# Patching and multiplicities of p-adic eigenforms

Eugen Hellmann, Valentin Hernandez, Benjamin Schraen

► To cite this version:

Eugen Hellmann, Valentin Hernandez, Benjamin Schraen. Patching and multiplicities of p-adic eigen-
forms. 2024. hal-04607508

HAL Id: hal-04607508

https://hal.science/hal-04607508

Preprint submitted on 10 Jun 2024

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Distributed under a Creative Commons Attribution 4.0 International License
# Patching and multiplicities of p-adic eigenforms

Eugen Hellmann, Valentin Hernandez, Benjamin Schraen

► To cite this version:

Eugen Hellmann, Valentin Hernandez, Benjamin Schraen. Patching and multiplicities of p-adic eigenforms. Tunisian Journal of Mathematics, 2024, 7 (3-4), pp.831-908. 10.2140/tunis.2025.7.831. hal-04607508

HAL Id: hal-04607508

https://hal.science/hal-04607508v1

Submitted on 10 Jun 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
<table>
  <tbody>
    <tr>
      <td>3</td>
      <td>The Emerton-Jacquet functor</td>
      <td>20</td>
    </tr>
    <tr>
      <td>3.1</td>
      <td>Families of locally analytic representations of the Borel subgroup</td>
      <td>20</td>
    </tr>
    <tr>
      <td>3.2</td>
      <td>Families of locally analytic representations of G</td>
      <td>23</td>
    </tr>
    <tr>
      <td>3.3</td>
      <td>The case of Banach representations with coefficients</td>
      <td>27</td>
    </tr>
    <tr>
      <td>3.4</td>
      <td>Comparison with the parabolic Jacquet functor</td>
      <td>29</td>
    </tr>
    <tr>
      <td>4</td>
      <td>Quasi-trianguline local deformation rings</td>
      <td>30</td>
    </tr>
    <tr>
      <td>4.1</td>
      <td>Local models</td>
      <td>30</td>
    </tr>
    <tr>
      <td>4.2</td>
      <td>Partially de Rham deformation rings</td>
      <td>34</td>
    </tr>
    <tr>
      <td>5</td>
      <td>Global construction</td>
      <td>36</td>
    </tr>
    <tr>
      <td>5.1</td>
      <td>Classical and p-adic automorphic forms</td>
      <td>36</td>
    </tr>
    <tr>
      <td>5.2</td>
      <td>Patching the completed cohomology</td>
      <td>39</td>
    </tr>
    <tr>
      <td>6</td>
      <td>Patching functors</td>
      <td>41</td>
    </tr>
    <tr>
      <td>6.1</td>
      <td>Locally analytic patching functors</td>
      <td>42</td>
    </tr>
    <tr>
      <td>6.2</td>
      <td>A factorization property</td>
      <td>43</td>
    </tr>
    <tr>
      <td>6.3</td>
      <td>Bi-module structure on the patched functor</td>
      <td>47</td>
    </tr>
    <tr>
      <td>6.4</td>
      <td>Computation of a support</td>
      <td>50</td>
    </tr>
    <tr>
      <td>7</td>
      <td>Main results</td>
      <td>51</td>
    </tr>
    <tr>
      <td>7.1</td>
      <td>Sheaves and supports.</td>
      <td>52</td>
    </tr>
    <tr>
      <td>7.2</td>
      <td>Recollection on Bezrukavnikov's functor</td>
      <td>56</td>
    </tr>
    <tr>
      <td>7.3</td>
      <td>A detail study of local models when n = 3</td>
      <td>58</td>
    </tr>
    <tr>
      <td>7.4</td>
      <td>The case of dual Vermas</td>
      <td>62</td>
    </tr>
    <tr>
      <td>7.5</td>
      <td>The case of the antidominant projective</td>
      <td>65</td>
    </tr>
    <tr>
      <td>7.6</td>
      <td>Duality</td>
      <td>67</td>
    </tr>
    <tr>
      <td>7.7</td>
      <td>Consequences</td>
      <td>71</td>
    </tr>
    <tr>
      <td>8</td>
      <td>Existence of very critical classical modular forms</td>
      <td>76</td>
    </tr>
    <tr>
      <td>8.1</td>
      <td>Choosing a Hecke character</td>
      <td>78</td>
    </tr>
    <tr>
      <td>8.2</td>
      <td>Galois induction</td>
      <td>79</td>
    </tr>
    <tr>
      <td>8.3</td>
      <td>Construction of an explicit set of Hecke characters</td>
      <td>81</td>
    </tr>
    <tr>
      <td>8.4</td>
      <td>Automorphic Induction and base change</td>
      <td>82</td>
    </tr>
  </tbody>
</table>
<table>
  <tbody>
    <tr>
      <td>3</td>
      <td>The Emerton-Jacquet functor</td>
      <td>20</td>
    </tr>
    <tr>
      <td>3.1</td>
      <td>Families of locally analytic representations of the Borel subgroup</td>
      <td>20</td>
    </tr>
    <tr>
      <td>3.2</td>
      <td>Families of locally analytic representations of G</td>
      <td>23</td>
    </tr>
    <tr>
      <td>3.3</td>
      <td>The case of Banach representations with coefficients</td>
      <td>27</td>
    </tr>
    <tr>
      <td>3.4</td>
      <td>Comparison with the parabolic Jacquet functor</td>
      <td>29</td>
    </tr>
    <tr>
      <td>4</td>
      <td>Quasi-trianguline local deformation rings</td>
      <td>30</td>
    </tr>
    <tr>
      <td>4.1</td>
      <td>Local models</td>
      <td>30</td>
    </tr>
    <tr>
      <td>4.2</td>
      <td>Partially de Rham deformation rings</td>
      <td>34</td>
    </tr>
    <tr>
      <td>5</td>
      <td>Global construction</td>
      <td>36</td>
    </tr>
    <tr>
      <td>5.1</td>
      <td>Classical and p-adic automorphic forms</td>
      <td>36</td>
    </tr>
    <tr>
      <td>5.2</td>
      <td>Patching the completed cohomology</td>
      <td>39</td>
    </tr>
    <tr>
      <td>6</td>
      <td>Patching functors</td>
      <td>41</td>
    </tr>
    <tr>
      <td>6.1</td>
      <td>Locally analytic patching functors</td>
      <td>42</td>
    </tr>
    <tr>
      <td>6.2</td>
      <td>A factorization property</td>
      <td>43</td>
    </tr>
    <tr>
      <td>6.3</td>
      <td>Bi-module structure on the patched functor</td>
      <td>47</td>
    </tr>
    <tr>
      <td>6.4</td>
      <td>Computation of a support</td>
      <td>50</td>
    </tr>
    <tr>
      <td>7</td>
      <td>Main results</td>
      <td>51</td>
    </tr>
    <tr>
      <td>7.1</td>
      <td>Sheaves and supports.</td>
      <td>52</td>
    </tr>
    <tr>
      <td>7.2</td>
      <td>Recollection on Bezrukavnikov's functor</td>
      <td>56</td>
    </tr>
    <tr>
      <td>7.3</td>
      <td>A detail study of local models when n = 3</td>
      <td>58</td>
    </tr>
    <tr>
      <td>7.4</td>
      <td>The case of dual Vermas</td>
      <td>62</td>
    </tr>
    <tr>
      <td>7.5</td>
      <td>The case of the antidominant projective</td>
      <td>65</td>
    </tr>
    <tr>
      <td>7.6</td>
      <td>Duality</td>
      <td>67</td>
    </tr>
    <tr>
      <td>7.7</td>
      <td>Consequences</td>
      <td>71</td>
    </tr>
    <tr>
      <td>8</td>
      <td>Existence of very critical classical modular forms</td>
      <td>76</td>
    </tr>
    <tr>
      <td>8.1</td>
      <td>Choosing a Hecke character</td>
      <td>78</td>
    </tr>
    <tr>
      <td>8.2</td>
      <td>Galois induction</td>
      <td>79</td>
    </tr>
    <tr>
      <td>8.3</td>
      <td>Construction of an explicit set of Hecke characters</td>
      <td>81</td>
    </tr>
    <tr>
      <td>8.4</td>
      <td>Automorphic Induction and base change</td>
      <td>82</td>
    </tr>
  </tbody>
</table>
# 1 Introduction

The aim of this paper is to unravel (and explain) a new phenomenon in the theory of $p$-adic automorphic forms. Given a reductive group $G$ over a number field (overconvergent) $p$-adic automorphic forms are $p$-adic avatars of automorphic forms on $G$. We usually refer to the latter as classical automorphic forms in order to distinguish them from their $p$-adic limits. Additional structures on spaces of automorphic forms, such as the Hecke-action, naturally extend to the $L$-vector spaces of overconvergent $p$-adic automorphic forms $S^\dagger(K^p), S_\kappa^\dagger(K^p)$, where the field of coefficients $L$ is a finite extension of $\mathbb{Q}_p$ and $K^p \subset G(\mathbb{A}^p)$ is a compact open subgroup (referred to as the *tame level*) and $\kappa$ is a *weight*. A central question about $p$-adic automorphic forms is to clarify whether a given overconvergent $p$-adic automorphic form (of algebraic weight) that is an eigenform for the Hecke action is a classical automorphic form. Often this question can be answered in terms of the Hecke eigenvalues. Coleman's *small slope implies classical* result [Col97] and generalizations thereof (see e.g. [Kas06], [Che11], [BPS16]) asserts that this question can be purely decided using the Hecke action at $p$ if the $p$-adic valuation of the Hecke eigenvalues at $p$ is small compared to the weight. Beyond the *numerically non critical slope* it is known that this fails. However, one can ask the same question taking into account the full Hecke action (as opposed to the Hecke action at $p$).

