I was telling you about representations of $G = \mathrm{GL}_ 2(\mathbb{Q}_ \ell)$ or $G = \mathrm{GL}_ 2(\mathbb{Q}_ \ell)^n$. Let $Z = \mathbb{Q}_ \ell^\times$ or $(\mathbb{Q}_ \ell^\times)^n$ and $G_0 = \mathrm{GL}_ 2(\mathbb{Z}_ \ell)$ or $\mathrm{GL}_ 2(\mathbb{Z}_ \ell)^n$.
Let $L/\mathbb{Q}_ \ell$ be finite with $\mathcal{O}/\lambda = \mathbb{F}$, and let $A$ be a complete Noetherian local $\mathcal{O}$-algebra with $A/\mathfrak{m} = \mathbb{F}$. We were looking at $\mathsf{Mod}_ G^\mathrm{sm}(A)$ the category of $G$ acting on an $A$-module $M$ where every element of $M$ is torsion and has open stabilizer. We looked at the subcategory $\mathsf{Mod}_ G^\mathrm{lfin}(A)$ of locally finite length representations, meaning for all $m \in M$ there exists $m \in N \subseteq M$ where $N$ is of finite length. For a (continuous) character $\chi \colon Z \to A^\times$, we have subcategories $\mathsf{Mod}_ {G,\chi}^\mathrm{lfin} \subseteq \mathsf{Mod}_ {G,\chi}$.
We stated that these are all abelian categories with exact inductive limits and has a single generator. This is enough to conclude that it has injective envelopes. More precisely, for every $M$ there exists an injective object $I$ and $M \hookrightarrow I$ satisfying the property that if $0 \neq N \subseteq I$ then $N \cap M \neq 0$.
The categories $\mathsf{Mod}_ G^\mathrm{lfin}$ and $\mathsf{Mod}_ {G,\chi}^\mathrm{lfin}$ also have a set of finite length generators. This is enough to produce a theory of blocks. For $\pi$ and $\sigma$ irreducible objects, we give this an equivalence relation $\pi \sim \sigma$ if and only if there is a sequence $\pi = \pi_0, \pi_1, \dotsc, \pi_r = \sigma$ of irreducible such that for all $i$ either $\pi_i \cong \pi_{i+1}$ or $\operatorname{Ext}^1(\pi_i, \pi_{i+1}) \neq 0$ or $\operatorname{Ext}^1(\pi_{i+1}, \pi_i) \neq 0$. An equivalence class is also called a block. For a block $\mathcal{B}$, we can consider the full subcategory $$ \mathsf{Mod}_ {G,(\chi)}^\mathrm{lfin}(A)_ \mathcal{B} \subseteq \mathsf{Mod}_ {G,(\chi)}^\mathrm{lfin}(A) $$ consisting of objects whose all irreducible subquotients are in $\mathcal{B}$. Then we have $$ \mathsf{Mod}_ {G,(\chi)}^\mathrm{lfin}(A) \cong \prod_{\mathcal{B}} \mathsf{Mod}_ {G,(\chi)}^\mathrm{lfin}(A)_ \mathcal{B} $$ where a tuple $(\pi_\mathcal{B})$ on the right side corresponds to $\bigoplus_{\mathcal{B}} \pi_\mathcal{B}$ on the left side.
Definition 1. Let $\mathsf{Mod}_ G^\mathrm{proaug}$ be the category of profinite $A$-modules $M$ together with an action of $A[G]$ such that for some (hence every) open compact subgroup $U \subseteq G$ the $A[U]$-action comes with an extension to a continuous $A[[U]]$-action.
There is then an anti-equivalence $$ \mathsf{Mod}_ G^\mathrm{sm}(A) \leftrightarrow \mathsf{Mod}_ G^\mathrm{proaug}(A); \quad M \mapsto \operatorname{Hom}_ \mathcal{O}(M, L/\mathcal{O}), \quad N \mapsto \operatorname{Hom}_ \mathcal{O}^\mathrm{cts}(N, L/\mathcal{O}). $$ The full subcategories $\mathsf{Mod}_ G^\mathrm{lfin}(A)$ and $\mathsf{Mod}_ {G,1}^\mathrm{lfin}(A)$ correspond to full subcategories $\mathcal{C}_ 1 \subseteq \mathcal{C} \subseteq \mathsf{Mod}_ G^\mathrm{proaug}(A)$, and then we still get a block decomposition $$ \mathcal{C} \cong \prod_{\mathcal{B}} \mathcal{C}_ \mathcal{B}, \quad \mathcal{C}_ 1 \cong \prod_{\mathcal{B}} \mathcal{C}_ {\mathcal{B},1}. $$
Fact 2. Consider the category $\mathsf{Mod}_ {G,1}^\mathrm{lfin}(A)$ or $\mathcal{C}_ 1$.
- If $\pi$ is irreducible then $\operatorname{End}(\pi)$ are finite over $\mathbb{F}$.
