Recall we had space $$ S^\mathrm{cts}(U_Q^\ell, L)_ \chi = S^\mathrm{cts}(U_Q^\ell, \mathcal{O})_ \chi[1/\ell], $$ which is a Banach representation of $\mathrm{GL}_ 2(F_\ell)$ and it is unitary admissible (because it is the dual of the quotient of something finite over the Iwasawa algebra). Let $k = (k_\sigma) \in \mathbb{Z}_ {\ge 1}^{\operatorname{Hom}(F, L)}$, and consider the representation $$ W_k = \bigotimes_{\sigma \in \operatorname{Hom}(F, L)} \operatorname{Sym}^{2k_\sigma-2}(L^2) \otimes \det^{1-k_\sigma}. $$ Because the center of $\mathrm{GL}_ 2$ acts trivially, this is an algebraic representation of $\mathrm{PGL}_ {2,L}$. We now consider the action of $\mathrm{GL}_ 2(F_\ell)$ where the action on the $\sigma$ parts is through $\sigma \colon F_\ell \to L$. We now have a map $$ \bigoplus_k \operatorname{Hom}_ {\mathrm{GL}_ 2(\mathcal{O}_ {F,\ell})}(W_k^\vee, S^\mathrm{cts}(U_Q^\ell, L)_ \chi) \otimes W_k^\vee \to S^\mathrm{cts}(U_Q^\ell, L)_ \chi, $$ which we claim has dense image.
Lemma 1. For $X$ a finite type affine scheme over $\mathbb{Z}_ \ell$, the ring $L[X]$ is dense in $C^\mathrm{cts}(X(\mathbb{Z}_ \ell), L)$.
In particular, this will tell us that $$ L[(\operatorname{Res}^F_ \mathbb{Q} \mathrm{PGL}_ {2,L})] = \bigoplus_k W_k \otimes W_k^\vee \subseteq C^\mathrm{cts}(\mathrm{PGL}_ 2(\mathcal{O}_ {F,\ell}), L) $$ is dense. We first check it for $X = \mathbb{A}^n$, and then note that $C^\mathrm{cts}(\mathbb{Z}_ \ell^n, L) \to C^\mathrm{cts}(X(\mathbb{Z}_ \ell), L)$ is surjective and continuous for $X \hookrightarrow \mathbb{A}^n$ a closed embedding.Proof.
We come back to our Hecke algebra $S^\mathrm{sm}(U_Q^\ell U_\ell, \lambda^{-n}/\mathcal{O})_ \chi$ with our usual Hecke operators $T_v$ for $v \notin R \cup Q \cup \lbrace v \mid \ell \rbrace$. Consider $$ \mathbb{T}(U_Q^\ell U_\ell, \mathcal{O}/\lambda^n)_ \chi \subseteq \operatorname{End}(S^\mathrm{sm}(U_Q^\ell U_\ell, \lambda^{-n}/\mathcal{O})_ \chi) $$ the subalgebra generated by $T_v$. For $n^\prime \gt n$, we have $$ S^\mathrm{sm}(U_Q^\ell U_\ell, \lambda^{-n}/\mathcal{O})_ \chi \hookrightarrow S^\mathrm{sm}(U_Q^\ell U_\ell, \lambda^{-n^\prime}/\mathcal{O})_ \chi, \quad \mathbb{T}(U_Q^\ell U_\ell, \mathcal{O}/\lambda^n)_ \chi \twoheadleftarrow \mathbb{T}(U_Q^\ell U_\ell, \mathcal{O}/\lambda^{n^\prime})_ \chi. $$ Similarly, for $U_\ell \supseteq U_\ell^\prime$ we get $$ S^\mathrm{sm}(U_Q^\ell U_\ell, \lambda^{-n}/\mathcal{O})_ \chi \hookrightarrow S^\mathrm{sm}(U_Q^\ell, U_\ell^\prime, \lambda^{-n}/\mathcal{O})_ \chi, \quad \mathbb{T}(U_Q^\ell U_\ell, \mathcal{O}/\lambda^n)_ \chi \twoheadleftarrow \mathbb{T}(U_Q^\ell U_\ell^\prime, \mathcal{O}/\lambda^n)_ \chi. $$ Taking the limit as $n$ grows and $U_\ell$ shrinks, we get $$ \mathbb{T}(U_Q^\ell, \mathcal{O})_ \chi = \varprojlim_{n,U_\ell} \mathbb{T}(U_Q^\ell U_\ell, \mathcal{O}/\lambda^n)_ \chi $$ a profinite algebra acting on $S^\mathrm{cts}(U_Q^\ell, \mathcal{O})_ \chi$ and $M(U_Q^\ell, \mathcal{O})_ \chi$ and commuting with the $\mathrm{GL}_ 2(F_\ell)$-actions.