Assume that we are in a situation where we can construct the Galois representation $\rho_f = \rho_\chi$ attached to a $p$-adic eigenform $f$, respectively to the Hecke character $\chi$ giving the system of Hecke eigenvalues of $f$. Then the Hecke action away from $p$ encodes all the information about the $p$-adic Galois representation $\rho_f$, including the $p$-adic Hodge theoretic information at places dividing $p$ (though this is encoded in a rather indirect and mysterious way). The naive generalization of the classicality question about over-convergent $p$-adic automorphic forms can hence be phrased as follows (though we phrase the question in a rather informal way):

**Question A:** Let $f$ be an overconvergent $p$-adic eigenform of dominant algebraic weight such that the corresponding Galois representation $\rho_f$ is de Rham at places dividing $p$. Is it true that $f$ is a classical automorphic form?

We note that a softer version of this question is the following expectation that is implied by the Fontaine–Mazur conjecture. Again we state the expectation in a rather informal way – it might fail without more precise assumptions on the group the level, etc. (see e.g. [BHS19, Conj. 5.1.1] for a precise formulation).

**Rough Expectation B:** Let $S_\kappa^\dagger(K^p)[\chi] \subset S_\kappa^\dagger(K^p)$ be an eigensystem (for the action of the full Hecke algebra $\mathbb{T}$ generated by Hecke operators at $p$ and away from $p$) in the space $S_\kappa^\dagger(K^p)$ of overconvergent $p$-adic automorphic forms of weight $\kappa$ on $G$. Assume that $\kappa$ is dominant algebraic and that the Galois representation $\rho_\chi$ associated to the Hecke character $\chi: \mathbb{T} \to L$ is de Rham at places dividing $p$. Then $S_\kappa^\dagger(K^p)[\chi]$ contains a classical automorphic form, i.e. its subspace $S_\kappa^{\text{cl}}(K^p)[\chi]$ of classical forms is non-zero.

Question A then can be rephrased as the question whether $S_\kappa^{\text{cl}}[\chi](K^p) = S_\kappa^\dagger(K^p)[\chi]$
# 1 Introduction

The aim of this paper is to unravel (and explain) a new phenomenon in the theory of $p$-adic automorphic forms. Given a reductive group $G$ over a number field (overconvergent) $p$-adic automorphic forms are $p$-adic avatars of automorphic forms on $G$. We usually refer to the latter as classical automorphic forms in order to distinguish them from their $p$-adic limits. Additional structures on spaces of automorphic forms, such as the Hecke-action, naturally extend to the $L$-vector spaces of overconvergent $p$-adic automorphic forms $S^\dagger(K^p), S_\kappa^\dagger(K^p)$, where the field of coefficients $L$ is a finite extension of $\mathbb{Q}_p$ and $K^p \subset G(\mathbb{A}^p)$ is a compact open subgroup (referred to as the *tame level*) and $\kappa$ is a *weight*. A central question about $p$-adic automorphic forms is to clarify whether a given overconvergent $p$-adic automorphic form (of algebraic weight) that is an eigenform for the Hecke action is a classical automorphic form. Often this question can be answered in terms of the Hecke eigenvalues. Coleman's *small slope implies classical* result [Col97] and generalizations thereof (see e.g. [Kas06], [Che11], [BPS16]) asserts that this question can be purely decided using the Hecke action at $p$ if the $p$-adic valuation of the Hecke eigenvalues at $p$ is small compared to the weight. Beyond the *numerically non critical slope* it is known that this fails. However, one can ask the same question taking into account the full Hecke action (as opposed to the Hecke action at $p$).

Assume that we are in a situation where we can construct the Galois representation $\rho_f = \rho_\chi$ attached to a $p$-adic eigenform $f$, respectively to the Hecke character $\chi$ giving the system of Hecke eigenvalues of $f$. Then the Hecke action away from $p$ encodes all the information about the $p$-adic Galois representation $\rho_f$, including the $p$-adic Hodge theoretic information at places dividing $p$ (though this is encoded in a rather indirect and mysterious way). The naive generalization of the classicality question about over-convergent $p$-adic automorphic forms can hence be phrased as follows (though we phrase the question in a rather informal way):

**Question A:** Let $f$ be an overconvergent $p$-adic eigenform of dominant algebraic weight such that the corresponding Galois representation $\rho_f$ is de Rham at places dividing $p$. Is it true that $f$ is a classical automorphic form?

We note that a softer version of this question is the following expectation that is implied by the Fontaine–Mazur conjecture. Again we state the expectation in a rather informal way – it might fail without more precise assumptions on the group the level, etc. (see e.g. [BHS19, Conj. 5.1.1] for a precise formulation).

**Rough Expectation B:** Let $S_\kappa^\dagger(K^p)[\chi] \subset S_\kappa^\dagger(K^p)$ be an eigensystem (for the action of the full Hecke algebra $\mathbb{T}$ generated by Hecke operators at $p$ and away from $p$) in the space $S_\kappa^\dagger(K^p)$ of overconvergent $p$-adic automorphic forms of weight $\kappa$ on $G$. Assume that $\kappa$ is dominant algebraic and that the Galois representation $\rho_\chi$ associated to the Hecke character $\chi: \mathbb{T} \to L$ is de Rham at places dividing $p$. Then $S_\kappa^\dagger(K^p)[\chi]$ contains a classical automorphic form, i.e. its subspace $S_\kappa^{\text{cl}}(K^p)[\chi]$ of classical forms is non-zero.

Question A then can be rephrased as the question whether $S_\kappa^{\text{cl}}[\chi](K^p) = S_\kappa^\dagger(K^p)[\chi]$
in Expectation B. It is known that Question A does not have an affirmative answer in general. Ludwig [Lud18] and Johansson-Ludwig [JL23] have shown that there are counterexamples for $SL_2$. The reason for these counterexamples however, is of global (endoscopic) nature and it remains a reasonable question to ask Question A for groups where these phenomena do not apply, e.g. for definite unitary groups.

Expectation B has been verified for $GL_2$ (this is basically [Kis03]), and generalizations of Kisins' result were proven by Bellaïche and his coauthors ([BC06], [Bel12] and [BD16]). For definite unitary groups, and under Taylor-Wiles assumptions, these results were vastly generalized in [BHS17a], [BHS19]. We point out that in the cases treated in [BHS17a] the results imply that $S_\kappa^{\text{cl}}(K^p)[\chi] = S_\kappa^\dagger(K^p)[\chi]$, while the more general case in [BHS19] only allows to construct some classical form in the eigensystem (though no counterexample to Question A is constructed in loc. cit.). The reason for this difference is due to a phenomenon in the geometry of eigenvarieties (i.e. rigid analytic spaces parametrizing the systems of Hecke eigenvalues in the space of overconvergent $p$-adic automorphic forms of finite slope), respectively in the geometry of their local Galois-theoretic counterparts (the so-called trianguline variety of [BHS17b]). In the case treated in [BHS17a] the trianguline variety is smooth at the Galois representations in question (and hence the eigenvariety is local complete intersection). In general the trianguline variety is not smooth, and as a consequence one can construct non-smooth points on the corresponding eigenvarieties, see [BHS19, Thm. 5.4.2]. It is this failure of smoothness that prevents [BHS19] from identifying $S_\kappa^{\text{cl}}(K^p)[\chi]$ and $S_\kappa^\dagger(K^p)[\chi]$.

In this paper we prove that the answer to Question A is no for definite unitary groups in three variables (see Theorem 1.2 below for a more precise formulation).

**Theorem 1.1.** There exists a unitary group in three variables $U$, a tame level $K^p$, a dominant algebraic weight $\kappa$ and a Hecke character $\chi: \mathbb{T} \to L$ that occurs in the space $S_\kappa^\dagger(K^p)_{\text{fs}}$ of overconvergent automorphic forms of finite slope and weight $\kappa$ such that the eigenspace $S_\kappa^\dagger(K^p)[\chi]$ contains classical as well as non-classical eigenforms.

The construction of this example also clarifies the role of the singularities of the trianguline variety $X_{\text{tri}}$. The precise results we prove suggest that the answer to Question A is no, whenever the dualizing sheaf $\omega_{X_{\text{tri}}}$ is not locally free at the point defined by $\rho$ (and the refinement associated to $\chi$), i.e. whenever $X_{\text{tri}}$ is non-Gorenstein at this point (we refer to Theorem 1.3 below for the precise link with $\omega_{X_{\text{tri}}}$). In the three dimensional case, this results in a precise comparison of the dimensions of the eigenspaces $S_\kappa^{\text{cl}}(K^p)[\chi] \subset S_\kappa^\dagger(K^p)[\chi]$.

We point out that, in contrast to [Lud18] and [JL23] this is a purely local $p$-adic phenomenon. Moreover, the theorem implies that the usual invariants (i.e. the Hecke action, respectively the $p$-adic Hodge theoretic information of the associated Galois representation) can not distinguished between classical and non-classical forms. We like to refer to the non-classical forms in such eigensystems as *undercover* automorphic forms.

The main result, and in particular the occurrence of the dualizing sheaf $\omega_{X_{\text{tri}}}$ therein, is inspired by the categorical point of view in the $p$-adic Langlands program, see [EGH23].
in Expectation B. It is known that Question A does not have an affirmative answer in general. Ludwig [Lud18] and Johansson-Ludwig [JL23] have shown that there are counterexamples for $SL_2$. The reason for these counterexamples however, is of global (endoscopic) nature and it remains a reasonable question to ask Question A for groups where these phenomena do not apply, e.g. for definite unitary groups.