- The blocks are finite.
- If $N \in \mathsf{Mod}_ {G,1}^\mathrm{proaut}(A)$ is finitely generated over $A[[U]]$ for one (hence every) open compact subgroup $U \subseteq G$, then $N \in \mathcal{C}_ 1$.
Let $\mathcal{B} = \lbrace \pi_1 \dotsc, \pi_r \rbrace$ be a block. Consider the projective envelope $P_i \twoheadrightarrow \pi_i^\vee$ and consider $$ P_\mathcal{B} = \bigoplus P_i, \quad E_\mathcal{B} = \operatorname{End}(P_\mathcal{B}). $$ This has a topology; if $\pi_M \colon P_\mathcal{B} \twoheadrightarrow M$ where $M$ is of finite length, we consider $$ I_M = \lbrace \alpha \in E_\mathcal{B} : \pi_M \circ \alpha = 0 \rbrace. $$ We take these to be a basis of open neighborhoods. We also set $$ \mathfrak{m}_ \mathcal{B} = I_{(P_\mathcal{B} \twoheadrightarrow \bigoplus_i \pi_i^\vee)}. $$ Then $E_\mathcal{B}$ is compact and $\mathfrak{m}_ \mathcal{B}$ is its Jacobson radical. We also have an equivalence $$ \mathcal{C}_ {\mathcal{B},1} \cong \lbrace \text{profinite left } E_\mathcal{B} \text{-modules} \rbrace, $$ where we send $M \mapsto \operatorname{Hom}_ {\mathcal{C}_ 1}(P_\mathcal{B}, M)$ and $N \mapsto P_ \mathcal{B} \otimes_{E_\mathcal{B}} N$.
Blocks for $\mathrm{GL}_ 2(\mathbb{Q}_ \ell)$
For $G = \mathrm{GL}_ 2(\mathbb{Q}_ \ell)$, Paskunas wrote down all possible blocks.
In the supersingular case, we have $\mathcal{B} = \lbrace \pi \rbrace$. Here, $\pi = \pi_{r,\eta}$ where $0 \le r \le \ell-1$ is even and $\eta \colon \mathbb{Q}_ \ell^\times \to \mathbb{F}^\times$ satisfies $\eta^2 = \omega^{-r}$ where $\omega \colon \mathbb{Q}_ \ell^\times \to \mathbb{F}^\times$ sends $\ell \mapsto 1$ and $\mathbb{Z}_ \ell^\times \ni a \mapsto a \bmod{\ell}$. Then we have $$ \pi = \pi_{r,\eta} = (\operatorname{cInd}_ {Z G_0}^G \mathrm{Sym}^r (\mathbb{F}^2) / T) \otimes (\eta \circ \mathrm{det}). $$ Here, $Z$ acts trivially, and the compact induction has endomorphism ring $\mathbb{F}[T]$.
Let $B$ be an upper triangular Borel. For two characters $\chi_1, \chi_2$, we have $$ \chi_1 \times \chi_2 \colon B \to \mathbb{F}^\times; \quad \begin{pmatrix} a & b \br 0 & d \end{pmatrix} \mapsto \chi_1(a) \chi_2(d). $$ Then we have $$ \mathcal{B} = \lbrace \operatorname{Ind}_ B^G(\chi \times \chi^{-1}), \operatorname{Ind}_ B^G(\omega \chi^{-1} \times \chi \omega^{-1}) \rbrace $$ as long as $\chi^2 \neq 1$ and $(\chi \omega)^2 \neq 1$.
(In the above case, we can actually have $\chi$ valued in some finite extension of $\mathbb{F}$. We need to do this to get the complete list, but we can also just pass to a bigger extension.)
There is the block $\mathcal{B} = \lbrace 1, \mathrm{Sp}, \operatorname{Ind}_ B^G (\omega \times \omega^{-1}) \rbrace \otimes \chi$ for $\chi^2 = 1$.
In all these cases, we can make the following definition.
Definition 3. We define $\bar{r}_ \mathcal{B}^\vee \colon G_{\mathbb{Q}_ \ell} \to \mathrm{GL}_ 2(\mathbb{F})$ as
- $(\operatorname{Ind}_ {G_{\mathbb{Q}_ {\ell^2}}}^{G_{\mathbb{Q}_ \ell}} \omega_2^{r+1}) \otimes \eta$ where $\omega_2 \colon \mathbb{Q}_ {\ell^2}^\times \to \mathbb{F}^\times$ is the second fundamental character $\ell \mapsto 1$, $\mathbb{Z}_ {\ell^2}^\times \ni a \mapsto a \bmod{\ell}$,
- $\chi \oplus \omega \chi^{-1}$,
- $\chi \oplus \omega \chi^{-1}$.
