We had two facts we needed. Recall we have a section of $\tilde{W}_ \chi \to \Omega_\chi$, which we denote by $\beta \mapsto w^\beta$.
Proposition 1. The category ${}_ \chi \mathcal{H}_ {\mathrm{mon},\chi}^\mathrm{tilt}$ is generated as an idempotent-complete monoidal additive category by the objects $T_{w^\beta,\chi}^\mathrm{mon}$ and $T_{r,\chi}^\mathrm{mon}$ where $\beta \in \Omega_\chi$ and $r \in \tilde{W}_ \chi^0$ is a simple reflection.
Lemma 2. For every $w \in {}_ \chi \tilde{W}_ {\chi^\prime}$, the object $T_{w,\chi}^\mathrm{mon} \in {}_ \chi \mathcal{H}_ {\mathrm{mon},\chi^\prime}^\mathrm{tilt}$ has both a standard filtration and a costandard filtration. If $\Delta_v^\mathrm{mon}(\mathrm{Ch}_ \chi)$ appears in the standard filtration of $T_{w,\chi}^\mathrm{mon}$ then $v \le_\beta w$, and similarly for the costandard filtration.
Here, recall that we have $$ {}_ \chi \tilde{W}_ {\chi^\prime}^\beta = \tilde{W}_ \chi^0 w^\beta = w^\beta {}_ {\chi^\prime} \tilde{W}_ {\chi^\prime}^0 $$ and then this Bruhat order $\le_\beta$ is the Bruhat order on either $\tilde{W}_ {\chi^\prime}^0$ or $\tilde{W}_ \chi^0$.
The techniques for proving these statements are the same. Let’s look at the proof of the second fact. We induct on $\ell(w)$. If $\ell(w) = 0$ then this is clear. Otherwise we can write $w = us$ where $\ell(w) = \ell(u) + 1$ and $s \in \tilde{W}$ is a simple reflection. If $s \notin {}_ {\chi^\prime} \tilde{W}_ {\chi^\prime}^0$ then $s$ has $\beta$-length zero, which means that $s = w^\beta$ for some $\beta \in {}_ {s\chi^\prime} \Omega_{\chi^\prime}$. Then $$ \Delta_s^\mathrm{mon}(\mathrm{Ch}_ {s\chi^\prime\mathrm{-mon}}) = \nabla_s^\mathrm{mon}(\mathrm{Ch}_ {s\chi^\prime\mathrm{-mon}}) $$ and convolution by this is an equivalence. Then we see that $$ T_{w,\chi}^\mathrm{mon} = T_{u,s\chi^\prime}^\mathrm{mon} \star \Delta_s^\mathrm{mon}(\mathrm{Ch}_ {s\chi^\prime\mathrm{-mon}}). $$ Now we have the inductive hypothesis on $u$ and use the fact that $$ {}_ \chi \tilde{W}_ {s\chi^\prime}^{\beta\gamma} s = {}_ \chi \tilde{W}_ {\chi^\prime}^\beta $$ preserving partial orders.
There is also the case when $s \in {}_ {\chi^\prime} \tilde{W}_ {\chi^\prime}^0$. This means that $s$ is a simple reflection in the group ${}_ {\chi^\prime} \tilde{W}_ {\chi^\prime}^0$. But by construction, $T_{w,\chi}^\mathrm{mon}$ appears as a direct summand of $$ T_{u,s\chi^\prime}^\mathrm{mon} \star T_{s,\chi^\prime}^\mathrm{mon}. $$ Again, we can use induction on $u$ and we also know the support of $T_{s,\chi^\prime}^\mathrm{mon}$ to be $\lbrace 1, s \rbrace$.
We now prove the first fact. Here we really have to work with $\chi = \chi^\prime$ because otherwise we don’t have a monoidal category. It is enough to show that if $w = ur$ in ${}_ \chi \tilde{W}_ \chi^\beta$ with $\ell_\beta(w) = \ell_\beta(u) + 1$, then we need to show that $T_{w,\chi}^\mathrm{mon}$ appears as a direct summand of $T_{u,\chi}^\mathrm{mon} \star T_{r,\chi}^\mathrm{mon}$. Here the problem is that we don’t necessarily have $\ell(w) = \ell(u) + \ell(r)$. But at least if $r = s$ is a simple reflection in $\tilde{W}$ then this equality is true and hence this is fine.