Expectation B has been verified for $GL_2$ (this is basically [Kis03]), and generalizations of Kisins' result were proven by Bellaïche and his coauthors ([BC06], [Bel12] and [BD16]). For definite unitary groups, and under Taylor-Wiles assumptions, these results were vastly generalized in [BHS17a], [BHS19]. We point out that in the cases treated in [BHS17a] the results imply that $S_\kappa^{\text{cl}}(K^p)[\chi] = S_\kappa^\dagger(K^p)[\chi]$, while the more general case in [BHS19] only allows to construct some classical form in the eigensystem (though no counterexample to Question A is constructed in loc. cit.). The reason for this difference is due to a phenomenon in the geometry of eigenvarieties (i.e. rigid analytic spaces parametrizing the systems of Hecke eigenvalues in the space of overconvergent $p$-adic automorphic forms of finite slope), respectively in the geometry of their local Galois-theoretic counterparts (the so-called trianguline variety of [BHS17b]). In the case treated in [BHS17a] the trianguline variety is smooth at the Galois representations in question (and hence the eigenvariety is local complete intersection). In general the trianguline variety is not smooth, and as a consequence one can construct non-smooth points on the corresponding eigenvarieties, see [BHS19, Thm. 5.4.2]. It is this failure of smoothness that prevents [BHS19] from identifying $S_\kappa^{\text{cl}}(K^p)[\chi]$ and $S_\kappa^\dagger(K^p)[\chi]$.

In this paper we prove that the answer to Question A is no for definite unitary groups in three variables (see Theorem 1.2 below for a more precise formulation).

**Theorem 1.1.** There exists a unitary group in three variables $U$, a tame level $K^p$, a dominant algebraic weight $\kappa$ and a Hecke character $\chi : \mathbb{T} \to L$ that occurs in the space $S_\kappa^\dagger(K^p)_{\text{fs}}$ of overconvergent automorphic forms of finite slope and weight $\kappa$ such that the eigenspace $S_\kappa^\dagger(K^p)[\chi]$ contains classical as well as non-classical eigenforms.

The construction of this example also clarifies the role of the singularities of the trianguline variety $X_{\text{tri}}$. The precise results we prove suggest that the answer to Question A is no, whenever the dualizing sheaf $\omega_{X_{\text{tri}}}$ is not locally free at the point defined by $\rho$ (and the refinement associated to $\chi$), i.e. whenever $X_{\text{tri}}$ is non-Gorenstein at this point (we refer to Theorem 1.3 below for the precise link with $\omega_{X_{\text{tri}}}$). In the three dimensional case, this results in a precise comparison of the dimensions of the eigenspaces $S_\kappa^{\text{cl}}(K^p)[\chi] \subset S_\kappa^\dagger(K^p)[\chi]$.

We point out that, in contrast to [Lud18] and [JL23] this is a purely local $p$-adic phenomenon. Moreover, the theorem implies that the usual invariants (i.e. the Hecke action, respectively the $p$-adic Hodge theoretic information of the associated Galois representation) can not distinguished between classical and non-classical forms. We like to refer to the non-classical forms in such eigensystems as *undercover* automorphic forms.

The main result, and in particular the occurrence of the dualizing sheaf $\omega_{X_{\text{tri}}}$ therein, is inspired by the categorical point of view in the $p$-adic Langlands program, see [EGH23].
The space of overconvergent *p*-adic automorphic forms of finite slope $S^\dagger(K^p)_{\text{fs}}$ can be viewed as the topological dual of the global sections of a coherent sheaf (that we simply refer to as the *sheaf of *p*-adic automorphic forms*) on the rigid analytic generic fiber of the universal deformation space of Galois representations (more precisely, on the product of this space with the space of continuous characters of a maximal torus $\mathcal{T}(\mathbb{Q}_p) \subset \mathcal{G}(\mathbb{Q}_p)$ at *p*). The support of this sheaf is, by definition, the corresponding eigenvariety. The local-global-compatibility conjectures [EGH23, Conj. 9.6.8 and Conj. 9.6.16] give a precise description of this sheaf in terms of the geometry of moduli stacks of $(\varphi, \Gamma)$-modules (that are closely related to the trianguline variety). More precisely, the categorical approach to the *p*-adic Langlands program asks for a functor from certain (locally analytic) representations of $\mathcal{G}(\mathbb{Q}_p)$ to sheaves on stacks of $(\varphi, \Gamma)$-modules, and the sheaf of *p*-adic automorphic forms is the globalization of the evaluation of this functor on a specific representation. One of the punchlines of [EGH23] (see section 1.6 therein for a more detailed discussion) is that avatars of the envisioned functor have been around in number theory during the past decades in the context of the Taylor-Wiles patching method, in particular *patching functors* as used for example in [EGS15] (or also in [BHS19, 5.]) A crucial point in the proof of the main theorem is the identification of such a patching functor with an explicit local functor, see Theorem 1.4 below. This partially confirms expectations in the categorical picture, see [EGH23, Expectation 6.2.27].

Note that the multiplicity result in Theorem 1.2 has some striking consequence for the *p*-adic Langlands Program for $\text{GL}_3(\mathbb{Q}_p)$. It implies that the locally analytic representation of $\text{GL}_3(\mathbb{Q}_p)$ on the Hecke eigenspace of overconvergent *p*-adic modular forms over $\mathcal{G}$ corresponding to a Galois representation $\rho$ as in Theorem 1.2 contains locally analytic vectors which are *not* in the socle of the representation (see Remark 7.29). After finishing this works, the authors learned that Ding also proved examples of this penroma for generic Galois representations (see [Din]).

We now describe our results in more detail. Let $F$ be a totally real number field and let $E/F$ be a CM (imaginary) quadratic extension in which every place $v|p$ in $F$ splits in $E$. Let $U$ be a unitary group (over $\mathbb{Q}$) in $n$ variables for the quadratic extension $E/F$ which is compact at infinity. By the hypothesis on $p$ the group $U_{\mathbb{Q}_p}$ is a product of general linear groups over finite extensions of $\mathbb{Q}_p$ and we denote $\mathcal{T}$ a maximal torus of $U_{\mathbb{Q}_p}$. We also fix a finite extension $L/\mathbb{Q}_p$ which is big enough to split $E$. Let $\mathcal{O}_L \subset L$ be its ring of integers, $\pi_L$ a uniformizer and $k_L$ its residue field.

For any continuous character $\delta : \mathcal{T}(\mathbb{Q}_p) \to L^\times$, we can define a weight $\kappa$ (which is given by the derivative of $\delta$ at 1) and a character of the Atkin-Lehner ring $\mathcal{A}(p)$ (the ring of Hecke-operators at $p$, see Definition 5.4) that we still denote by $\delta$. We will assume that $\delta|_{T^0}$ is algebraic where $T^0 \subset \mathcal{T}(\mathbb{Q}_p)$ is the maximal compact subgroup. Let $K^p \subset U(\mathbb{A}^p)$ be a tame level and let $S$ be a finite set, containing places above $p$, away from which $K^p$ is hyperspecial. We write $\mathbb{T}^S$ for the unramified Hecke algebra at places not in $S$ and $\mathbb{T} = \mathbb{T}^S \otimes_{\mathbb{Z}} \mathcal{A}(p)$. Associated to these data we consider the spaces $S_\kappa^\dagger(K^p)$ and $S_\kappa^\text{cl}(K^p)$, see Definition 5.7 for the precise definition, which come equipped with an action of $\mathbb{T}^S$ and $\mathcal{A}(p)$.
The space of overconvergent *p*-adic automorphic forms of finite slope $S^\dagger(K^p)_{\text{fs}}$ can be viewed as the topological dual of the global sections of a coherent sheaf (that we simply refer to as the *sheaf of *p*-adic automorphic forms*) on the rigid analytic generic fiber of the universal deformation space of Galois representations (more precisely, on the product of this space with the space of continuous characters of a maximal torus $\mathcal{T}(\mathbb{Q}_p) \subset \mathcal{G}(\mathbb{Q}_p)$ at *p*). The support of this sheaf is, by definition, the corresponding eigenvariety. The local-global-compatibility conjectures [EGH23, Conj. 9.6.8 and Conj. 9.6.16] give a precise description of this sheaf in terms of the geometry of moduli stacks of $(\varphi, \Gamma)$-modules (that are closely related to the trianguline variety). More precisely, the categorical approach to the *p*-adic Langlands program asks for a functor from certain (locally analytic) representations of $\mathcal{G}(\mathbb{Q}_p)$ to sheaves on stacks of $(\varphi, \Gamma)$-modules, and the sheaf of *p*-adic automorphic forms is the globalization of the evaluation of this functor on a specific representation. One of the punchlines of [EGH23] (see section 1.6 therein for a more detailed discussion) is that avatars of the envisioned functor have been around in number theory during the past decades in the context of the Taylor-Wiles patching method, in particular *patching functors* as used for example in [EGS15] (or also in [BHS19, 5.]) A crucial point in the proof of the main theorem is the identification of such a patching functor with an explicit local functor, see Theorem 1.4 below. This partially confirms expectations in the categorical picture, see [EGH23, Expectation 6.2.27].

Note that the multiplicity result in Theorem 1.2 has some striking consequence for the *p*-adic Langlands Program for $\text{GL}_3(\mathbb{Q}_p)$. It implies that the locally analytic representation of $\text{GL}_3(\mathbb{Q}_p)$ on the Hecke eigenspace of overconvergent *p*-adic modular forms over $\mathcal{G}$ corresponding to a Galois representation $\rho$ as in Theorem 1.2 contains locally analytic vectors which are *not* in the socle of the representation (see Remark 7.29). After finishing this works, the authors learned that Ding also proved examples of this penroma for generic Galois representations (see [Din]).