It turns out that this runs over all semisimple mod-$\ell$ representations of $G_{\mathbb{Q}_ \ell}$ with simple factors defined over $\mathbb{F}$ and $\det \bar{r}_ \mathcal{B} = \bar{\epsilon}_ \ell^{-1}$.
Theorem 4 (Paskunas). 1. The endomorphism ring $E_\mathcal{B}$ is finite over its center.
- The center is $Z(E_\mathcal{B}) \cong R_{\tr r_\mathcal{B}, \epsilon_\ell^{-1}}^\mathrm{ps}$ the universal pseudo-deformation ring with fixed character.
This also extends to $G = \mathrm{GL}_ 2(\mathbb{Q}_ \ell)^n$. The irreducibles are products of irreducibles and blocks are products of blocks. Then for $\mathcal{B} = \prod_{i=1}^n \mathcal{B}_ i$, we have a map $$ \bigotimes^\wedge R_{\tr r_{\mathcal{B}_ i}, \epsilon_\ell^{-1}}^\mathrm{ps} \to Z(E_\mathcal{B}) $$ that is known to be finite. (One may hope that this is an isomorphism but it is not known. It turns out that $P_\mathcal{B}$ is just the completed tensor product of the $P_{\mathcal{B},i}$ and similarly for $E_\mathcal{B}$, but the problem is that the center of the tensor product is not the tensor product of the centers. There is a counterexample on MathOverflow.)
Consider a maximal ideal $\mathfrak{p} \subseteq R_{\tr r_\mathcal{B}, \epsilon_\ell^{-1}}^\mathrm{ps}[1/\ell]$ so that $k(\mathfrak{p}) / L$ is finite. Then we have $$ r \colon G_{\mathbb{Q}_ \ell} \to \mathrm{GL}_ 2(\mathcal{O}_ {k(\mathfrak{p})}) $$ that is crystalline with Hodge–Tate weights $\lbrace w, 1-w \rbrace$ and $\tr r = T^\mathrm{univ} \bmod{\mathfrak{p}}$. Consider the Frobenius action on $(r \otimes B_\mathrm{cris})^{G_{\mathbb{Q}_ \ell}}$ with eigenvalues $\alpha, \beta$. Suppose that $\alpha/\beta \neq \ell^{\pm 1}$. Then we have $$ \tilde{V}_ \mathfrak{p} = (\operatorname{Sym}^{2w}(k(\mathfrak{p})^{\oplus 2}) \otimes \mathrm{det}^w)^\vee \otimes \operatorname{smInd}_ B^G(\mu_{\alpha/\ell} \times \mu_\beta) $$ where $\mu_\gamma \colon \mathbb{Q}_ \ell^\times / \mathbb{Z}_ \ell^\times \to \mathbb{F}^\times$ sends $\ell \mapsto \gamma$. This is some locally algebraic representation, and now we consider its universal completion $\tilde{V}_ \mathfrak{p}$, i.e., the limit of quotients of $\tilde{V}_ \mathfrak{p}$ by its finitely generated $\mathcal{O}[G]$-submodules. This is universal in the sense that if $W$ is a unitary Banach representation of $G$ and $W \supseteq \tilde{V}_ \mathfrak{p}^{\oplus n}$ is dense then we get a surjection $V_\mathfrak{p}^{\oplus n} \twoheadrightarrow W$.
Theorem 5. The representation $V_\mathfrak{p}$ is topologically irreducible, nontrivial, admissible unitary Banach representation of $G$. (Unitary means that there is a unit ball $V_\mathfrak{p} \supseteq V_\mathfrak{p}^\circ$ that is $G$-stable, and admissible means that for all $U \subseteq G$ compact open, $(V_\mathfrak{p}^\circ / \lambda)^U$ is finite-dimensional over $\mathbb{F}$.)
Then we can check that $$ M_\mathfrak{p} = \operatorname{Hom}_ \mathcal{O}(V_\mathfrak{p}^\circ, \mathcal{O}) \in \operatorname{ob}(\mathcal{C}_ 1). $$
Theorem 6 (Paskunas). The $R_{\tr r_\mathcal{B}, \epsilon_\ell^{-1}}^\mathrm{ps}$-action on $\operatorname{Hom}(P_\mathcal{B}, M_\mathfrak{p})[1/\ell]$ factors through $R_{\tr r_\mathcal{B},\epsilon_\ell^{-1}}^\mathrm{ps}[1/\ell] / \mathfrak{p}$.
The problem we are going to face is that $$ \operatorname{Spec} R_{\bar{r} \vert_{G_{\mathbb{Q}_ v}}, \epsilon_\ell^{-1}}^ {\mathrm{crys}, \lbrace w, 1-w \rbrace, \square} $$ has lots of irreducible components if $w \gg 0$. The solution to this is to use the entire deformation ring $R_{\bar{r} \vert_{G_{\mathbb{Q}_ \ell}}, \epsilon_\ell^{-1}}^\square$, which is often irreducible and even a power series ring because the obstruction for lifting vanishes.