Hecke correspondence

Definition

Let $X$ be a smooth projective curve. The Hecke stack $\operatorname{Hecke}_G$ classifies tuples $(x,E,E',\beta)$ , where $x\in X$ , $E,E'\in\operatorname{Bun}_G(X)$ , and $\beta$ is a Hecke modification from $E$ to $E'$ at $x$ .

It forms a correspondence

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

Interchanging “left” and “right” is a convention; formulas must use one choice consistently.

Bounded correspondence

For each dominant coweight $\lambda$ , a closed substack $\operatorname{Hecke}_{G,\leq\lambda}$ bounds the relative position by $\lambda$ . More generally, a spherical sheaf on the affine Grassmannian provides a kernel on the Hecke stack.

Use

Pull-push along this correspondence defines a geometric Hecke functor. Because these maps are maps of stacks, derived pullback, pushforward, and shifts are part of the construction.

References

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