We had this space $M(U_Q^\ell, \mathcal{O})_ {\chi,\mathfrak{m}}$. This is admissible, so is in $\mathcal{C}_ {\mathrm{GL}_ 2(F_\ell, 1}^\mathrm{adm}(\mathcal{O})$. We stated the following theorem last time.
Theorem 1 (Pan). 1. This $M(U_Q^\ell, \mathcal{O})_ {\chi,\mathfrak{m}}$ is contained in the block $\mathcal{C}_ {\mathrm{GL}_ 2(F_\ell), 1}^\mathrm{adm}(\mathcal{O})_ {\mathcal{B}_ \mathfrak{m}}$ where $\mathcal{B}_ \mathfrak{m} = \prod_{v \mid \ell} B_{\bar{r}_ \mathfrak{m} \vert_{G_{F_v}}^\mathrm{ss}}$ is the block that Paskunas associates.
- Then we can construct $\operatorname{Hom}(P_\mathfrak{m}, M(U_Q^\ell, \mathcal{O})_ {\chi,\mathfrak{m}})$ with an action of both $E_{B_\mathfrak{m}}$ and $\mathbb{T}(U_Q^\ell, \mathcal{O})_ {\chi,\mathfrak{m}}$. Then the two induced actions $R_{B_\mathfrak{m}}^\mathrm{ps}$ agree.
It suffices to prove that for all $v \mid \ell$ and all open compact $U_\ell^v \subseteq \mathrm{GL}_ 2(F_\ell^v)$ that
- $M(U_Q^\ell U_\ell^v, \mathcal{O})_ {\chi,\mathfrak{m}}$ is in the block for $\mathcal{B}_ {\bar{r}_ \mathfrak{m} \vert_{G_{F_v}}^\mathrm{ss}}$,
- the two actions of $R_{\operatorname{tr} \bar{r}_ \mathfrak{m} \vert_{G_{F_v}}, \epsilon_\ell^{-1}}$ agree.
To see this, we recall that we had a map $$ \bigoplus_k \operatorname{Hom}_ {\mathrm{GL}_ 2(\mathcal{O}_ {F,v})}(W_k^\vee, S^\mathrm{cts}(U_Q^\ell U_\ell^v, \mathcal{O})_ {\chi,\mathfrak{m}}) \otimes W_k^\vee \to S^\mathrm{cts}(U_Q^\ell U_\ell^v, \mathcal{O})_ {\chi,\mathfrak{m}} $$ that has dense image. Now for $U_v \subseteq \mathrm{GL}_ 2(F_v)$ some open compact subgroup, we have $\mathbb{T}(U_Q^\ell U_\ell^v, \mathcal{O})_ \chi[1/\ell]$ acting on $$ \Hom_{U_v}(W_k^v, S^\mathrm{cts}(U_Q^\ell U_\ell^v, L)_ \chi) = \biggl\lbrace \varphi \colon D^\times \backslash \mathrm{GL}_ 2(\mathbb{A}_ F^\infty) / (\mathbb{A}_ F^\infty)^\times \to W_k : \begin{matrix} \varphi(gu) = \chi(u^\ell) u_v^{-1} \varphi(g) \br \text{for } u \in U_Q^\ell U_\ell^v U_v \end{matrix} \biggr\rbrace. $$ This gives us $$ \mathbb{T}(U_Q^\ell, \mathcal{O})_ \chi[1/\ell] \twoheadrightarrow \mathbb{T}_ k(U_Q^\ell U_\ell^v U_v, L)_ \chi. $$ After choosing $i \colon L \hookrightarrow \mathbb{C}$ we have an isomorphism between these spaces and the complex automorphic forms of weight $k$.
Definition 2. We say that a maximal ideal $\mathfrak{p}$ of $\mathbb{T}(U_Q^i U_\ell^v, \mathcal{O})_ \chi[1/\ell]$ is algebraic if it arises for some $k$ and $U_v$ from $\mathbb{T}_ k(U_Q^\ell U_\ell^v U_v, L)_ \chi$.
For each algebraic maximal ideal $\mathfrak{p}$, we then have $\mathrm{GL}_ 2(F_v)$ acting smoothly on $$ \varinjlim_{U_v} \operatorname{Hom}_ {U_v}(W_k^v, S^\mathrm{cts}(U_Q^\ell U_\ell^v, L)_ \chi)[\mathfrak{p}] $$ and this actually will decompose as $\pi_\mathfrak{p}^{\oplus d_\mathfrak{p}}$ for $\pi_\mathfrak{p}$ some irreducible smooth representation of $\mathrm{GL}_ 2(F_v)$.
We can say that $\mathfrak{p}$ is unramified if $\pi_\mathfrak{p}$ is. Then we have seen above that $$ \bigoplus_k \bigoplus_\mathfrak{p} \varinjlim_{U_v} \operatorname{Hom}(U_v, W_k^\vee, S^\mathrm{cts}(U_Q^\ell U_\ell^v, L)_ {\chi,\mathfrak{m}}[\mathfrak{p}]) \otimes W_k^\vee) \to S^\mathrm{cts}(U_Q^\ell U_\ell^v, L)_ {\chi,\mathfrak{m}} $$ is dense. That is, we have $$ \mathcal{M}(U_Q^\ell U_\ell^v, \mathcal{O})_ {\chi,\mathfrak{m}} \hookrightarrow \prod \pi(\mathfrak{p})^\vee, $$ where $\pi(\mathfrak{p})$ is the closure of the image of the terms corresponding to $\mathfrak{p}$.