We now describe our results in more detail. Let $F$ be a totally real number field and let $E/F$ be a CM (imaginary) quadratic extension in which every place $v|p$ in $F$ splits in $E$. Let $U$ be a unitary group (over $\mathbb{Q}$) in $n$ variables for the quadratic extension $E/F$ which is compact at infinity. By the hypothesis on $p$ the group $U_{\mathbb{Q}_p}$ is a product of general linear groups over finite extensions of $\mathbb{Q}_p$ and we denote $\mathcal{T}$ a maximal torus of $U_{\mathbb{Q}_p}$. We also fix a finite extension $L/\mathbb{Q}_p$ which is big enough to split $E$. Let $\mathcal{O}_L \subset L$ be its ring of integers, $\pi_L$ a uniformizer and $k_L$ its residue field.

For any continuous character $\delta : \mathcal{T}(\mathbb{Q}_p) \to L^\times$, we can define a weight $\kappa$ (which is given by the derivative of $\delta$ at 1) and a character of the Atkin-Lehner ring $\mathcal{A}(p)$ (the ring of Hecke-operators at $p$, see Definition 5.4) that we still denote by $\delta$. We will assume that $\delta|_{T^0}$ is algebraic where $T^0 \subset \mathcal{T}(\mathbb{Q}_p)$ is the maximal compact subgroup. Let $K^p \subset U(\mathbb{A}^p)$ be a tame level and let $S$ be a finite set, containing places above $p$, away from which $K^p$ is hyperspecial. We write $\mathbb{T}^S$ for the unramified Hecke algebra at places not in $S$ and $\mathbb{T} = \mathbb{T}^S \otimes_{\mathbb{Z}} \mathcal{A}(p)$. Associated to these data we consider the spaces $S_\kappa^\dagger(K^p)$ and $S_\kappa^\text{cl}(K^p)$, see Definition 5.7 for the precise definition, which come equipped with an action of $\mathbb{T}^S$ and $\mathcal{A}(p)$.
Given a character $\chi^S: T^S \to L$ let $\chi = \chi^S \otimes \delta$ and consider the eigenspaces $S_\kappa^\dagger(K^p)[\chi]$ and $S_\kappa^{\text{cl}}(K^p)[\chi]$. We note that the classical subspace $S_\kappa^{\text{cl}}(K^p)[\chi]$ is zero unless $\kappa$ is dominant algebraic. To an eigenvector $f \in S_\kappa^\dagger(K^p)[\chi]$ we can associate a Galois representation $\rho = \rho_f = \rho_\chi: \text{Gal}_E := \text{Gal}(\bar{E}/E) \to \text{GL}_n(\mathbb{Q}_p)$. For the precise form of the main result we introduce the following (strong) Taylor-Wiles hypothesis. Let $\bar{\rho}: \text{Gal}_E \to \text{GL}_n(k_L)$ be the semisimplification of the reduction modulo the maximal ideal of $O_L$ of $\rho$. We assume that (see Hypothesis 5.9 in the text)

$$ \left\{ \begin{array}{ll} \bullet & p > 2, \\ \bullet & E/F \text{ is unramified and } \zeta_p \notin E, \\ \bullet & U \text{ is quasi-split at all finite places of } F, \\ \bullet & \text{if a place } v \text{ of } F \text{ is inert in } E, \text{ then } K_v \text{ is hyperspecial,} \\ \bullet & \bar{\rho} \text{ is absolutely irreducible and } \bar{\rho}(\text{Gal}_E(\zeta_p)) \text{ is adequate.} \end{array} \right. \qquad (1) $$

For simplicity of the exposition we assume now that that $p$ is totally split in $F$ (in the core of the paper we work in the general case). If the representation $\rho$ is crystalline at $v|p$, it can be described by its associated filtered isocrystal which is a finite dimensional $L$-vector space $D_{\text{cris}}(\rho_v)$ endowed with a linear automorphism $\varphi \in \text{GL}(D_{\text{cris}}(\rho_v))$ and a complete flag $D^\bullet$, called the Hodge-Tate filtration (in our case, this is a complete flag as $\rho_v$ has necessarily regular Hodge-Tate weights). We say that $\rho_v$ is $\varphi$-generic if the ratio of two of its eigenvalues is not in $\{1, p\}$. In this case the character $\delta$ determines an order of the eigenvalues of $\varphi$ (that is called a refinement of $\rho_v$) which in turn (using the fact that the $\varphi$-eigenvalues are pairwise distinct) defines another complete flag $\mathcal{F}_\bullet$ on $D_{\text{cris}}(\rho_v)$ which is $\varphi$-stable. We denote $w_{\rho,\delta,v} \in \mathfrak{S}_n$ the relative position of the flags $\mathcal{F}_\bullet$ and $D^\bullet$ in the flag variety of $D_{\text{cris}}(\rho_v)$. When $w_{\rho,\delta,v} = w_0$ is the longest element of $\mathfrak{S}_n$, i.e. when the two flags $D^\bullet$ and $\mathcal{F}_\bullet$ are in generic position, we say that $f$ is non-critical at $v$. The "most critical case" is the case where $w_{\rho,\delta,v} = 1$, i.e. when the two flags coincides. In this case we say that $f$ is very critical at $v$.

**Theorem 1.2.** Assume $n = 3$. Let $\delta: \mathcal{T}(\mathbb{Q}_p) \to L^\times$ be a continuous character of weight $\kappa$ dominant algebraic. Let $\chi^S: T^S \to L$ be a character and let $\chi = \chi^S \otimes \delta$. We assume that the eigenspace $S_\kappa^\dagger(K^p)[\chi]$ is non-zero and that for any $v|p$ the local Galois representation $\rho_v = \rho_\chi|_{\text{Gal}_{E_v}}: \text{Gal}_{E_v} \to \text{GL}_3(\mathbb{Q}_p)$ is crystalline with distinct Hodge-Tate weights and is $\varphi$-generic. Assume moreover that the Taylor-Wiles hypothesis (1) is satisfied. Let $r$ be the number of places $v|p$ in $F$ such that $w_{\rho_\chi,\delta,v} = 1$. Then

$$ \dim S_\kappa^\dagger(K^p)[\chi] = 2^r \dim S_\kappa^{\text{cl}}(K^p)[\chi]. $$

We refer to Corollary 7.25 for a more general statement where $p$ is not necessarily totally split in $F$.

Theorem 1.2 would be vacuous without proving the existence of characters $\chi$ and $\delta$ (and a group $U$ and a tame level $K^p$) such that the corresponding eigenspace $S_\kappa^{\text{cl}}(K^p)[\chi]$ is non-zero and consists of very critical forms. As there exist only countably many classical automorphic forms, but uncountably many flags it doesn't seem very easy to
Given a character $\chi^S: T^S \to L$ let $\chi = \chi^S \otimes \delta$ and consider the eigenspaces $S_\kappa^\dagger(K^p)[\chi]$ and $S_\kappa^{\text{cl}}(K^p)[\chi]$. We note that the classical subspace $S_\kappa^{\text{cl}}(K^p)[\chi]$ is zero unless $\kappa$ is dominant algebraic. To an eigenvector $f \in S_\kappa^\dagger(K^p)[\chi]$ we can associate a Galois representation $\rho = \rho_f = \rho_\chi: \text{Gal}_E := \text{Gal}(\bar{E}/E) \to \text{GL}_n(\mathbb{Q}_p)$. For the precise form of the main result we introduce the following (strong) Taylor-Wiles hypothesis. Let $\bar{\rho}: \text{Gal}_E \to \text{GL}_n(k_L)$ be the semisimplification of the reduction modulo the maximal ideal of $O_L$ of $\rho$. We assume that (see Hypothesis 5.9 in the text)

$$ \left\{ \begin{array}{ll} \bullet & p > 2, \\ \bullet & E/F \text{ is unramified and } \zeta_p \notin E, \\ \bullet & U \text{ is quasi-split at all finite places of } F, \\ \bullet & \text{if a place } v \text{ of } F \text{ is inert in } E, \text{ then } K_v \text{ is hyperspecial,} \\ \bullet & \bar{\rho} \text{ is absolutely irreducible and } \bar{\rho}(\text{Gal}_E(\zeta_p)) \text{ is adequate.} \end{array} \right. \qquad (1) $$

For simplicity of the exposition we assume now that that $p$ is totally split in $F$ (in the core of the paper we work in the general case). If the representation $\rho$ is crystalline at $v|p$, it can be described by its associated filtered isocrystal which is a finite dimensional $L$-vector space $D_{\text{cris}}(\rho_v)$ endowed with a linear automorphism $\varphi \in \text{GL}(D_{\text{cris}}(\rho_v))$ and a complete flag $D^\bullet$, called the Hodge-Tate filtration (in our case, this is a complete flag as $\rho_v$ has necessarily regular Hodge-Tate weights). We say that $\rho_v$ is $\varphi$-generic if the ratio of two of its eigenvalues is not in $\{1, p\}$. In this case the character $\delta$ determines an order of the eigenvalues of $\varphi$ (that is called a refinement of $\rho_v$) which in turn (using the fact that the $\varphi$-eigenvalues are pairwise distinct) defines another complete flag $\mathcal{F}_\bullet$ on $D_{\text{cris}}(\rho_v)$ which is $\varphi$-stable. We denote $w_{\rho,\delta,v} \in \mathfrak{S}_n$ the relative position of the flags $\mathcal{F}_\bullet$ and $D^\bullet$ in the flag variety of $D_{\text{cris}}(\rho_v)$. When $w_{\rho,\delta,v} = w_0$ is the longest element of $\mathfrak{S}_n$, i.e. when the two flags $D^\bullet$ and $\mathcal{F}_\bullet$ are in generic position, we say that $f$ is non-critical at $v$. The "most critical case" is the case where $w_{\rho,\delta,v} = 1$, i.e. when the two flags coincides. In this case we say that $f$ is very critical at $v$.

