For $H$ a connected linear algebraic group acting on $X$, we had this $\mathsf{Shv}(H)$ acting on $\mathsf{Shv}(X)$ as $(\mathscr{F}, \mathscr{G}) \mapsto m_\ast(\mathscr{F} \boxtimes \mathscr{G})$. On the other hand, we have inclusions $$ \mathsf{Mod}_ \Lambda \to \mathsf{Shv}_ {\chi\mathrm{-mon}}(H) \to \mathsf{Shv}_ \mathrm{mon}(H) \to \mathsf{Shv}(H) $$ that have right adjoints. We then defined $$ \begin{align} \mathsf{Shv}_ {H\mathrm{-mon}}(X) &= \mathsf{Shv}_ \mathrm{mon}(H) \otimes_{\mathsf{Shv}(H)} \mathsf{Shv}(X), \br \mathsf{Shv}((H,\chi) \backslash X) &= (\mathsf{Mod}_ \Lambda)_ \chi \otimes_{\mathsf{Shv}(H)} \mathsf{Shv}(X). \end{align} $$ Then we have functors $$ \mathsf{Shv}((H,\chi) \backslash X) \to \mathsf{Shv}_ {H\mathrm{-mon}}(X) \hookrightarrow \mathsf{Shv}(X) $$ where the forgetful functor has a right adjoint $\mathsf{Shv}_ {H\mathrm{-mon}}(X) \to \mathsf{Shv}((H,\chi) \backslash X)$.
Example 1. In the case of $\chi = u$, peuple call this $\mathsf{Shv}_ {H,u\mathrm{-mon}}(X)$ the full subcategory of unipotent monodromic sheaves. Given any $\mathscr{F} \in \mathsf{Shv}_ {H,u\mathrm{-mon}}(X)$ we have a canonical isomorphism $\mathrm{Ch}_ {u\mathrm{-mon}} \star \mathscr{F} \cong \mathscr{F}$, and so the ring $$ \End(\mathrm{Ch}_ {u\mathrm{-mon}}) \cong \Lambda[[x-1]] $$ acts on $\mathscr{F}$.
Example 2. When $H = \mathbb{G}_ a$ or more generally unipotent, fix a character $\phi \colon H(\mathbb{F}_ q) \to \Lambda^\times$. Then there is an equivalence $$ \mathsf{Shv}((H,\phi) \backslash X) \simeq \mathsf{Shv}_ {H,\phi\mathrm{-mon}}(X). $$
Proposition 3. For $\chi = u$ trivial, we have $\mathsf{Shv}((H,u) \backslash X) \cong \mathsf{Shv}(H \backslash X)$.
Proof.
The left hand side is by definition $$ (\mathsf{Mod} \Lambda)_ u \otimes_{\mathsf{Shv}_ \mathrm{mon}(H)} \mathsf{Shv}_ {H\mathrm{-mon}}(X). $$ But how do we compute this relative tensor product? Using the bar resolution, we see that this is given by the colimit of $$ \dotsb \mathsf{Shv}_ \mathrm{mon}(H) \otimes \mathsf{Shv}_ \mathrm{mon}(H) \otimes \mathsf{Shv}_ {H\mathrm{-mon}}(X) \rightrightarrows\to \mathsf{Shv}_ \mathrm{mon}(H) \otimes \mathsf{Shv}_ {H\mathrm{-mon}}(X) \rightrightarrows \mathsf{Shv}_ {H\mathrm{-mon}}(X). $$ Because $\ast$-pushforward is the same as $!$-pushforward for monodromic sheaves, these all admit continuous right adjoints, so we can compute this as the limit of $$ \dotsb \mathsf{Shv}_ \mathrm{mon}(H) \otimes \mathsf{Shv}_ \mathrm{mon}(H) \otimes \mathsf{Shv}_ {H\mathrm{-mon}}(X) \leftleftarrows\leftarrow \mathsf{Shv}_ \mathrm{mon}(H) \otimes \mathsf{Shv}_ {H\mathrm{-mon}}(X) \leftleftarrows \mathsf{Shv}_ {H\mathrm{-mon}}(X). $$ This fully faithfully embeds into the limit of $$ \dotsb \mathsf{Shv}(H \times H \times X) \leftleftarrows\leftarrow \mathsf{Shv}(H \times X) \leftleftarrows \mathsf{Shv}(X), $$ which is just $\mathsf{Shv}(H \backslash X)$ by descent. One can also check that this is essentially surjective.