In the general case, we can write $r = sts$ for $s$ a simple reflection in $\tilde{W}$ and $t$ a simple reflection in ${}_ {s\chi} \tilde{W}_ {s\chi}^0$ so that $\ell(r) = \ell(t) + 2$. Then $s \notin {}_ \chi \tilde{W}_ \chi^0$ and therefore $\Delta_s^\mathrm{mon}(\mathrm{Ch}_ {s\chi\mathrm{-mon}}) \star (-) \star \Delta_s^\mathrm{mon}(\mathrm{Ch}_ {\chi\mathrm{-mon}})$ is an equivalence from ${}_ {s\chi} \mathcal{H}_ {s\chi}^{\mathrm{tilt},\beta} \to {}_ \chi \mathcal{H}_ \chi^{\mathrm{tilt},\gamma\beta\gamma}$, where $\gamma \in {}_ \chi \Omega_{s\chi}$ corresponds to $s$. Then we an do induction on the length.
Theorem 3. The Soergel bimodule functor defines an equivalence $$ \mathbb{V} \colon {}_ \chi \mathcal{H}_ {\mathrm{mon},\chi}^\mathrm{tilt} \xrightarrow{\cong} \mathsf{SBim}_ \chi \subseteq \mathsf{IndCoh}(\hat{\chi}^2)^\heartsuit, $$ where $\mathsf{SBim}_ \chi$ is the idempotent-complete additive monoidal subcategory generated by $\omega_{\Gamma_{w^\beta}}$ and $\omega_{\hat{\chi} \times_{\hat{\chi} // \lbrace 1, r \rbrace} \hat{\chi}}$.
Theorem 4. There exists a fully faithful monoidal functor $$ G_0 \colon K^b({}_ \chi \mathcal{H}_ {\mathrm{mon},\chi}^\mathrm{tilt}) \hookrightarrow {}_ \chi \mathcal{H}_ {\mathrm{mon},\chi} $$ that induces a functor $$ G \colon \mathsf{Ind}(K^b({}_ \chi \mathcal{H}_ {\mathrm{mon},\chi}^\mathrm{tilt})) \to {}_ \chi \mathcal{H}_ {\mathrm{mon},\chi}. $$ This moreover admits a fully faithful left adjoint $F$.
For each $w \in {}_ \chi \tilde{W}_ \chi$, we let $K^b({}_ \chi \mathcal{H}_ {\mathrm{mon},\chi}^\mathrm{tilt})_ {\le w}$ be the full idempotent-complete subcategory generated by $T_{v,\chi}^\mathrm{mon}$ for $v \le_\beta w$, and define similarly $K^b({}_ \chi \mathcal{H}_ {\mathrm{mon},\chi})_ {\lt w}$. Then there is a fully faithful embedding $$ \iota_{\lt w} \colon K^b({}_ \chi \mathcal{H}_ {\mathrm{mon},\chi}^\mathrm{tilt})_ {\lt w} \hookrightarrow K^b({}_ \chi \mathcal{H}_ {\mathrm{mon},\chi}^\mathrm{tilt}) $$ with a left adjoint $(\iota_{\lt w})^\ast$ and right adjoint $(\iota_{\lt w})^!$. Then we have an exact triangle $$ (\iota_{\lt w})_ \ast (\iota_{\lt w})^! T_{w,\chi}^\mathrm{mon} \to T_{w,\chi}^\mathrm{mon} \to \nabla_{w,\chi}^\mathrm{mon}. $$ This shows that the cofree (co)standard objects are in the essential image of $G_0$.
Lemma 5. The subcategory $({}_ \chi \mathcal{H}_ {\mathrm{mon},\chi})^\omega$ of compact objects is contained in the essential image of $G_0$.
To see this, we note that the objects $\Delta_w^\mathrm{mon}(\mathscr{L})$ for $\mathscr{L} \in \mathrm{Ch}(\tilde{T})$ form a set of compact generators. Now we note that there is a finite Koszul resolution $$ 0 \to \mathrm{Ch}_ \chi \to \tilde{\mathrm{Ch}}_ {\chi\mathrm{-mon}} \to \dotsb $$ corresponding to $$ 0 \to \Lambda \to \Lambda[[x_1-1, \dotsc, x_n-1]] \to \bigoplus_{i=1}^n \Lambda[[x_1-1, \dotsc, x_n-1]] \to \dotsb. $$ Because each $\Delta_w^\mathrm{mon}(\tilde{\mathrm{Ch}}_ {\chi\mathrm{-mon}})$ is in the essential image of $G_0$, so is $\Delta_w^\mathrm{mon}(\mathrm{Ch}_ \chi)$.
