ScalingStacks

As an illustration, we briefly mention a concrete application of the above theorem to a fundamental object in geometric representation theory.

Fix a reductive group $G$, and consider the Grothendieck-Springer resolution $\tilde{G}\to G$ of pairs of a group element and a Borel subgroup containing it. Let $\mathcal{H}_{G}^{\it{aff}}$ be the quasi-coherent affine Hecke $\infty$-category of $G$-equivarant quasi-coherent sheaves on the Steinberg variety

Work of Bezrukavnikov [Be] and others places $\mathcal{H}_{G}^{\it{aff}}$ at the heart of many recent developments in geometric representation theory.

Let us apply the above theorem with $X=\tilde{G}/G$ and $Y=G/G=\mathcal{L}BG$, where $G$ acts via conjugation. Observe that the iterated loop space $\mathcal{L}Y=\mathcal{L}(G/G)=\mathcal{L}(\mathcal{L}BG)$ is nothing more than the derived moduli stack

of $G$-local systems on the torus. Concretely, ${\mathcal{L}oc}_{G}(T^{2})$ is the “commuting variety” (or rather, commuting derived stack) parameterizing pairs of commuting elements in $G$ up to simultaneous conjugation.

A similar statement holds replacing the Grothendieck-Springer resolution $\tilde{G}\to G$ by the Springer resolution $T^{*}G/B\to{\mathcal{N}}$ of the nilpotent cone. In this case, the center is equivalent to the $\infty$-category of sheaves on the derived stack of pairs of a nilpotent and a commuting group element up to simultaneous conjugation. In this linear version, one can also work $\mathbb{G}_{m}$-equivariantly via the natural dilation action.