**Theorem 1.2.** Assume $n = 3$. Let $\delta: T(\mathbb{Q}_p) \to L^\times$ be a continuous character of weight $\kappa$ dominant algebraic. Let $\chi^S: T^S \to L$ be a character and let $\chi = \chi^S \otimes \delta$. We assume that the eigenspace $S_\kappa^\dagger(K^p)[\chi]$ is non-zero and that for any $v|p$ the local Galois representation $\rho_v = \rho_\chi|_{\text{Gal}_{E_v}}: \text{Gal}_{E_v} \to \text{GL}_3(\mathbb{Q}_p)$ is crystalline with distinct Hodge-Tate weights and is $\varphi$-generic. Assume moreover that the Taylor-Wiles hypothesis (1) is satisfied. Let $r$ be the number of places $v|p$ in $F$ such that $w_{\rho_\chi,\delta,v} = 1$. Then

$$ \dim S_\kappa^\dagger(K^p)[\chi] = 2^r \dim S_\kappa^{\text{cl}}(K^p)[\chi]. $$

We refer to Corollary 7.25 for a more general statement where $p$ is not necessarily totally split in $F$.

Theorem 1.2 would be vacuous without proving the existence of characters $\chi$ and $\delta$ (and a group $U$ and a tame level $K^p$) such that the corresponding eigenspace $S_\kappa^{\text{cl}}(K^p)[\chi]$ is non-zero and consists of very critical forms. As there exist only countably many classical automorphic forms, but uncountably many flags it doesn't seem very easy to
construct an $f$ with $w_{\rho_f, \delta} = 1$. This is Corollary 8.13, the main result of section 8, which uses global automorphic methods that are rather disjoint from the methods of the other parts of the paper. The Galois representation corresponding to the constructed Hecke character is induced from a degree 3 extension of $E$.

We finally discuss the relation of these results with patching functors and the categorical approach to a p-adic Langlands correspondence. Assume that $\delta = \delta_\lambda \delta_R^{\text{sm}}$ is the product of a dominant algebraic character $\delta_\lambda$ and a smooth unramified character $\delta_R^{\text{sm}}$ (which is in fact implied by the assumption that $\rho_v$ is crystalline). As the notation suggests, the character $\delta_R^{\text{sm}}$ corresponds to the choice of a refinement $\mathcal{R}$ of $\rho_p := (\rho_v)_{v|p}$. Let $\mathcal{X}_{\rho_p} = \text{Spec}(R_{\rho_p})$ be the scheme associated to the universal deformation ring of $\rho_p$. Using results of [BHS19], we can construct a subscheme

$$ \mathcal{X}_{\rho_p, \mathcal{R}}^{\text{qtri}} = \text{Spec}(R_{\rho_p, \mathcal{R}}^{\text{qtri}}) \subset \mathcal{X}_{\rho_p} $$

of “quasi-trianguline” deformations of $\rho_p$ associated to the refinement $\mathcal{R}$. By loc. cit. this scheme has a local model modeled on the Steinberg variety (or rather its “Grothendieck–Springer” variant) and its irreducible components $\mathcal{X}_{\rho, \mathcal{R}}^{\text{qtri},w}$ are labeled by the Weyl group $W$ of $\prod_{v|p} \text{GL}_3$. It is known that these irreducible components are normal and Cohen–Macaulay.

Let's denote $\lambda = \delta|_{T^0}(= \delta_{\lambda|_{T^0}})$, this is a dominant algebraic character. Using hypothesis (1) the Taylor–Wiles method, as extended to the setting of completed cohomology in [CEG$^+$16], can be used ([BHS19, 5.] to construct coherent sheaves $\mathcal{M}_\infty(L(\lambda))$ and $\mathcal{M}_\infty(M(w \cdot \lambda))$ for $w \in W$ over $\mathcal{X}_{\infty, \rho, \mathcal{R}}^{\text{qtri}} = \text{Spec}(R_{\rho_p, \mathcal{R}}^{\text{qtri}}[[x_1, \dots, x_g]])$ for some $g \ge 0$, that “patch” the duals of the spaces of classical, respectively p-adic, automorphic forms. More precisely

$$ \begin{aligned} \mathcal{M}_\infty(L(\lambda)) \otimes k(\rho_p) &= \text{Hom}_L(S_\lambda^{\text{cl}}(K^p)[\chi], L), \\ \mathcal{M}_\infty(M(w \cdot \lambda)) \otimes k(\rho_p) &= \text{Hom}_L(S_{w \cdot \lambda}^{\dagger}(K^p)[\chi], L). \end{aligned} $$

These coherent sheaves are in a certain precise sense associated to the $U(\mathfrak{g})$-modules $L(\lambda)$ (the algebraic representation of highest weight $\lambda$) respectively the Verma modules $M(w \cdot \lambda)$, where $\mathfrak{g}$ is the Lie algebra of $U_L \cong \prod_{v|p} \text{GL}_3$. The results of [BHS19] show that the coherent sheaves $\mathcal{M}_\infty(M(w \cdot \lambda))$ have generic rank (when nonzero) equal to $\dim_L S_\lambda^{\text{cl}}(K^p)[\chi]$. Denote $\mathcal{X}_{\infty, \rho, \mathcal{R}}^{\text{qtri},w} := \mathcal{X}_{\infty, \rho, \mathcal{R}}^{\text{qtri}} \times_{\mathcal{X}_{\rho, \mathcal{R}}^{\text{qtri}}} \mathcal{X}_{\rho, \mathcal{R}}^{\text{qtri},w}$. The key to the proof of Theorem 1.2 is the following result:

**Theorem 1.3.** Under the assumptions of Theorem 1.2, let $m = \dim_L S_\lambda^{\text{cl}}[\chi]$. For any $w \in W$, there is an isomorphism

$$ \mathcal{M}_\infty(M(w \cdot \lambda)) \cong \omega_{\overline{\mathcal{X}}_{\infty, \rho, \mathcal{R}}^{\text{qtri}, w w_0}}^{\oplus m}. $$

Here $\omega_{\overline{\mathcal{X}}_{\infty, \rho, \mathcal{R}}^{\text{qtri}, w w_0}}$ is the dualizing sheaf of a complete intersection $\overline{\mathcal{X}}_{\infty, \rho, \mathcal{R}}^{\text{qtri}, w w_0} \subset \mathcal{X}_{\infty, \rho, \mathcal{R}}^{\text{qtri}, w w_0}$.
construct an $f$ with $w_{\rho_f, \delta} = 1$. This is Corollary 8.13, the main result of section 8, which uses global automorphic methods that are rather disjoint from the methods of the other parts of the paper. The Galois representation corresponding to the constructed Hecke character is induced from a degree 3 extension of $E$.

We finally discuss the relation of these results with patching functors and the categorical approach to a p-adic Langlands correspondence. Assume that $\delta = \delta_\lambda \delta_R^{\text{sm}}$ is the product of a dominant algebraic character $\delta_\lambda$ and a smooth unramified character $\delta_R^{\text{sm}}$ (which is in fact implied by the assumption that $\rho_v$ is crystalline). As the notation suggests, the character $\delta_R^{\text{sm}}$ corresponds to the choice of a refinement $\mathcal{R}$ of $\rho_p := (\rho_v)_{v|p}$. Let $\mathcal{X}_{\rho_p} = \text{Spec}(R_{\rho_p})$ be the scheme associated to the universal deformation ring of $\rho_p$. Using results of [BHS19], we can construct a subscheme

$$ \mathcal{X}_{\rho_p, \mathcal{R}}^{\text{qtri}} = \text{Spec}(R_{\rho_p, \mathcal{R}}^{\text{qtri}}) \subset \mathcal{X}_{\rho_p} $$

of “quasi-trianguline” deformations of $\rho_p$ associated to the refinement $\mathcal{R}$. By loc. cit. this scheme has a local model modeled on the Steinberg variety (or rather its “Grothendieck–Springer” variant) and its irreducible components $\mathcal{X}_{\rho, \mathcal{R}}^{\text{qtri},w}$ are labeled by the Weyl group $W$ of $\prod_{v|p} \text{GL}_3$. It is known that these irreducible components are normal and Cohen–Macaulay.

