Let $H$ be a torus and let $\hat{H}$ be the dual torus over $\Lambda$. Let $F = k((\varpi))$.
Proposition 1. We have a functor $$ \mathrm{Ch} \colon \mathsf{IndCoh}(\mathcal{R}_ {I_F^t}, \hat{H}) \cong \mathsf{Shv}_ \mathrm{mon}(H), $$ which is moreover t-exact, monoidal, functorial in $H$ in the sense that $f^\ast$ corresponds to $\hat{f}_ \ast^\mathrm{IndCoh}$ and $f_\ast^\mathrm{mon}$ corresponds to $\hat{f}^{\mathrm{IndCoh}!}$.
Note that after fixing a topological generator of $I_F^t$, we have $\mathsf{IndCoh}(\mathcal{R}_ {I_F^t,\hat{H}}) \hookrightarrow \mathsf{QCoh}(\hat{H})$. Note that we have an equivalence $$ \mathsf{Coh}(\mathcal{R}_ {I_F^t,\hat{H}})^\heartsuit \cong \mathsf{Shv}_ \mathrm{mon}(H)^{\omega,\heartsuit}; \quad \mathscr{O}_ \chi \mapsto \mathrm{Ch}(\mathscr{O}_ \chi) = \mathrm{Ch}_ \chi. $$ Now the point is that by the $K(\pi,1)$-property of $H$, we have isomorphisms $$ \begin{align} R\Hom_\mathsf{Coh}(\mathscr{F}_ 1, \mathscr{F}_ 2) &\cong R\Hom_{\pi_1^\mathrm{c}(H)\mathrm{-mod}}(\mathscr{F}_ 1, \mathscr{F}_ 2) \cong R\Hom_{\pi_1^\mathrm{et}(H)}(\mathscr{F}_ 1, \mathscr{F}_ 2) \br &\cong R\Hom_{\mathsf{Shv}(H)}(\mathrm{Ch}(\mathscr{F}_ 1), \mathrm{Ch}(\mathscr{F}_ 2)). \end{align} $$Proof.
Now we want to understand the dualizing sheaf. Under this equivalence, $\omega_{\mathcal{R}_ {I_F^t,\hat{H}}}$ corresponds to $\tilde{\mathrm{Ch}}$. So we have $$ \End_{\mathsf{IndCoh}(\mathcal{R}_ {I_F^t,\hat{H}})} \omega \cong \End_{\mathsf{Shv}_ \mathrm{mon}}(\tilde{\mathrm{Ch}}) = \Hom(\tilde{\mathrm{Ch}}, \delta_1) = ((\lbrace 1 \rbrace \to H)^\ast \tilde{\mathrm{Ch}}). $$
Lemma 2. Let $\mathfrak{X}$ be an ind-scheme. Then we have $\End(\omega_\mathfrak{X}) \cong R\Gamma(\mathfrak{X}, \mathscr{O}_ \mathfrak{X})$.
Proof.
Let us write $\mathfrak{X} = \varinjlim X_i$. Then we compute $$ \begin{align} R\Hom(\omega_\mathfrak{X}, \omega_\mathfrak{X}) &= R\Hom(\varinjlim_i \omega_{X_i}, \omega_\mathfrak{X}) = \varprojlim_i R\Hom(\omega_{X_i}, \omega_\mathfrak{X}) = \varprojlim_i R\Hom(\omega_{X_i}, \omega_{X_i}) \br &= \varprojlim_i R\Hom(\mathscr{O}_ {X_i}, \mathscr{O}_ {X_i}) = \varprojlim_i R\Gamma(X_i, \mathscr{O}_ {X_i}) = R\Gamma(\mathfrak{X}, \mathscr{O}_ \mathfrak{X}). \end{align} $$
Let us look at the case of $\Lambda = \mathbb{Z}_ \ell$. Here, the ind-scheme $\mathcal{R}_ {I_F^t, \hat{H}}$ decomposes in terms of $\bar{\chi} \colon \pi_1^\mathrm{c}(H) \to E^\times$ for $E/\mathbb{F}_ \ell$ finite. Then we have $$ \omega_{\mathcal{R}_ {I_F^t,\hat{H}}} = \bigoplus_{\bar{\chi}} \omega_{\mathcal{R}_ {I_F^t,\hat{H}}, \bar{\chi}}, $$ and so we have $$ \End(\omega_{\mathcal{R}_ {I_F^t,\hat{H}}}) = \prod_{\bar{\chi}} \End(\omega_{\mathcal{R}_ {I_F^t,\hat{H}}, \bar{\chi}}). $$ For the trivial character $\bar{u}$, we can compute that $$ \End(\omega_{\mathcal{R}_ {I_F^t,\hat{H}}}) = \mathbb{Z}_ \ell[[x_1-1, \dotsc, x_n-1]], $$ because we have $$ \varprojlim \mathrm{Fun}(X_i) = \varprojlim_{d,i} \mathrm{Fun}(X_i)/\ell^d, $$ and so we are seeing the entire formal completion of $1$ in $\hat{H}$.
