Jeremy Taylor: universal monodromic Bezrukavnikov equivalence
Let $G$ be a reductive group over $\mathbb{C}$. Let $\mathring{I}\subseteq I$ be the pro-unipotent radical of Iwahori.
Let $\widetilde {\mathrm{FL}}= LG/\mathring{I}$ be the enhanced affine flag variety, which is a $T$-torsor over $\mathrm{FL}=LG/I$.
Let $\mathscr{H}:=\mathrm{Shv}_ {(I)}(\widetilde {\mathrm{FL}} )$ weakly constructible category (locally constructible on $I$-orbits without finiteness condition on stalks), in analytic topology.
On dual side, $\check{G}/\mathbb{k}$ the dual group over $\mathbb{k}$ with $\text{char}(\mathbb{k})=0$.
Let $\widetilde {\check{G}}:= \check{G}\times^{\check{B}}\check{B}$ group version of Grothendieck alteration and $\mathrm{St}:=\widetilde {\check{G}}\times_ {\check{G}}\widetilde {\check{G}}$.
Theorem 1 (Taylor-Dhillon). There is a monoidal equivalence $$\mathscr{H}\cong \mathrm{IndCoh}(\mathrm{St}/\check{G}).$$
This is a family of equivalences over $\check{T}\times \check{T}$. Fiber at $(1,1)$ recovers Bezrukavnikov’s equivalence.
Torus case:
- Betti version: $\operatorname{Shv}_ {\text{loc.const}}(T\times\mathbb{X}_ {\ast}(T))\cong \operatorname{QCoh}(\check{T}\times \text{pt}/\check{T})$.
- De Rham version: $\mathscr{D}\text{-Mod}(T\times\mathbb{X}_ {\ast}(T))\cong \operatorname{QCoh}(\check{\mathfrak{t}}/\check{T}\times \text{pt}/\check{T}).$
The big tilting object is the object $\Xi$ which corresponds to structure sheaf under$\mathscr{H}\cong \operatorname{IndCoh}(\mathrm{St}/\check{G})$. It turns out that $\Xi$ is supported on $G/U$.
Theorem 2 (Taylor). There exists $\Xi\in \operatorname{Shv}_ {(B)}(G/U)$, satisfying
- $\operatorname{Hom}(\Xi,-): \mathrm{Shv}_ {(B)}(G/U)\rightarrow \operatorname{QCoh}(\check{T}\times \check{T})$ is monoidal.
- The $!$ and $\ast$-restriction to any $B$-orbit is free over $\check{T}$ and concentrated in perverse degree $0$.
- $\operatorname{End}(\Xi)\cong \mathscr{O}(\mathrm{St}/\check{G})$ and isomorphic to $\mathscr{O}(\check{T}\times_ {\check{T}/W}\check{T})$ if $Z(G)$ is connected.
The Iwahori-Whittaker category
By monoidality, one can define that $${}_ {\chi}\mathscr{H}:=\operatorname{QCoh}(\check{T})\otimes_ {\operatorname{Shv}_ {(B)}(G/U)}\mathscr{H},$$
where $\mathrm{Shv}_ {(B)}(G/U)\xrightarrow{\operatorname{Hom}(\Xi,-)}\operatorname{QCoh}(\check{T}\times\check{T})$ acts on $\operatorname{QCoh}(\check{T})$.
Similarly one defines bi-Whittaker ${}_ {\chi}\mathscr{H}_ {\chi}:=\operatorname{QCoh}(\check{T})\otimes_ {\operatorname{Shv}_ {(B)}(G/U)}\mathscr{H}\otimes_ {\operatorname{Shv}_ {(B)}(G/U)}\operatorname{QCoh}(\check{T}).$
Our goal is to match commuting action of ${}_ {\chi}\mathscr{H}_ {\chi}$ and $\mathscr{H}$ on ${}_ {\chi}\mathscr{H}$ with commuting action of $\operatorname{QCoh}(\widetilde { \check{G} }/\check{G})$ and $\operatorname{QCoh}(\mathrm{St}/\check{G})$ on $\operatorname{QCoh}(\widetilde {\check{G}}/\check{G})$.
The universal monodromic Arkhipov-Bezrukavnikov equivalence
Construct a monoidal functor
Two approaches to fully faithfulness:
- AB equivalence localizes in nilpotent directions $\operatorname{Shv}_ {(I,\chi)}(\mathrm{FL})/\langle \operatorname{IC}_ {w}:\ell(w)>0 \rangle\cong \operatorname{QCoh}(\check{N}^{\text{reg}}/\check{B})$.