Let's denote $\lambda = \delta|_{T^0}(= \delta_{\lambda|_{T^0}})$, this is a dominant algebraic character. Using hypothesis (1) the Taylor–Wiles method, as extended to the setting of completed cohomology in [CEG$^+$16], can be used ([BHS19, 5.] to construct coherent sheaves $\mathcal{M}_\infty(L(\lambda))$ and $\mathcal{M}_\infty(M(w \cdot \lambda))$ for $w \in W$ over $\mathcal{X}_{\infty, \rho, \mathcal{R}}^{\text{qtri}} = \text{Spec}(R_{\rho_p, \mathcal{R}}^{\text{qtri}}[[x_1, \dots, x_g]])$ for some $g \ge 0$, that “patch” the duals of the spaces of classical, respectively p-adic, automorphic forms. More precisely

$$ \begin{aligned} \mathcal{M}_\infty(L(\lambda)) \otimes k(\rho_p) &= \text{Hom}_L(S_\lambda^{\text{cl}}(K^p)[\chi], L), \\ \mathcal{M}_\infty(M(w \cdot \lambda)) \otimes k(\rho_p) &= \text{Hom}_L(S_{w \cdot \lambda}^{\dagger}(K^p)[\chi], L). \end{aligned} $$

These coherent sheaves are in a certain precise sense associated to the $U(\mathfrak{g})$-modules $L(\lambda)$ (the algebraic representation of highest weight $\lambda$) respectively the Verma modules $M(w \cdot \lambda)$, where $\mathfrak{g}$ is the Lie algebra of $U_L \cong \prod_{v|p} \text{GL}_3$. The results of [BHS19] show that the coherent sheaves $\mathcal{M}_\infty(M(w \cdot \lambda))$ have generic rank (when nonzero) equal to $\dim_L S_\lambda^{\text{cl}}(K^p)[\chi]$. Denote $\mathcal{X}_{\infty, \rho, \mathcal{R}}^{\text{qtri},w} := \mathcal{X}_{\infty, \rho, \mathcal{R}}^{\text{qtri}} \times_{\mathcal{X}_{\rho, \mathcal{R}}^{\text{qtri}}} \mathcal{X}_{\rho, \mathcal{R}}^{\text{qtri},w}$. The key to the proof of Theorem 1.2 is the following result:

**Theorem 1.3.** Under the assumptions of Theorem 1.2, let $m = \dim_L S_\lambda^{\text{cl}}[\chi]$. For any $w \in W$, there is an isomorphism

$$ \mathcal{M}_\infty(M(w \cdot \lambda)) \cong \omega_{\overline{\mathcal{X}}_{\infty, \rho, \mathcal{R}}^{\text{qtri}, w w_0}}^{\oplus m}. $$

Here $\omega_{\overline{\mathcal{X}}_{\infty, \rho, \mathcal{R}}^{\text{qtri}, w w_0}}$ is the dualizing sheaf of a complete intersection $\overline{\mathcal{X}}_{\infty, \rho, \mathcal{R}}^{\text{qtri}, w w_0} \subset \mathcal{X}_{\infty, \rho, \mathcal{R}}^{\text{qtri}, w w_0}$.
In order to prove Theorem 1.3, we extend $\mathcal{M}_\infty$ to a functor on the whole category $\mathcal{O}_\lambda$, the block of the BGG category $\mathcal{O}$ containing $L(\lambda)$. This is the *patching functor* alluded to above. More precisely, assuming that $\rho_p$ is crystalline with regular Hodge-Tate weights, and $\delta$ is $\varphi$-generic, we construct an exact functor

$$ \mathcal{M}_{\infty} : \mathcal{O}_{\lambda} \to \mathrm{Coh}(\mathcal{X}_{\infty, \rho, \mathcal{R}}^{\mathrm{qtri}}), $$

such that, for every $M \in \mathcal{O}_\lambda$ the sheaf $\mathcal{M}_\infty(M)$ is Cohen-Macaulay of the expected dimension.

In spirit of the categorical approach to the p-adic Langlands correspondence the functor $\mathcal{M}_\infty$ should be a “local” functor, that is (up to multiplicities coming from contributions at the places away from $p$) the functor $\mathcal{M}_\infty$ should be the pullback, denoted $\mathcal{B}_\infty$, of a functor

$$ \mathcal{B}_p : \mathcal{O}_\lambda \to \mathrm{Coh}(\mathcal{X}_{\rho_p, \mathcal{R}}^{\mathrm{qtri}}). $$

This functor $\mathcal{B}_p$ can be written down explicitly using the local model for $\mathcal{X}_{\rho_p, \mathcal{R}}^{\mathrm{qtri}}$ and a functor constructed by Bezrukavnikov [Bez16], see 7.2 for details. Our main local result compares $\mathcal{M}_\infty$ and $\mathcal{B}_p$ (see Corollary 7.23 for the general version):

**Theorem 1.4.** Under the assumptions of Theorem 1.2, let $m = \dim_L S_\lambda^{\mathrm{cl}}[\chi]$. Then there is an isomorphism of functors $\mathcal{M}_\infty \simeq \mathcal{B}_\infty^{\oplus m}$. As a consequence, we have

$$ 1) \text{ for all } w \in W, \mathcal{M}_\infty(M(w \cdot \lambda)^\vee) \simeq \mathcal{O}_{\mathcal{X}_{\infty, \rho, \mathcal{R}}^{\mathrm{qtri}, ww_0}}^{\oplus m} ; $$

$$ 2) \text{ for all } w \in W, \mathcal{M}_\infty(M(w \cdot \lambda)) \simeq \omega_{\mathcal{X}_{\infty, \rho, \mathcal{R}}^{\mathrm{qtri}, ww_0}}^{\oplus m} ; $$

3) for all $M \in \mathcal{O}$, we have $\mathcal{M}_\infty(M^\vee) \simeq \mathcal{M}_\infty(M)^\vee$ where $(\cdot)^\vee$ denote both the dual in $\mathcal{O}_\lambda$ and the Serre dual in the category of coherent sheaves.

*Remark 1.5.* We can only prove Theorem 1.4 in the three dimensional case. However, we expect an isomorphism $\mathcal{M}_\infty \simeq \mathcal{B}_\infty^{\oplus m}$ for higher dimensional definite unitary groups as well.

In fact $\mathcal{B}_p$ should factor through the category of locally analytic representations, and is expected to extend to a functor with values in coherent sheaves on the stack of all $(\varphi, \Gamma)$-modules (compare [EGH23, Conjecture 6.2.4 and Expectation 6.2.27]). Theorem 1.4 should be viewed as some partial evidence for these expectations.

The key to proving Theorem 1.4 is to extend the functor $\mathcal{M}_\infty$ to a larger category $\mathcal{O}_{\mathrm{alg}}^\infty$ and to a deformation $\tilde{\mathcal{O}}_{\mathrm{alg}}$ as introduced in [Soe92], which we think of as a *deformed version* of $\mathcal{O}_{\mathrm{alg}}$. We would like to emphasize that we first prove 1) and we deduce the isomorphism $\mathcal{M}_\infty \simeq \mathcal{B}_\infty^{\oplus m}$ from this in a second time. The proof of 1) is based on a *dévissage* whose has its origin in the paper [EGS15]. We first prove the result in the case where $\mathcal{X}_{\infty, \mathcal{R}}^{\mathrm{qtri}, w}$ is smooth and then proceed inductively. Note that the *existence* of Bezrukavnikov’s functor $\mathcal{B}_\infty$ plays a key role in this induction. The second main input
In order to prove Theorem 1.3, we extend $\mathcal{M}_\infty$ to a functor on the whole category $\mathcal{O}_\lambda$, the block of the BGG category $\mathcal{O}$ containing $L(\lambda)$. This is the *patching functor* alluded to above. More precisely, assuming that $\rho_p$ is crystalline with regular Hodge-Tate weights, and $\delta$ is $\varphi$-generic, we construct an exact functor

$$ \mathcal{M}_{\infty} : \mathcal{O}_{\lambda} \to \mathrm{Coh}(\mathcal{X}_{\infty, \rho, \mathcal{R}}^{\mathrm{qtri}}), $$

such that, for every $M \in \mathcal{O}_\lambda$ the sheaf $\mathcal{M}_\infty(M)$ is Cohen-Macaulay of the expected dimension.

In spirit of the categorical approach to the p-adic Langlands correspondence the functor $\mathcal{M}_\infty$ should be a “local” functor, that is (up to multiplicities coming from contributions at the places away from $p$) the functor $\mathcal{M}_\infty$ should be the pullback, denoted $\mathcal{B}_\infty$, of a functor

$$ \mathcal{B}_p : \mathcal{O}_\lambda \to \mathrm{Coh}(\mathcal{X}_{\rho_p, \mathcal{R}}^{\mathrm{qtri}}). $$

This functor $\mathcal{B}_p$ can be written down explicitly using the local model for $\mathcal{X}_{\rho_p, \mathcal{R}}^{\mathrm{qtri}}$ and a functor constructed by Bezrukavnikov [Bez16], see 7.2 for details. Our main local result compares $\mathcal{M}_\infty$ and $\mathcal{B}_p$ (see Corollary 7.23 for the general version):

**Theorem 1.4.** Under the assumptions of Theorem 1.2, let $m = \dim_L S_\lambda^{\mathrm{cl}}[\chi]$. Then there is an isomorphism of functors $\mathcal{M}_\infty \simeq \mathcal{B}_\infty^{\oplus m}$. As a consequence, we have

$$ \text{1) for all } w \in W, \mathcal{M}_\infty(M(w \cdot \lambda)^\vee) \simeq \mathcal{O}_{\mathcal{X}_{\infty, \rho, \mathcal{R}}^{\mathrm{qtri}, ww_0}}^{\oplus m} ; $$

$$ \text{2) for all } w \in W, \mathcal{M}_\infty(M(w \cdot \lambda)) \simeq \omega_{\mathcal{X}_{\infty, \rho, \mathcal{R}}^{\mathrm{qtri}, ww_0}}^{\oplus m} ; $$

3) for all $M \in \mathcal{O}$, we have $\mathcal{M}_\infty(M^\vee) \simeq \mathcal{M}_\infty(M)^\vee$ where $(\cdot)^\vee$ denote both the dual in $\mathcal{O}_\lambda$ and the Serre dual in the category of coherent sheaves.

