Computation of the Soergel bimodule

• Computing the Soergel bimodule functor

Fix $\chi \in \mathcal{R}_ {I_F^t,\hat{\tilde{T}}}(\Lambda)$ (where $\Lambda$ is a field, but we can work integrally). We defined the full subcategory $$ \mathsf{SBim}_ \chi \subseteq \mathsf{IndCoh}(\hat{\chi}^2)^\heartsuit $$ that is the idempotent complete monoidal additive subcategory generated by the objects

• the dualizing sheaf $\omega_{\Gamma_{w^\beta}}$ of the graph of $w_\beta \in \tilde{W}_ \chi$ corresponding to $\beta \in \Omega_\chi$,
• the dualizing sheaf $\omega_{\hat{\chi} \times_{\hat{\chi} // W_r} \hat{\chi}}$ where $r$ is a simple reflection of $\tilde{W}_ \chi^0$ and $W_r = \lbrace 1, r \rbrace$.

Implicit in this definition is that the convolutions of these sheaves are still concentrated in degree $0$, but this will actually follow from the tilting property and the Soergel bimodule construction.

Theorem 1. We have an equivalence $$ \mathbb{V} \colon {}_ \chi \mathcal{H}_ {\mathrm{mon},\chi}^\mathrm{tilt} \cong \mathsf{SBim}_ \chi. $$

This will follow from the following two facts.

Proposition 2. We have $\mathbb{V}(T_{r,\chi}^\mathrm{mon}) = \omega_{\hat{\chi} \times_{\hat{\chi} // \lbrace 1, r \rbrace} \hat{\chi}}$ and $\mathbb{V}(T_{w^\beta,\chi}^\mathrm{mon}) = \omega_{\Gamma_{w^\beta}}$.

Proposition 3. The category ${}_ \chi \mathcal{H}_ {\mathrm{mon},\chi}^\mathrm{tilt}$ is generated by $T_{w^\beta,\chi}^\mathrm{mon}$ and $T_{r,\chi}^\mathrm{mon}$ as an idempotent complete monoidal additive category.

Computing the Soergel bimodule functor

Let us start with the computation of $\mathbb{V}$. We first translate the usual Bruhat order and length to $$ (\tilde{W}_ \chi^\beta = \tilde{W}_ \chi^0 w^b = w^b \tilde{W}_ \chi^0, \le_\beta, \ell_\beta). $$

Lemma 4. Let $w \in \tilde{W}_ \chi^\beta$. Then in the associated grading of the standard and costandard filtration of $T_{w,\chi}^\mathrm{mon}$, only $$ \lbrace \Delta_v^\mathrm{mon}(\mathrm{Ch}_ {\chi\mathrm{-mon}}) \rbrace_{v \le_\beta w}, \quad \lbrace \nabla_v^\mathrm{mon}(\mathrm{Ch}_ {\chi\mathrm{-mon}}) \rbrace_{v \le_\beta w} $$ appear.

Corollary 5. For $w = w^\beta$ minimal length in $\tilde{W}_ \chi^\beta$, we have $$ T_{w,\chi}^\mathrm{mon} = \Delta_w^\mathrm{mon}(\mathrm{Ch}_ {\chi\mathrm{-mon}}) = \nabla_w^\mathrm{mon}(\mathrm{Ch}_ {\chi\mathrm{-mon}}). $$

Remark 6. In general, we have $\ell(w^\beta) \neq 0$. This is why this is not by definition.

Let us first prove this corollary and then the lemma. To prove the corollary, we induct on $\ell(w)$. Here, let us assume that $w = w^\beta$. If $\ell(w) = 0$ then we are done. Otherwise we have $\ell(w) = \ell(v) + 1$ for some $w = vs$. This acturally forces $s \notin \tilde{W}_ \chi^0$ and $s \notin \tilde{W}_ {s\chi}^0$, and so we have $$ \Delta_s^\mathrm{mon}(\mathrm{Ch}_ {\chi\mathrm{-mon}}) = \nabla_s^\mathrm{mon}(\mathrm{Ch}_ {\chi\mathrm{-mon}}) $$ and therefore the functor $$ \Delta_s^\mathrm{mon}(\mathrm{Ch}_ {s\chi\mathrm{-mon}}) \star (-) \star \Delta_s^\mathrm{mon}(\mathrm{Ch}_ {\chi\mathrm{mon}}) \colon {}_ \chi \mathcal{H}_ {\mathrm{mon},\chi} \to {}_ {s\chi} \mathcal{H}_ {\mathrm{mon},s\chi} $$ sends standard objects to standard objects (if the lengths add, then we are find, if not, we can replace $\Delta_s$ by $\nabla_s$) and similarly costandard objects to costandard objects. It follows that it sends tilting objects to tilting objects. It is also an equivalence because this functor is invertible, and hence respects indecomposable objects on both sides. By looking at the support, we conclude that it sends $T_{w,\chi}^\mathrm{mon}$ to $T_{sws,\chi}^\mathrm{mon}$.

