Tilting objects

• The homotopy category
• The Soelgel bimodule functor

Theorem 1. 1. For each $\chi$ and $w \in \tilde{W}$, there exists a unique (up to non-unique isomorphism) cofree $\chi$-monodromic tilting sheaf $T_{w,\chi}^\mathrm{mon}$ which is

• indecomposable,
• supported on $\operatorname{Supp}(T_{w,\chi}^\mathrm{mon}) \subseteq I^+ \backslash \widetilde{LG}_ {\le w} / I^+$ with $T_{w,\chi}^\mathrm{mon} \vert_{I^+ \backslash \widetilde{LG}_ w / I^+} \cong \mathrm{Ch}_ {\chi\mathrm{-mon}}$.

1. Let $\mathcal{H}_ \mathrm{mon}^\mathrm{tilt}$ be the full subcategory of cofree $\chi$-monodromic tilting objects. Then every object is a finite direct sum of $T_{w,\chi}^\mathrm{mon}$.
2. The category $\mathcal{H}_ \mathrm{mon}^\mathrm{tilt}$ is an additive monoidal subcategory of $\mathcal{H}_ \mathrm{mon}$.

The proof will be some complicated induction. When $\ell(w) = 0$, we will just have $$ T_{\dot{w},\chi}^\mathrm{mon} = \Delta_{\dot{w}}^\mathrm{mon}(\mathrm{Ch}_ {\chi\mathrm{-mon}}) = \nabla_{\dot{w}}^\mathrm{mon}(\mathrm{Ch}_ {\chi\mathrm{-mon}}). $$ This is indecomposable because $\mathrm{Ch}_ {\chi\mathrm{-mon}}$ has endomorphism ring that is some completed local ring of a dual torus, which doesn’t have idempotents. When $w = s$ is a simple reflection, we have the triangle $$ \Delta_{\dot{e}}^\mathrm{mon}(\tilde{\mathrm{Ch}}_ s) \to \Delta_{\dot{s}}^\mathrm{mon}(\tilde{\mathrm{Ch}}) \to \nabla_s^\mathrm{mon}(\tilde{\mathrm{Ch}}). $$ Here, if $\hat{\alpha}_ s \colon \hat{\tilde{T}} \to \mathbb{G}_ m$ is the coroot associated to $s$, then we have $$ \omega_{\ker(\hat{\alpha}_ s)} \in \operatorname{IndCoh}(\mathcal{R}_ {I_F^\mathrm{t}, \hat{\tilde{T}}}), \quad \tilde{\mathrm{Ch}}_ s = \mathrm{Ch}(\omega_{\ker(\hat{\alpha}_ s)}). $$ So we have some triangle $$ \omega_{\ker(\hat{\alpha}_ s)} \to \omega \xrightarrow{x-1} \omega \to 0. $$ We can now define $T_{\dot{s}}^\mathrm{mon}$ as the pushout $$ \begin{CD} \Delta_{\dot{e}}^\mathrm{mon}(\tilde{\mathrm{Ch}}_ s) @>>> \Delta_{\dot{s}}^\mathrm{mon}(\tilde{\mathrm{Ch}}) @>>> \nabla_s^\mathrm{mon}(\tilde{\mathrm{Ch}}) \br @VVV @VVV @| \br \Delta_e^\mathrm{mon}(\tilde{\mathrm{Ch}}) @>>> T_{\dot{s}}^\mathrm{mon} @>>> \nabla_s^\mathrm{mon}(\tilde{\mathrm{Ch}}) \br @VVV @VVV \br \Delta_e^\mathrm{mon}(\tilde{\mathrm{Ch}}) @>>> \Delta_e^\mathrm{mon}(\tilde{\mathrm{Ch}}). \end{CD} $$

Remark 2. When $s \notin \tilde{W}_ \chi^0$, we will still have $$ T_{\dot{s},\chi}^\mathrm{mon} \cong \nabla_{\dot{s}}^\mathrm{mon}(\mathrm{Ch}_ {\chi\mathrm{-mon}}) \cong \Delta_{\dot{s}}^\mathrm{mon}(\mathrm{Ch}_ {\chi\mathrm{-mon}}). $$

Lemma 3. If $\mathscr{F}$ admits a finite filtration by cofree (co)standard objects, so does $T_{\dot{s},\chi}^\mathrm{mon} \star^u \mathscr{F}$.