Remark 4. The category $\mathsf{Shv}_ \mathrm{mon}(H)$ as a right $\mathsf{Shv}(H)$-module admits a dual given by $\mathsf{Shv}_ \mathrm{mon}(H)$ as left $\mathsf{Shv}(H)$-module. In particular, there are unit and counit functors $$ \mathsf{Mod}_ \Lambda \to \mathsf{Shv}_ \mathrm{mon}(H) \otimes_{\mathsf{Shv}(H)} \mathsf{Shv}_ \mathrm{mon}(H), \quad \mathsf{Shv}_ \mathrm{mon}(H) \otimes \mathsf{Shv}_ \mathrm{mon}(H) \to \mathsf{Shv}(H). $$ Informally we have $$ \mathsf{Shv}_ \mathrm{mon}(H) \cong \mathsf{Fun}^L(\mathsf{Shv}_ \mathrm{mon}(H), \mathsf{Mod}_ \Lambda) \cong \mathsf{Fun}^L_{\mathsf{RMod}_ {\mathsf{Shv}(H)}}(\mathsf{Shv}_ \mathrm{mon}(H), \mathsf{Shv}(H)) $$ compatible with the actions of $\mathsf{Shv}(H)$, so we have $$ \begin{align} \mathsf{Shv}_ {H\mathrm{-mon}}(X) &\cong \mathsf{Fun}^L_{\mathsf{Shv}(H)\mathrm{-mod}}(\mathsf{Shv}_ \mathrm{mon}(H), \mathsf{Shv}(X)), \br \mathsf{Shv}((H,\chi) \backslash X) &\cong \mathsf{Fun}^L_{\mathsf{Shv}(H)\mathrm{-mod}}((\mathsf{Mod}_ \Lambda)_ \chi, \mathsf{Shv}(X)). \end{align} $$ But we don’t need this because it is more useful to regard of these categories as coinvariants rather than invariants.
The affine Hecke category
Let us return to our setting of $(G, B, T)$ with $LG \supseteq I \supseteq I^u$ where $I/I^u \cong T$. Then we have the affine Hecke category $$ \mathsf{Shv}(I \backslash LG / I). $$ This is naturally monoidal for the convolution product $\star$, which is pullback-pushfoward along $$ (I \backslash LG / I) \times (I \backslash LG / I) \leftarrow I \backslash LG \times^I LG / I \xrightarrow{m} I \backslash LG / I. $$ Its unit is given by $(\Delta_1)_ \ast \delta$ where $\delta$ is the unit of $\mathsf{Shv}(I \backslash I / I)$ and $\Delta_1 \colon I \backslash I / I \to I \backslash LG / I$. Here, we note that $m_\ast = m_!$ because $m$ is ind-proper. This is the categorical analogue of the Iwahori–Hecke algebra $\mathcal{H}_ I = \End(\operatorname{cInd}_ I^{G(F)} \mathbb{C})$.
But we now want to look at the category $$ \mathsf{Shv}(I^u \backslash LG / I^u) $$ which would be the analogue of $\End(\operatorname{cInd}_ {I^u}^{G(F)} \mathbb{C})$. But not quite. We can still define the convolution product $\star^u$ by looking at $$ I^u \backslash LG / I^u \times I^u \backslash LG / I^u \leftarrow I^u \backslash LG \times^{I^u} LG / I^u \xrightarrow{m^u} I^u \backslash LG / I^u $$ and defining $$ \mathscr{F} \star^u \mathscr{G} = (m^u)_ \ast (\mathscr{F} \tilde{\boxtimes} \mathscr{G}). $$ But here, $m^u$ no longer is ind-proper, so $(m^u)_ \ast$ is genuinely different from $(m^u)_ !$.