Lemma 2. If $U_\ell$ is pro-$\ell$, then the map $$ \mathbb{T}(U_Q^\ell U_\ell^\prime, \mathcal{O}/\lambda^{n^\prime})_ \chi \twoheadrightarrow \mathbb{T}(U_Q^\ell U_\ell, \mathcal{O}/\lambda^n)_ \chi $$ is a bijection on prime ideals.
Proof.
Todo
Hence $\mathbb{T}(U_Q^\ell, \mathcal{O})_ \chi$ has finitely many maximal ideals and so we can decompose $$ \mathbb{T}(U_Q^\ell, \mathcal{O})_ \chi = \prod_\mathfrak{m} \mathbb{T}(U_Q^\ell, \mathcal{O})_ {\chi,\mathfrak{m}} $$ for each one a complete local ring.
Lemma 3. If $A_i$ are Artinian rings and $A_i \twoheadrightarrow A_{i-1}$ satisfies $\operatorname{Spec} A_i \cong \operatorname{Spec} A_i$ then $A_\infty = \varprojlim A_i$ has the same set of maximal ideals and $A_\infty = \prod_\mathfrak{m} A_{\infty,\mathfrak{m}}$.
The map from R to T
We now have a psuedorepresentation $$ T^\mathrm{mod} \colon G_{F, Q \cup R \cup \lbrace v \mid \ell \rbrace} \to \mathbb{T}(U_Q^\ell U_\ell, \bar{L})_ \chi $$ such that
- if $v$ is a good prime then $T^\mathrm{mod}(\mathrm{Frob}_ v) = T_v$,
- $\det T^\mathrm{mod} = \epsilon_\ell^{-1}$.
This means that by Chebotarev, this lands in $\mathbb{T}(U_Q^\ell U_\ell, \mathcal{O})$, and so we can reduce this modulo $\lambda$. These also are compatible as we change $U_\ell$. Hence taking the limit we get a pseudorepresentation $$ T^\mathrm{mod} \colon G_{F,Q \cup R \cup \lbrace v \mid \ell \rbrace} \to \mathbb{T}(U_Q^\ell, \mathcal{O}). $$ It follows that we have a map $$ R_{T^\mathrm{mod} \bmod{\mathfrak{m}}}^\mathrm{ps} \to \mathbb{T}(U_Q^\ell, \mathcal{O})_ \mathfrak{m}, \quad T^\mathrm{univ}(\mathrm{Frob}_ v) \mapsto T_v. $$
Remark 4. If $S \supseteq R \cup Q \cup \lbrace v \mid \ell \rbrace$ is any finite set of places of $F$ then $\mathbb{T}(U_Q^\ell, \mathcal{O})_ \chi$ is topologically generated by $T_v$ for $v \notin S$.
Modulo $\mathfrak{m}$ we have $T^\mathrm{mod} \bmod{\mathfrak{m}}$ that is $\tr \bar{r}_ \mathfrak{m}$, where $r_\mathfrak{m}$ is semisimple. If $\bar{r}_ \mathfrak{m}$ is absolutely irreducible, then from the local theory we have $$ r_\mathfrak{m} \colon G_F \to \mathrm{GL}_ 2(\mathbb{T}(U_Q^\ell, \mathcal{O})_ {\chi,\mathfrak{m}}). $$