Now it suffices to show that for $\mathfrak{p}$ unramified algebraic with $\mathfrak{p}^c \subseteq \mathfrak{m}$, then
- $\pi(\mathfrak{p})$ lies in the block of $\mathsf{Ban}_ {\mathrm{GL}_ 2(F_v),1}^\mathrm{adm,fl}(L)$ corresponding to $\bar{r}_ \mathfrak{m} \vert_{G_{F_v}}^\mathrm{ss}$,
- this actually corresponds to a prime ideal $\tilde{\mathfrak{p}}$ of $R_{\mathrm{tr} \bar{r}_ \mathfrak{m} \vert_{G_{F_v}}, \epsilon_\ell^{-1}}^\mathrm{ps}[1/\ell]$ and then this is the preimage of $\mathfrak{p} \subset \mathbb{T}(U_Q^\ell U_\ell^v, \mathcal{O})_ {\chi,\mathfrak{m}}[1/\ell]$.
The point is that there is a crystalline representation $r_\mathfrak{p} \colon G_F \to \mathrm{GL}_ 2(k(\mathfrak{p}))$ of Hodge–Tate weights $k$ and $1-k$. Let $\alpha, \beta$ be the Frobenius eigenvalues, and for $\alpha/\beta \neq \ell^{\pm 1}$, we have $$ \pi(\mathfrak{p}) \supseteq (\operatorname{Ind}_ {B(F_v)}^{\mathrm{GL}_ 2(F_v)}(\mu_{\alpha/\ell} \times \mu_\beta) \otimes W_k^\vee)^{\oplus d_\mathfrak{p}} $$ dense, and therefore for $V_\mathfrak{p}$ a universal unitary completion of $\operatorname{Ind}_ {B(F_v)}^{\mathrm{GL}_ 2(F_v)}(\mu_{\alpha/\ell} \times \mu_\beta) \otimes W_k^\vee$ we get a surjection $$ V_\mathfrak{p}^{\oplus d_\mathfrak{p}} \twoheadrightarrow \pi(\mathfrak{p}). $$ At this point, it is a purely local calculation that $V_\mathfrak{p} \in \mathsf{Ban}_ {\mathrm{GL}_ 2(F_v),1}(L)$ at the correct prime ideal.
Classicality
Theorem 3. Suppose we have a maximal ideal $\mathfrak{p} \subseteq \mathbb{T}(U_Q^\ell, \mathcal{O})_ {\chi,\mathfrak{m}}[1/\ell]$ and suppose that $r \colon G_F \to \mathrm{GL}_ 2(k(\mathfrak{p}))$ has correct pseudorepresentation $\tr r = T_\mathfrak{m} \bmod{\mathfrak{p}}$. Suppose that $r \vert_{G_{F_v}}$ is de Rham and absolutely irreducible with Hodge–Tate numbers $k_v$ and $1-k_v$ for $k_v \in \mathbb{Z}_ {\gt 0}$. Then $\mathfrak{p}$ pulls back from a maximal ideal of $\mathbb{T}_ k(U_Q^\ell U_\ell, L)_ \chi$.
For $v \mid \ell$, it suffices to show that $$ \operatorname{Hom}_ {U_\ell}(W_k^\vee, S^\mathrm{cts}(U_Q^\ell, L)_ {\chi,\mathfrak{m}}[\mathfrak{p}]) \neq 0. $$ But we know from local-global compatibility that this Banach representation given by $\mathfrak{p}$-torsion is in the correct block with the correct prime ideal. Then because $r \vert_{G_{F_v}}$ is absolutely irreducible, this category has a unique irreducible object $\pi_{\tilde{\mathfrak{p}}}$. Now we can check non-vanishing for this.
Theorem 4. The Hecke algebra $\mathbb{T}(U_Q^\ell, \mathcal{O})_ {\chi,\mathfrak{m}}$ is finite over $R_{\mathcal{B}_ \mathfrak{m}}^\mathrm{ps} = \bigotimes_{v \mid \ell}^\wedge R_{\tr \bar{r}_ \mathfrak{m} \vert_{G_{F_v}}, \epsilon_\ell^{-1}}^\mathrm{ps}$ and has dimension $\ge 1+2[F:\mathbb{Q}]$.
To see this, we consider $$ \mathcal{M}(U_Q^\ell, \mathcal{O})_ {\chi,\mathfrak{m}} = \operatorname{Hom}(P_\mathfrak{m} M(U_Q^\ell, \mathcal{O})_ {\chi,\mathfrak{m}}) $$ which has a faithful action of $\mathbb{T}(U_Q^\ell, \mathcal{O})_ {\chi,\mathfrak{m}}$. Here, $M(U_Q^\ell, \mathcal{O})_ {\chi,\mathfrak{m}}$ is finitely generated over $\mathcal{O}[[U_\ell]]$ and so this is finitely generated over $E_{B_\mathfrak{m}}$. The finiteness follows from this.
For the dimension, we know that the $M(U_Q^\ell, \mathcal{O})_ {\chi,\mathfrak{m}}$ is finite free over $\mathcal{O}[[U_\ell]]$ and this has dimension $1 + 3 [F:\mathbb{Q}]$. Then we see from the local calculation that $$ 1 + 2[F:\mathbb{Q}] \le \dim_{\mathbb{T}} M(U_Q^\ell, \mathcal{O})_ {\chi,\mathfrak{m}}. $$