Now we can construct $F$ and $G$. We have $$ {}_ \chi \mathcal{H}_ {\mathrm{mon},\chi}^\omega \hookrightarrow K^b({}_ \chi \mathcal{H}_ {\mathrm{mon},\chi}^\mathrm{tilt}) \hookrightarrow {}_ \chi \mathcal{H}_ {\mathrm{mon},\chi}. $$ From this we can now define $$ {}_ \chi \mathcal{H}_ {\mathrm{mon},\chi} \xrightarrow{F} \mathrm{Ind}(K^b({}_ \chi \mathcal{H}_ {\mathrm{mon},\chi}^\mathrm{tilt})) \xrightarrow{G} {}_ \chi \mathcal{H}_ {\mathrm{mon},\chi} $$ that composes to the identity.
Applications
Let $G/F = k((\varpi))$ be a split connected reductive group with $k = \bar{k}$ and $Z_G$ connected, and let $\chi$ be a character sheaf on $T$. We consider $\tilde{T} = T \times \mathbb{G}_ m$ and a character $\tilde{\chi} = (\chi, u)$. We have this Hecke category $$ {}_ {\tilde{\chi}} \mathcal{H}_ {\mathrm{mon},\tilde{\chi}}, $$ which we can actually recover the combinatorial data of $\tilde{W}$ acting on $\hat{\tilde{T}}$. This is because we can describe $\mathsf{SBim}_ \chi$ combinatorially, which gives a combinatorial description of ${}_ \chi \mathcal{H}_ {\mathrm{mon},\chi}^\mathrm{tilt}$.
Given this $\chi \in \hat{T}$, we have $\hat{H} = Z_{\hat{G}}(\chi)$ that is connected because $Z_G$ is connected, and have a corresponding endoscopic group $H$ for $(G, \chi)$. Then we have $\tilde{W}_ \chi = \tilde{W}_ H$ and there is a corresponding category $$ {}_ {\tilde{\chi}} \mathcal{H}_ {H,\mathrm{mon},\tilde{\chi}}. $$
Theorem 6. There is a perverse t-exact monoidal equivalence of categories $$ {}_ {\tilde{\chi}} \mathcal{H}_ {G,\mathrm{mon},\tilde{\chi}} \cong {}_ {\tilde{\chi}} \mathcal{H}_ {H,\mathrm{mon},\tilde{\chi}} $$ sending (co)standard objects to (co)standard objects.
Remark 7. Of course, this depends on some choices, e.g., the affine generic characters.
On the other hand, $H$ has a larger center, so we can untwist this part to get an equivalence $$ {}_ {\tilde{\chi}} \mathcal{H}_ {H,\mathrm{mon},\tilde{\chi}} \cong {}_ u \mathcal{H}_ {H,\mathrm{mon},u} $$ This still has a central monodromy, but if we apply $(-) \otimes{\mathsf{Shv}_ {u\mathrm{-mon}}} \mathsf{Mod}_ u$, we get an equivalence $$ \mathsf{Shv}_ \mathrm{mon}((I^+, \chi) \backslash LG / (I^+,\chi)) \cong \mathsf{Shv}_ \mathrm{mon}((I^+, u) \backslash LH / (I^+,u)). $$ By the Bezrukavnikov equivalence, this is going to be equivalent to $$ \mathsf{IndCoh}((B_\hat{H}/B_\hat{H})_ \hat{u} \times_{\hat{H}/\hat{H}} (B_\hat{H}/B_\hat{H})_ \hat{u}). $$ On the other hand, this stack literally is the same stack as $$ (B_\hat{G} / B_\hat{G})_ \hat{\chi} \times_{\hat{G}/\hat{G}} (B_\hat{G} / B_\hat{G})_ \hat{\chi}. $$
Theorem 8. There is an equivalence of categories $$ \mathsf{Shv}_ \mathrm{mon}((I^+,\chi) \backslash LG / (I^+,\chi)) \cong \mathsf{IndCoh}((B_\hat{G}/B_\hat{G})_ \hat{\chi} \times_{\hat{G}/\hat{G}} (B_\hat{G}/B_\hat{G})_ \hat{\chi}). $$
Another application is on quantum Langlands. Let us work with $k = \Lambda = \mathbb{C}$. If we use the character $(u, \kappa)$ on $\tilde{T} = T \times \mathbb{G}_ m$, we get a category $$ \mathsf{Shv}_ \kappa(I^+ \backslash \tilde{LG} / I^+) $$ that decategorifies to the Hecke algebra for the metaplectic group. Again, we will have $\tilde{W}_ {G,\kappa} \cong \tilde{W}_ {G^\vee,\kappa^\vee}$ acting on $(\hat{\tilde{T}}, \kappa) \cong (\hat{\tilde{\hat{T}}}, \hat{\kappa})$.
Theorem 9. There is an equivalence $$ \mathsf{Shv}_ \kappa(I^+ \backslash \tilde{LG} / I^+) \cong \mathsf{Shv}_ {\kappa^\vee}(I^+ \backslash LG^\vee / I^+). $$