Corollary 4. If $w = s_{i_1} \dotsm s_{i_n} \omega$ with $\ell(\omega) = 0$ then $$ T’ = T_{s_{i_1},\chi}^\mathrm{mon} \star^u T_{s_{i_2}, s_{i_1}(\chi)}^\mathrm{mon} \star^u \dotsb \star^u T_{\omega, (s_{i_1} \dotsm s_{i_n})(\chi)}^\mathrm{mon} $$ is a cofree monodromic tilting sheaf supported on $I^+ \backslash \widetilde{LG}_ {\le w} / I^+$.

This has the property that $$ T^\prime \vert_{I^+ \backslash \widetilde{LG}_ w / I^+} \cong \mathrm{Ch}_ {\chi\mathrm{-mon}}. $$ Consider the set $S$ of all direct summands $T^{\prime\prime} \hookrightarrow T^\prime$ such that $T^{\prime\prime} \vert_{I^+ \backslash \widetilde{LG}_ w / I^+} \cong \mathrm{Ch}_ {\chi\mathrm{-mon}}$. If we choose a minimal such one $T^{\prime\prime} \in S$, then this is indecomposable. This will be our construction in general.

Lemma 5. Suppose we have $T_1, T_2$ cofree $\chi$-monodromic tilting sheaves supported on $I^+ \backslash \widetilde{LG}_ {\le w} / I^+$. Then $\Hom(T_1, T_2)$ admits a filtration with associated graded being $$ \Hom((i_{w^\prime})^\ast T_1^\prime, (i_{w^\prime})^! T_2^\prime), $$ where $w^\prime \le w$.

Here, we have that each term is a finite free module over $\mathcal{R}_ \chi = \End(\mathrm{Ch}_ {\chi\mathrm{-mon}})$.

Corollary 6. We have a surjection $\End(T) \twoheadrightarrow \End(T \vert_{I^+ \backslash \tilde{LG}_ w / I^+})$.

Now if $T$ is a cofree $\chi$-monodromic tilting sheaf supported on $I^+ \backslash \widetilde{LG}_ {\le w} / I^+$, we can find a map $$ T_{\dot{w},\chi}^\mathrm{mon} \vert_{I^+ \backslash \widetilde{LG}_ {\le w} / I^+} \to T \vert_{I^+ \backslash \widetilde{LG}_ {\le w} / I^+} \to T_{\dot{w},\chi}^\mathrm{mon} \vert_{I^+ \backslash \widetilde{LG}_ {\le w} / I^+} $$ that composes to the identity. Now using this surjectivity, we can lift this to $$ T_{\dot{w},\chi}^\mathrm{mon} \to T \to T_{\dot{w},\chi}^\mathrm{mon}. $$ Because the endomorphism ring $\mathcal{R}$ is a local ring, this composition is still an automorphism. This shows that this is a direct summand. Using this, we get both uniqueness of $T_{\dot{w},\chi}^\mathrm{mon}$ and also the direct sum decomposition of $\mathcal{H}_ \mathrm{mon}^\mathrm{tilt}$.

The homotopy category

We now consider the bounded homotopy category $K^b(\mathcal{H}_ \mathrm{mon}^\mathrm{tilt})$, viewed as a dg-category meaning the objects are literal bounded chain complexes and Hom sets are upgraded to chain complexes. If we take its homotopy category, we obtain a monoidal stable category, and then we have $\operatorname{Ind}(K^b(\mathcal{H}_ \mathrm{mon}^\mathrm{tilt}))$.