*Remark 1.5.* We can only prove Theorem 1.4 in the three dimensional case. However, we expect an isomorphism $\mathcal{M}_\infty \simeq \mathcal{B}_\infty^{\oplus m}$ for higher dimensional definite unitary groups as well.

In fact $\mathcal{B}_p$ should factor through the category of locally analytic representations, and is expected to extend to a functor with values in coherent sheaves on the stack of all $(\varphi, \Gamma)$-modules (compare [EGH23, Conjecture 6.2.4 and Expectation 6.2.27]). Theorem 1.4 should be viewed as some partial evidence for these expectations.

The key to proving Theorem 1.4 is to extend the functor $\mathcal{M}_\infty$ to a larger category $\mathcal{O}_{\mathrm{alg}}^\infty$ and to a deformation $\tilde{\mathcal{O}}_{\mathrm{alg}}$ as introduced in [Soe92], which we think of as a *deformed version* of $\mathcal{O}_{\mathrm{alg}}$. We would like to emphasize that we first prove 1) and we deduce the isomorphism $\mathcal{M}_\infty \simeq \mathcal{B}_\infty^{\oplus m}$ from this in a second time. The proof of 1) is based on a *dévissage* whose has its origin in the paper [EGS15]. We first prove the result in the case where $\mathcal{X}_{\infty, \mathcal{R}}^{\mathrm{qtri}, w}$ is smooth and then proceed inductively. Note that the *existence* of Bezrukavnikov’s functor $\mathcal{B}_\infty$ plays a key role in this induction. The second main input
into this induction is the computation of $\mathcal{M}_\infty(M_I(w\cdot\lambda))$ where $M_I(w\cdot\lambda)$ is a generalized Verma module (corresponding to some parabolic $P_I$). These sheaves, that are related to sheaves of $p$-adic automorphic forms on the partial eigenvarieties constructed by Wu [Wu], are supported on "partially de Rham quasi-trianguline" deformation spaces $X_{\rho_p, \mathcal{R}}^{I-q\text{tri}}$ which have been studied by Breuil and Ding in [BD].

We finally note that the component $X_{\infty, \rho, \mathcal{R}}^{\text{qtri}, w_0}$ is not Gorenstein and its dualizing sheaf has a $2^r$-dimensional fiber at $\rho_p$, which is the reason for the factor $2^r$ in Theorem 1.2.

We now describe the content of the article. In section 2 we introduce the category $\mathcal{O}_{\text{alg}}$ and its deformed versions. Section 3 studies Emerton's Jacquet functor and gives the abstract framework to construct patching functors. In section 4, we recall the quasi-trianguline deformation spaces of [BHS19], their local models, and their parabolic version ([BD, Wu]). Section 5 recalls the definitions of the global objects like completed cohomology, overconvergent automorphic forms and their patched versions. Section 6 is devoted to the further study of the functor $\mathcal{M}_\infty$ and its factorization through $X_{\infty, \rho, \mathcal{R}}^{\text{qtri}}$, the (global) quasi-trianguline deformation space. In section 7, we study the supports of the sheaves $\mathcal{M}_\infty(M)$ for specific objects of $\mathcal{O}_{\text{alg}}$ (and their deformed version), and we recall results on Bezrukavnikov's functor before deducing Theorem 1.4 (in the three dimensional case). Finally, in section 8 we explain how to explicitly construct very critical forms satisfying the assumptions in Theorem 1.2 for $n=3$.

## Acknowledgements

We are very grateful to Roman Bezrukavnikov, George Boxer, Christophe Breuil, Matthew Emerton, Toby Gee, Arthur-Cesar Le Bras, Simon Riche and Olivier Schiffmann for discussions related to this work. We also would like to thank the team maintaining the software Macaulay2 [GS] which has been very useful to make preliminary computations related to this work. E.H. was supported by Germany's Excellence Strategy EXC 2044-390685587 "Mathematics Münster: Dynamics-Geometry-Structure" and by the CRC 1442 "Geometry: Deformations and Rigidity" of the DFG. Part of this work was done during the Trimester Program 2023 on the arithmetic of the Langlands Program in HIM Bonn where the three authors were invited.

## Notations

Let $p$ be a prime number. When $K$ is a field, we write $\text{Gal}_K = \text{Gal}(K^{\text{sep}}/K)$ for its absolute Galois group. We fix $L$ a finite extension of $\mathbb{Q}_p$ which will be chosen sufficiently large in the text.
into this induction is the computation of $\mathcal{M}_\infty(M_I(w\cdot\lambda))$ where $M_I(w\cdot\lambda)$ is a generalized Verma module (corresponding to some parabolic $P_I$). These sheaves, that are related to sheaves of $p$-adic automorphic forms on the partial eigenvarieties constructed by Wu [Wu], are supported on "partially de Rham quasi-trianguline" deformation spaces $X_{\rho_p, \mathcal{R}}^{I-q\text{tri}}$ which have been studied by Breuil and Ding in [BD].

We finally note that the component $X_{\infty, \rho, \mathcal{R}}^{q\text{tri}, w_0}$ is not Gorenstein and its dualizing sheaf has a $2^r$-dimensional fiber at $\rho_p$, which is the reason for the factor $2^r$ in Theorem 1.2.

We now describe the content of the article. In section 2 we introduce the category $\mathcal{O}_{\text{alg}}$ and its deformed versions. Section 3 studies Emerton's Jacquet functor and gives the abstract framework to construct patching functors. In section 4, we recall the quasi-trianguline deformation spaces of [BHS19], their local models, and their parabolic version ([BD, Wu]). Section 5 recalls the definitions of the global objects like completed cohomology, overconvergent automorphic forms and their patched versions. Section 6 is devoted to the further study of the functor $\mathcal{M}_\infty$ and its factorization through $X_{\infty, \rho, \mathcal{R}}^{q\text{tri}}$, the (global) quasi-trianguline deformation space. In section 7, we study the supports of the sheaves $\mathcal{M}_\infty(M)$ for specific objects of $\mathcal{O}_{\text{alg}}$ (and their deformed version), and we recall results on Bezrukavnikov's functor before deducing Theorem 1.4 (in the three dimensional case). Finally, in section 8 we explain how to explicitly construct very critical forms satisfying the assumptions in Theorem 1.2 for $n=3$.

## Acknowledgements

We are very grateful to Roman Bezrukavnikov, George Boxer, Christophe Breuil, Matthew Emerton, Toby Gee, Arthur-Cesar Le Bras, Simon Riche and Olivier Schiffmann for discussions related to this work. We also would like to thank the team maintaining the software Macaulay2 [GS] which has been very useful to make preliminary computations related to this work. E.H. was supported by Germany's Excellence Strategy EXC 2044-390685587 "Mathematics Münster: Dynamics-Geometry-Structure" and by the CRC 1442 "Geometry: Deformations and Rigidity" of the DFG. Part of this work was done during the Trimester Program 2023 on the arithmetic of the Langlands Program in HIM Bonn where the three authors were invited.

## Notations

Let $p$ be a prime number. When $K$ is a field, we write $\text{Gal}_K = \text{Gal}(K^{\text{sep}}/K)$ for its absolute Galois group. We fix $L$ a finite extension of $\mathbb{Q}_p$ which will be chosen sufficiently large in the text.
## 2 Variants of the BGG-category $\mathcal{O}$

In this section, we fix $L$ to be a field of characteristic 0. Let $\mathcal{G}$ be a split reductive group over $L$. Let $\mathcal{B}$ be a Borel subgroup, $\mathcal{T}$ a maximal split torus of $\mathcal{G}$ contained in $\mathcal{B}$ and $\mathcal{N}$ the radical of $\mathcal{B}$. We use the notation $\mathfrak{g}$, $\mathfrak{b}$, $\mathfrak{t}$, $\mathfrak{n}$... for the Lie algebras of $\mathcal{G}$, $\mathcal{B}$, $\mathcal{T}$, $\mathcal{N}$... We denote by $X^*(\mathcal{T})$ the finite free abelian group $\text{Hom}(\mathcal{T}, \mathbb{G}_{m,L})$ of characters of $\mathcal{T}$. This abelian group can be identified with a $\mathbb{Z}$-lattice in $\mathfrak{t}^* := \text{Hom}_L(\mathfrak{t}, L)$. For $\lambda \in X^*(\mathcal{T})$, we also write $\lambda$ for the character of $\mathfrak{t}$ induced by $\lambda$. Let $\Phi$ be the set of roots of the pair $(\mathcal{G}, \mathcal{T})$ and let $\Phi^+ \subset \Phi$ be the subset of positive roots with respect to $\mathcal{B}$ and $\Delta \subset \Phi^+$ the subset of simple roots. As usual we write $\delta_G \in X^*(\mathcal{T}) \otimes_{\mathbb{Z}} \mathbb{Q}$ for the half sum of positive roots. Let $W$ be the Weyl group of $(\mathcal{G}, \mathcal{T})$. For $w \in W$, we write $\lambda \mapsto w \cdot \lambda$ for the dot action of $W$ on $X^*(\mathcal{T})$ (with respect to $\mathcal{B}$, that is $w \cdot \lambda := w(\lambda + \delta_G) - \delta_G$). We equip $W$ with the Bruhat order corresponding to the choice of $\mathcal{B}$ and we denote $w_0 \in W$ the longest element for this order.