Remark 7. At this point, we want to say that $\ell(sws) \le \ell(w)$, but maybe this is not true. Instead, we need to prove a more general statement that if $\beta \in {}_ \chi \Omega_{\chi^\prime}$ and $w = w^\beta \in {}_ \chi \tilde{W}_ {\chi^\prime}$ corresponds to it, then $$ T_{w,\chi}^\mathrm{mon} \cong \Delta_w^\mathrm{mon}(\mathrm{Ch}_ {\chi\mathrm{-mon}}) \cong \nabla_w^\mathrm{mon}(\mathrm{Ch}_ {\chi\mathrm{-mon}}). $$ The proof here works, because we don’t have to conjugate by $s$ but only convolve $\Delta_s^\mathrm{mon}(\mathrm{Ch}_ {\chi\mathrm{-mon}})$ on one side.

Let us move on to the proof of the lemma. Given $w \in \tilde{W}_ \chi^\beta$, if $\ell_\beta(w) = 0$ this is just the corollary. Otherwise we can write $w = vr$ where $r$ is a simple reflection in $\tilde{W}_ \chi^0$.

Lemma 8. If $r \in \tilde{W}_ \chi^0$ is a simple reflection, then we have short exact sequences $$ \begin{align} 0 &\to \nabla_e^\mathrm{mon}(\mathrm{Ch}_ {\chi\mathrm{-mon}}) \to T_{r,\chi}^\mathrm{mon} \to \nabla_r^\mathrm{mon}(\mathrm{Ch}_ {\chi\mathrm{-mon}}) \to 0, \br 0 &\to \Delta_r^\mathrm{mon}(\mathrm{Ch}_ {\chi\mathrm{-mon}}) \to T_{r,\chi}^\mathrm{mon} \to \Delta_e^\mathrm{mon}(\mathrm{Ch}_ {\chi\mathrm{-mon}}) \to 0. \end{align} $$

Using this, we can now induct on the $\beta$-length of $w$ and obtain the desired support bound.

We can also compute $\mathbb{V}(T_{r,\chi}^\mathrm{mon})$ now. We have two exact sequences $$ 0 \to \omega_{\Gamma_e} \to \mathbb{V}(T_{r,\chi}^\mathrm{mon}) \to \omega_{\Gamma_r} \to 0, \quad 0 \to \omega_{\Gamma_r} \to \mathbb{V}(T_{r,\chi}^\mathrm{mon}) \to \omega_{\Gamma_e} \to 0. $$ The composition $$ \omega_{\Gamma_r} \to \mathbb{V}(T_{r,\chi}^\mathrm{mon}) \to \omega_{\Gamma_r} $$ is the identity. The other composition is not the identity. Recall that for $r = s$ a simple reflection in $\tilde{W}$ we deduced from an $\mathrm{SL}_ 2$-combination that $$ \begin{CD} 0 @>>> \Delta_e(\tilde{\mathrm{Ch}}_ s) @>>> \Delta_s^\mathrm{mon}(\tilde{\mathrm{Ch}}) @>>> \nabla_s^\mathrm{mon}(\tilde{\mathrm{Ch}}) @>>> 0 \br @. @VVV @VVV @| \br 0 @>>> \Delta_e(\tilde{\mathrm{Ch}}) @>>> \tilde{T}_ s^\mathrm{mon} @>>> \nabla_s^\mathrm{mon}(\tilde{\mathrm{Ch}}) @>>> 0 \br @. @VV{x-1}V @VVV \br @. \Delta_e(\tilde{\mathrm{Ch}}) @= \Delta_e(\tilde{\mathrm{Ch}}). \end{CD} $$ This will still be true for general $r$.

Now this $\mathbb{V}(T_{r,\chi}^\mathrm{mon})$ is a $R_\chi \hat{\otimes} R_\chi$-module, and we have a map $$ \mathbb{V}(T_{r,\chi}^\mathrm{mon}) \to R_\chi(e) \oplus R_\chi(s) $$ as bimodules. Now if we have such an extension, it forces $\mathbb{V}(T_{r,\chi}^\mathrm{mon})$ to be the union of $R_\chi(e)$ and $R_\chi(s)$. So we get what we want.