nLab Hecke category

Idea

Generally, for $X \longrightarrow Y$ a suitable map of schemes or algebraic stacks, then the corresponding Hecke category is the category (derived category, (infinity,1)-category) $\mathcal{D}(X \times_Y X)$ of D-modules on the (homotopy) fiber product, regarded as a monoidal category by regarding its objects as specifying integral transforms in $Fun_{\mathcal{D}(Y)}(\mathcal{D}(X), \mathcal{D}(X))$ (Ben-Zvi & Nadler 09, section 5.1).

For $X\to Y$ the inclusion $\mathbf{B}B \to \mathbf{B}G$ of the delooping of a Borel subgroup of a complex reductive group, then (generally for maps of delooped groups like this, see here at homotopy limit) the homotopy fiber product is $X \times_Y X \simeq B \backslash \backslash G // B$. The Hecke category for this case is the default case of Hecke categories used in geometric representation theory.

The concept of Hecke category is a categorification of that of Hecke algebra.

Related concepts

• Hecke correspondence

• Hecke eigensheaf

References

• David Ben-Zvi, David Nadler, section 5.1 of The Character Theory of a Complex Group (arXiv:0904.1247)

A sruvey is around slide 15 (40 of 77) in

• David Ben-Zvi, Loop Groups, Characters and Elliptic Curves, talk (2012)