In general, we consider $$ R_\bar{\chi} = \mathbb{Z}_ {\ell^d}[[x_1 - \bar{\chi}(x_1), \dotsc, x_n - \bar{\chi}(x_n)]] $$ the formal completion of $\bar{\chi} \in \hat{H}$ and also write $R = R_\bar{u}$. Then there is a functor $$ \mathsf{IndCoh}(\mathcal{R}_ {I_F^t,\bar{H}})_ \bar{\chi} \to \mathsf{Mod}(R_\bar{\chi}); \quad \mathscr{F} \mapsto R\Hom(\omega_{\mathcal{R}_ {I_F^t,\hat{H}}, \bar{\chi}}, \mathscr{F}). $$
Moreover, we have a monoidal functor $$ \mathsf{IndCoh}(\mathcal{R}_ {I_F^t,\hat{H}})_ \bar{u} \otimes \mathsf{IndCoh}(\mathcal{R}_ {I_F^t,\hat{H}})_ \bar{u} = \mathsf{IndCoh}(\mathcal{R}_ {I_F^t,\hat{H} \times \hat{H}})_ \bar{u} \to \mathsf{Mod}(R \hat{\otimes} R). $$
Definition 3. Let $\mathcal{A}$ be a monoidal category acting on $\mathsf{Shv}_ {\bar{u}\mathrm{-mon}}(H)$, we get a monoidal functor $\mathcal{A} \to \mathsf{IndCoh}((\mathcal{R}_ {I_F^t,\hat{H}})_ \bar{u})$, which induces a monoidal functor $$ \mathbb{V} \colon \mathcal{A} \to \mathsf{Mod}(R \hat{\otimes} R) $$ where the monoidal structure on the right hand side is $(-) \otimes_R (-)$. This is called the Soergel bimodule.
Monodromic and equivariant categories
Let us now go to the setting where $H$ is a connected algebraic group acting on a scheme $X$. Then we have an action of $\mathsf{Shv}(H)$ on $\mathsf{Shv}(X)$ where the action is given by sending $\mathscr{F} \in \mathsf{Shv}(H)$ and $\mathscr{G} \in \mathsf{Shv}(X)$ to $a_\ast(\mathscr{F} \boxtimes \mathscr{G})$.
Definition 4. We define $$ \mathsf{Shv}_ {H,\chi\mathrm{-mon}}(X) = \mathsf{Shv}_ {\chi\mathrm{-mon}}(H) \otimes_{\mathsf{Shv}(H)} \mathsf{Shv}(X), $$ which is a $\mathsf{Shv}_ {\chi\mathrm{-mon}}(H)$-module. We also define $$ \mathsf{Shv}((H,\chi) \backslash X) = (\mathsf{Mod}_ \Lambda)_ \chi \otimes_{\mathsf{Shv}(H)} \mathsf{Shv}(X). $$
Here are some remarks.
- The functor $\mathsf{Shv}_ {H(,\chi)\mathrm{-mon}}(X) \to \mathsf{Shv}(X)$ is a full subcategory.
- The equivariant category $\mathsf{Shv}((H, \chi) \backslash X)$ is not a subcategory of $\mathsf{Shv}(X)$ in general. But for $H = \mathbb{G}_ a$ (or more generally unipotent) we do have an equivalence $\mathsf{Shv}((H,\chi) \backslash X) \cong \mathsf{Shv}_ {H,\chi\mathrm{-mon}}(X)$.
- For $H = \mathbb{G}_ m$, there is an action of $\mathsf{Shv}_ {u\mathrm{-mon}}(\mathbb{G}_ m)$ on $\mathsf{Shv}_ {H,u\mathrm{-mon}}(X)$. Here, $\mathrm{Ch}_ {u\mathrm{-mon}}$ acts on every $\mathscr{F}$ trivially, so $\Lambda[[x-1]] = \End(\mathrm{Ch}_ {u\mathrm{-mon}})$ automatically acts on $\mathscr{F}$.
Proposition 5. For $\chi = u$ trivial, we have an equivalence $$ \mathsf{Shv}((H,u) \backslash X) \simeq \mathsf{Shv}(H \backslash X). $$