Subsection 1.4. (0NZ3) · ScalingStacks

1.4. Hecke categories

Let $X\to Y$ be a map of perfect stacks. Our main technical result gives an identification of (not necessarily symmetric) monoidal $\infty$ -categories

\qc(X\times_{Y}X)\simeq\Fun_{\qc(Y)}(\qc(X),\qc(X))

where the left hand side is equipped with convolution and the right hand side with the composition of functors. In other words, we have an identification of associative algebra objects in the $\infty$ -category ${\mathcal{P}r}^{\rm L}$ of presentable $\infty$ -categories with morphisms left adjoints.

In Section 5.2, we calculate the center of $\qc(X\times_{Y}X)$ in the following form. Recall that $\mathcal{L}Y$ denotes the derived loop space of $Y$ .

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Theorem 1.9. Suppose $p:X\to Y$ is a map of perfect stacks that satisfies descent. Then there is a canonical equivalence

\mathcal{Z}(\qc(X\times_{Y}X))\simeq\qc({\mathcal{L}Y})

in which the central functor

\qc({\mathcal{L}Y})\simeq\mathcal{Z}(\qc(X\times_{Y}X))\to\qc(X\times_{Y}X)

is given by pullback and pushforward along the correspondence

\mathcal{L}Y\longleftarrow\mathcal{L}Y\times_{Y}X\longrightarrow X\times_{Y}X.

When $p:X\to Y$ is also proper with invertible dualizing complex, we deduce an analogous statement identifying the trace of $\qc(X\times_{Y}X)$ with $\qc(\mathcal{L}Y)$ , conditional on the validity of Grothendieck duality in the derived setting.

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Remark 1.10. One can find a precursor to the above in the work of Müger [M] and Ostrik [Os].

Given a semisimple abelian monoidal category $\mathcal{C}$ and a module category $M$ , consider the monoidal category $\mathcal{C}_{M}^{*}$ consisting of $\mathcal{C}$ -linear endofunctors of $M$ . Then independently of $M$ , there is a canonical identification of the Drinfeld centers of $\mathcal{C}_{M}^{*}$ and $\mathcal{C}.$

A motivating example is when $H\subset G$ are finite groups, and one takes $\mathcal{C}=\operatorname{Rep}(G)$ and $M=\operatorname{Rep}(H)$ , so that $\mathcal{C}^{*}_{M}\simeq\operatorname{Vect}(H\backslash G/H)$ . Then independently of $H$ , the center of $\operatorname{Vect}(H\backslash G/H)$ is the category of adjoint equivariant vector bundles on $G$ .

The above theorem extends this picture from finite groups to algebraic groups.

1.4.1. Example: affine Hecke categories

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

{\mathcal{S}t}_{G}=\tilde{G}\times_{G}\tilde{G}.

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

{\mathcal{L}oc}_{G}(T^{2})=\Map(T^{2},BG)

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.

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Corollary 1.11. There is a canonical equivalence

\mathcal{Z}(\mathcal{H}_{G}^{\it{aff}})\simeq\qc(\mathcal{L}(\mathcal{L}BG))\simeq\qc((BG^{S^{1}})^{S^{1}})\simeq\qc({\mathcal{L}oc}_{G}(T^{2}))

between the center of the quasi-coherent affine Hecke $\infty$ -category and the $\infty$ -category of sheaves on the derived moduli stack of $G$ -local systems on the torus.

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.

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Remark 1.12. We will not return to specific applications to representation theory in this paper, but the interested reader will find further results along these lines in the paper [BN2] which studies integral transforms in the context of ${\mathcal{D}}$ -modules. It includes applications to the more familiar Hecke categories ${\mathcal{D}}(B\backslash G/B)$ of ${\mathcal{D}}$ -modules on flag varieties. In particular, their Drinfeld centers are identified with character sheaves on $G$ , resulting in a Langlands duality for character sheaves.

David Ben-Zvi, John Francis, David Nadler

Original source: arXiv:0805.0157v5