- localize in semi-simple directions: suffices to prove that for any $\mu,\lambda\in \mathbb{X}_ {\ast}(T)^{+}$, $$\operatorname{Hom}_ {\check{B}/\check{B}}(\mathscr{O}(-\mu),V_ {\lambda}\otimes \mathscr{O})\xrightarrow{\cong} \operatorname{Hom}(\Delta_ {-\mu}, Z_ {\lambda}).$$ Since restriction $\iota^{\ast}$ and $\text{gr}$ are faithful, we only need to check the images of $\iota^{\ast}$ and $\text{gr}$ coincide in $\operatorname{Hom}_ {\check{T}/\check{T}}(\mathscr{O}(-\mu), V_ {\lambda}\otimes\mathscr{O})$. Both are characterized by order vanishing condition along walls in $\check{T}$.
Example 3. Take $G=\operatorname{PGL}_ {2}$ and $\mathbb{X}_ {\ast}(T)=\mathbb{Z}$, $\mathscr{O}(\check{T})=\mathbb{k}[x^{\pm1}].$ Then $0\rightarrow \mathscr{O}(-1)\rightarrow V_ {1}\otimes\mathscr{O}\rightarrow \mathscr{O}(1)\rightarrow 0$ supported on diagonal corresponds to $0\rightarrow \Delta_ {-1}\rightarrow Z_ {1}\rightarrow \nabla_ {1}\rightarrow 0$ supported on $\mathrm{FL}_ {1}\cup \mathrm{FL}_ {-1}$. This extension is classified by $\operatorname{Ext}^{1}(\nabla_ {1}, \Delta_ {-1})\cong \mathbb{k}[x^{\pm 1}]/(x^{2}-1).$
$\operatorname{Hom}(\nabla_ {1},Z_ {1})\xrightarrow{\text{gr}} \operatorname{Hom}(\nabla_ {1},\nabla_ {1}) \rightarrow \operatorname{Ext}^{1}(\nabla_ {1},\Delta_ {-1}) =\mathbb{k}[x^{\pm 1}]/(x^{2}-1)$. Image of $\mathrm{gr}$ is the ideal generated by $(x^{2}-1)$.
Universal monodromic Bezrukavnikov equivalence
The commuting actions ${}_ {\chi}\mathscr{H}_ {\chi}\cong \operatorname{QCoh}(\widetilde {\check{G}}/\check{G})$ and $\mathscr{H}$ on ${}_ {\chi}\mathscr{H}\cong \operatorname{QCoh}(\widetilde {\check{G}}/\check{G})$ gives a functor $$\iota^{!}: \mathscr{H}\rightarrow \operatorname{End}_ {{}_ {\chi}\mathscr{H}_ {\chi}}({}_ {\chi}\mathscr{H}) \cong \operatorname{QCoh}(\mathrm{St}/\check{G}).$$
Claims:
- $\iota^{!}$ admits a fully faithful left adjoint $\iota_ {!}$,
- $\operatorname{Ker}(\iota^{!}) = \mathscr{H}^{\leq -\infty}$ (e.g. $\omega_ {\widetilde {\mathrm{FL}} }$).
- $\iota^{!}$ induces an equivalence $\mathscr{H}^{\omega}\xrightarrow{\cong}\operatorname{Coh}$.
Claim 1 follows by a general observation of [Ben-Zvi-Gunningham-Orem]. It can also be proved similar to [Beraldo-Lin-Reeves]. The modules ${}_ {\chi}\mathscr{H}$ and $\mathscr{H}_ {\chi}$ are dual to each other and unit and counit
- $u: {}_ {\chi}\mathscr{H}_ {\chi}\rightarrow {}_ {\chi}\mathscr{H}\otimes_ {\mathscr{H}}\mathscr{H}_ {\chi}$,
- $c: {}_ {\chi}\mathscr{H}\otimes_ {{}_ {\chi}\mathscr{H}_ {\chi}}\mathscr{H}_ {\chi} \rightarrow \mathscr{H}$ preserve compact objects. Therefore, one obtains $u^{R}: {}_ {\chi}\mathscr{H}\otimes_ {\mathscr{H}}\mathscr{H}_ {\chi}\rightarrow {}_ {\chi}\mathscr{H}_ {\chi}$ which is fully faithful.
Triangle identities and $u u^{R}\cong \operatorname{id}$ shows that $c^{R}c\cong \operatorname{Id}.$