Recall that when we were studying the Hecke algebra, we broke it up into a direct sum $$ \End(\operatorname{cInd}_ {I^u}^{G(F)}(\mathbb{C})) = \bigoplus_{\chi,\chi^\prime} {}_ \chi \mathcal{H}_ {\chi^\prime}. $$ This motivates the following definition
Definition 5. Note that there are two actions of $T$ on $I^u \backslash LG / I^u$ from both the left and the right. We define $$ \mathsf{Shv}_ \mathrm{mon}(I^u \backslash LG / I^u) \subseteq \mathsf{Shv}(I^u \backslash LG / I^u) $$ the full subcategory of $(T \times T)$-monodromic sheaves. For $\chi, \chi^\prime \colon \pi_1^\mathrm{c}(T) \to \Lambda^\times$, we similarly define $$ \mathsf{Shv}_ {\chi,\chi^\prime\mathrm{-mon}}(I^u \backslash LG / I^u) \subseteq \mathsf{Shv}(I^u \backslash LG / I^u). $$
Then if $\Lambda = \bar{\mathbb{F}}_ \ell$ or $\Lambda = \bar{\mathbb{Q}}_ \ell$ we have a direct sum decomposition $$ \mathsf{Shv}_ \mathrm{mon}(I^u \backslash LG / I^u) \cong \bigoplus_{\chi,\chi^\prime} \mathsf{Shv}_ {\chi,\chi^\prime\mathrm{-mon}}(I^u \backslash LG / I^u). $$ We can also check that monodromic sheaves are closed under $\star^u$, because in general if $X \to Y$ is equivariant for the $H$-actions, then pullback and pushforward preserve monodromic sheaves. But here we note that the unit changes. In particular, this is something like $$ \mathrm{Av}^\mathrm{mon}((\Delta_1^u)_ \ast \delta_{I^u \backslash I^u / I^u}) = (I^u \backslash I / I^u \xrightarrow{\Delta_1} I^u \backslash LG / I^u)_ \ast \mathrm{Ch}_ \mathrm{mon}. $$ Similarly as before, we have actions $$ \mathsf{Shv}_ {\chi,\chi\mathrm{-mon}}(I^u \backslash LG / I^u) \curvearrowright \mathsf{Shv}_ {\chi,\chi^\prime\mathrm{-mon}}(I^u \backslash LG / I^u) \curvearrowleft \mathsf{Shv}_ {\chi^\prime,\chi^\prime\mathrm{-mon}}(I^u \backslash LG / I^u). $$
Remark 6. We can also define equivariant categories as $$ \mathsf{Shv}((I,\chi) \backslash LG / (I,\chi^\prime)) = (\mathsf{Mod}_ \Lambda)_ \chi \otimes_{\mathsf{Shv}_ \mathrm{mon}(T)} \mathsf{Shv}_ \mathrm{mon}(I^u \backslash LG / I^u) \otimes_{\mathsf{Shv}_ \mathrm{mon}(T)} (\mathsf{Mod}_ \Lambda)_ {\chi^\prime}. $$ Then we can check that $$ \mathsf{Shv}((I,u) \backslash LG / (I,u)) \simeq \mathsf{Shv}(I \backslash LG / I) $$ as before.
Kac–Moody central extension
In some sense, we want to recover the whole category $\mathsf{Shv}_ {u,u\mathrm{-mon}}(I^u \backslash LG / I^u)$ from a $\End(\mathrm{Ch}_ {u\mathrm{-mon}}) \times \End(\mathrm{Ch}_ {u\mathrm{-mon}})$-bimodule. But it turns out that these two tori are a bit small.
For $G = \GL_n$, we have the affine Grassmannian $\mathrm{Gr}_ G = LG / L^+G$ that classifies lattices $\Lambda \subseteq k((\varpi))^n$. There is the determinant line bundle $\mathscr{L}_ \mathrm{det} / \mathrm{Gr}_ G$ that sends $\Lambda$ to $$ \det(k[[\varpi]]^n / (k[[\varpi]] \cap \Lambda)) \otimes \det(\Lambda / (k[[\varpi]] \cap \Lambda))^{-1}. $$ It turns out that the action of $LG$ on $\mathrm{Gr}_ G$ does not lift to an action of $\mathscr{L}_ \mathrm{det}$. The problem is that when we try to identify the two fibers, we run into the problem that $g \Lambda_0$ is not $\Lambda_0$. If we record this, we can product a central extension $$ 1 \to \mathbb{G}_ m \to \widehat{LG} \to LG \to 1 $$ where now $\widehat{LG}$ acts on $\mathscr{L}_ \mathrm{det}$ equivariantly.
Remark 7. When $G$ is simple and simply connected, all possible such central extensions can be classified, and are labeled by integers.