Geometric Hecke functor

Core idea

Let $X$ be a smooth complex curve, let $G$ be a connected reductive complex group, and let $V$ be a representation of $\widehat G$ . Via geometric Satake, $V$ determines a spherical kernel $\mathcal K_V$ on the Hecke correspondence

\operatorname{Bun}_G \xleftarrow{\ h_{\mathrm{left}}\ } \operatorname{Hecke}_G \xrightarrow{\ (h_{\mathrm{right}},x)\ } \operatorname{Bun}_G\times X.

With right $D$ -modules and the standard Satake normalization, the geometric Hecke functor is the pull–tensor–push construction

H_V(\mathcal F)= (h_{\mathrm{right}},x)_* \bigl(h_{\mathrm{left}}^!\mathcal F \mathbin{\overset{!}{\otimes}}\mathcal K_V\bigr),

giving

H_V:D\text{-}\operatorname{mod}(\operatorname{Bun}_G) \longrightarrow D\text{-}\operatorname{mod}(\operatorname{Bun}_G\times X).

Tensor compatibility

The functors are compatible with tensor products: $H_{V\otimes W}$ is obtained from iterated Hecke modification and fusion. This factorization compatibility is what lets an eigenvalue be a $\widehat G$ -local system rather than an unrelated collection of vector bundles.

Other conventions

Left $D$ -modules replace the displayed operations by their corresponding left-module conventions. Reversing the two bundle projections gives the inverse-modification convention. Cohomological shifts are absorbed into the normalized Satake kernel $\mathcal K_V$ .

References

Alexander Beilinson and Vladimir Drinfeld, Quantization of Hitchin’s Integrable System and Hecke Eigensheaves, preprint. author manuscript.