Lemma 7. There exists a natural fully faithful monoidal functor $$ G \colon K^b(\mathcal{H}_ \mathrm{mon}^\mathrm{tilt}) \to \mathcal{H}_ \mathrm{mon}. $$

Theorem 8. The functor $G$ admits a fully faithful left adjoint $F \colon \operatorname{Ind}(K^b(\mathcal{H}_ \mathrm{mon}^\mathrm{tilt})) \to \mathcal{H}_ \mathrm{mon}$.

We will discuss this next time.

The Soelgel bimodule functor

Theorem 9. There exists a natural action of $\mathcal{H}_ \mathrm{mon}$ on $\mathsf{Shv}_ \mathrm{mon}(\tilde{T}) \cong \operatorname{IndCoh}(\mathcal{R}_ {I_F^t, \hat{\tilde{T}}})$, or equivalently a functor $$ \mathbb{V} \colon \mathcal{H}_ \mathrm{mon} \to \operatorname{IndCoh}(\mathcal{R}_ {I_F^t,\hat{\tilde{T}}} \times \mathcal{R}_ {I_F^t,\hat{\tilde{T}}}), $$ such that when we restrict to $\mathcal{H}_ \mathrm{mon}^\mathrm{tilt}$ we get a fully faithful functor $$ {}_ \chi \mathcal{H}_ {\mathrm{mon},\chi}^\mathrm{tilt} \to \mathsf{IndCoh}(\mathcal{R}_ {I_F^t,\hat{\tilde{T}}} \times \mathcal{R}_ {I_F^t,\hat{\tilde{T}}}) \to \mathsf{Mod}(\mathcal{R}_ \chi \hat{\otimes} \mathcal{R}_ \chi). $$

This is our version of the Soelgel bimodule. Combining with this fully faithful functor $\mathcal{H}_ \mathrm{mon} \to \mathsf{Ind} K^b(\mathcal{H}_ \mathrm{mon}^\mathrm{tilt})$, we see that this functor $\mathbb{V}$ contains a lot of information.

Let us start constructing this action of $\mathcal{H}_ \mathrm{mon}$ on $\mathsf{Shv}_ \mathrm{mon}(\tilde{T})$. We consider the map $$ \mathrm{Bun}_ G(\mathbb{P}^1)_ {(0,I^+), (\infty,I_\infty^{++})} \to \mathrm{Bun}_ G(\mathbb{P}^1)_ {(0,I), (\infty,I_\infty^+)} $$ that is a $T \times I_\infty^+/I_\infty^{++}$-torsor. Fixing a representation $G \hookrightarrow \GL(V)$, there is a determinant line bundle $$ \mathscr{L}_ V / \mathrm{Bun}_ G(\mathbb{P}^1) $$ whose fiber at a point is $\det R\Gamma(C, \mathscr{V})$. We can then pull it back to $\mathrm{Bun}_ G(\mathbb{P}^1)$ with these level structures and in particular have $$ \mathscr{L}_ V^\times \to \mathrm{Bun}_ G(\mathbb{P}^1)_ {(0,I), (\infty,I_\infty^+)} $$ that is a $\tilde{T} \times I_\infty^+/I_\infty^{++}$-torsor. Once we choose an affine pinning and an additive character $\psi$ as usual, we may consider the monodromic category $$ \mathsf{Shv}((\tilde{T}, \chi\mathrm{-mon}) \backslash \widetilde{\mathrm{Bun}}_ G(\mathbb{P}^1)_ {(0,I^+),(\infty,I_\infty^{++})} / (I_\infty^+/I_\infty^{++}, \psi)) $$ Because $\mathrm{Bun}_ G(\mathbb{P}^1)$ is homeomorphic with $\tilde{W}$ with the opposite Bruhat order, we will be able to identify this with $\mathsf{Shv}_ \mathrm{mon}(\tilde{M})$.