367–417. arXiv.