If $I \subset \Delta$ is a subset of simple roots, we denote by $\Phi_I \subset \Phi$ the subset of roots which are sums of elements of $I$ and $\mathcal{P}_I \supset \mathcal{B}$ be the standard parabolic subgroup of $\mathcal{G}$ such that $\mathfrak{p}_I = \mathfrak{b} + \sum_{\alpha \in \Phi_I} \mathfrak{g}_\alpha$. Let $\mathcal{L}_I$ be the standard Levi subgroup of $\mathcal{P}_I$ and $\mathcal{Z}_I$ be the center of $\mathcal{L}_I$. We say that a character $\lambda \in X^*(\mathcal{T})$ is dominant with respect to $\mathcal{P}_I$ if $\langle \lambda, \alpha^\vee \rangle \ge 0$ for $\alpha \in I$ and we denote $X^*(\mathcal{T})_I^+$ the set of such characters. When $I = \Delta$, we have $\mathcal{P}_\Delta = \mathcal{G}$ and we write $X^*(\mathcal{T})^+ = X^*(\mathcal{T})_\Delta^+$. We use the following order relation on $X^*(\mathcal{T})$, we say that $\lambda \ge \mu$ if and only if $\lambda - \mu \in \sum_{\alpha \in \Phi^+} \mathbb{N}\alpha$.

We write $W_I$ for the Weyl group of the Levi $\mathcal{L}_I$ of $\mathcal{P}_I$; it is the subgroup of $W$ generated by the simple reflexions $s_\alpha$ for $\alpha \in I$. Given $w \in W$, we denote $w^{\min}$ (resp. $w^{\max}$) the unique minimal (resp. maximal) element for the Bruhat order having the same class as $w$ in $W_I \setminus W$. This definition depends on $I$ (and on the fact that the quotient is on the left) but we hope our notation will cause no confusion. As usual, we write $w_0 \in W$ for the longest element in $W$. Then we have $(ww_0)^{\min} = w^{\max}w_0$ and $(ww_0)^{\max} = w^{\min}w_0$ for any $w \in W$. Finally, we write $IW$ for the set of minimal length representatives of $W_I \setminus W$ in $W$.

If $\mathfrak{h}$ is a Lie algebra we note $\mathfrak{h}^{\text{ss}}$ its derived Lie algebra.

### 2.1 Recollections

For $I \subset \Delta$, we consider the full subcategory $\mathcal{O}^{I,\infty}$ of the category $U(\mathfrak{g})$-mod of $U(\mathfrak{g})$-modules that consists of all finitely generated $U(\mathfrak{g})$-modules $M$ such that

* for any $m \in M$, the $L$-vector space $U(\mathfrak{p}_I)m$ is finite dimensional;

* for any $h \in \mathfrak{t}$ and any $h$-stable finite dimensional $L$-vector subspace $V \subset M$, the characteristic polynomial of $h|_V$ is split in $L[X]$.

This is the category $\mathcal{O}^{\mathfrak{p}_I, \infty}$ in [AS22, §3.1].
## 2 Variants of the BGG-category $\mathcal{O}$

In this section, we fix $L$ to be a field of characteristic 0. Let $\mathcal{G}$ be a split reductive group over $L$. Let $\mathcal{B}$ be a Borel subgroup, $\mathcal{T}$ a maximal split torus of $\mathcal{G}$ contained in $\mathcal{B}$ and $\mathcal{N}$ the radical of $\mathcal{B}$. We use the notation $\mathfrak{g}$, $\mathfrak{b}$, $\mathfrak{t}$, $\mathfrak{n}$... for the Lie algebras of $\mathcal{G}$, $\mathcal{B}$, $\mathcal{T}$, $\mathcal{N}$... We denote by $X^*(\mathcal{T})$ the finite free abelian group $\text{Hom}(\mathcal{T}, \mathbb{G}_{m,L})$ of characters of $\mathcal{T}$. This abelian group can be identified with a $\mathbb{Z}$-lattice in $\mathfrak{t}^* := \text{Hom}_L(\mathfrak{t}, L)$. For $\lambda \in X^*(\mathcal{T})$, we also write $\lambda$ for the character of $\mathfrak{t}$ induced by $\lambda$. Let $\Phi$ be the set of roots of the pair $(\mathcal{G}, \mathcal{T})$ and let $\Phi^+ \subset \Phi$ be the subset of positive roots with respect to $\mathcal{B}$ and $\Delta \subset \Phi^+$ the subset of simple roots. As usual we write $\delta_G \in X^*(\mathcal{T}) \otimes_{\mathbb{Z}} \mathbb{Q}$ for the half sum of positive roots. Let $W$ be the Weyl group of $(\mathcal{G}, \mathcal{T})$. For $w \in W$, we write $\lambda \mapsto w \cdot \lambda$ for the dot action of $W$ on $X^*(\mathcal{T})$ (with respect to $\mathcal{B}$, that is $w \cdot \lambda := w(\lambda + \delta_G) - \delta_G$). We equip $W$ with the Bruhat order corresponding to the choice of $\mathcal{B}$ and we denote $w_0 \in W$ the longest element for this order.

If $I \subset \Delta$ is a subset of simple roots, we denote by $\Phi_I \subset \Phi$ the subset of roots which are sums of elements of $I$ and $\mathcal{P}_I \supset \mathcal{B}$ be the standard parabolic subgroup of $\mathcal{G}$ such that $\mathfrak{p}_I = \mathfrak{b} + \sum_{\alpha \in \Phi_I} \mathfrak{g}_{\alpha}$. Let $\mathcal{L}_I$ be the standard Levi subgroup of $\mathcal{P}_I$ and $\mathcal{Z}_I$ be the center of $\mathcal{L}_I$. We say that a character $\lambda \in X^*(\mathcal{T})$ is dominant with respect to $\mathcal{P}_I$ if $\langle \lambda, \alpha^{\vee} \rangle \ge 0$ for $\alpha \in I$ and we denote $X^*(\mathcal{T})_I^+$ the set of such characters. When $I = \Delta$, we have $\mathcal{P}_{\Delta} = \mathcal{G}$ and we write $X^*(\mathcal{T})^+ = X^*(\mathcal{T})_{\Delta}^+$. We use the following order relation on $X^*(\mathcal{T})$, we say that $\lambda \ge \mu$ if and only if $\lambda - \mu \in \sum_{\alpha \in \Phi^+} \mathbb{N}\alpha$.

We write $W_I$ for the Weyl group of the Levi $\mathcal{L}_I$ of $\mathcal{P}_I$; it is the subgroup of $W$ generated by the simple reflexions $s_{\alpha}$ for $\alpha \in I$. Given $w \in W$, we denote $w^{\min}$ (resp. $w^{\max}$) the unique minimal (resp. maximal) element for the Bruhat order having the same class as $w$ in $W_I \setminus W$. This definition depends on $I$ (and on the fact that the quotient is on the left) but we hope our notation will cause no confusion. As usual, we write $w_0 \in W$ for the longest element in $W$. Then we have $(ww_0)^{\min} = w^{\max}w_0$ and $(ww_0)^{\max} = w^{\min}w_0$ for any $w \in W$. Finally, we write $IW$ for the set of minimal length representatives of $W_I \setminus W$ in $W$.

If $\mathfrak{h}$ is a Lie algebra we note $\mathfrak{h}^{\text{ss}}$ its derived Lie algebra.

### 2.1 Recollections

For $I \subset \Delta$, we consider the full subcategory $\mathcal{O}^{I,\infty}$ of the category $U(\mathfrak{g})$-mod of $U(\mathfrak{g})$-modules that consists of all finitely generated $U(\mathfrak{g})$-modules $M$ such that

* for any $m \in M$, the $L$-vector space $U(\mathfrak{p}_I)m$ is finite dimensional;

* for any $h \in \mathfrak{t}$ and any $h$-stable finite dimensional $L$-vector subspace $V \subset M$, the characteristic polynomial of $h|_V$ is split in $L[X]$.

This is the category $\mathcal{O}^{\mathfrak{p}_I, \infty}$ in [AS22, §3.1].
For $\mu \in \operatorname{Hom}_L(t, L)$, we write $M^\mu \subset M$ for the $L$-subspace of those $v \in M$ such that, for any $h \in t$, $(h - \mu(h))^n \cdot v = 0$ for some $n \ge 1$. We have

$$M = \bigoplus_{\mu \in \operatorname{Hom}_L(t, L)} M^{\mu}.$$

We write $O_{\mathrm{alg}}^{I,\infty}$ for the full subcategory of $O^{I,\infty}$ whose objects $M$ satisfy $M^\mu = 0$ for $\mu \notin X^*(\mathcal{T})$.

Moreover, we write $O_{\mathrm{alg}}^I \subset O_{\mathrm{alg}}^{I,\infty}$ for the full subcategory whose objects are direct sums of finitely generated semisimple $U(l_I)$-modules (when seen as $U(l_I)$-modules). This coincides with the usual parabolic (algebraic) category $\mathcal{O}$, which is denoted $O_{\mathrm{alg}}^{p_I}$ in [OS15]). When $I = \emptyset$ we simply use the notations $O_{\mathrm{alg}}^\infty$ and $O_{\mathrm{alg}}^{\emptyset,\infty}$ and $O_{\mathrm{alg}}^\emptyset$. Note that $O_{\mathrm{alg}}^{I,\infty} \subset O_{\mathrm{alg}}^\infty$ for any $I \subset \Delta$. As these categories depend on the choices of $\mathfrak{g}$ and $\mathfrak{b}$ we write $O_{\mathrm{alg}}^{\mathfrak{g},\mathfrak{b}}$ (with additional decorations) instead of $\mathcal{O}$, when the context is unclear.

These categories are stable by subobject and quotients in the category of $U(\mathfrak{g})$-modules. Moreover the category $O_{\mathrm{alg}}^{I,\infty}$ is stable under extensions.

For any character $\lambda \in X^*(\mathcal{T})_I^+$, we write $L_I(\lambda)$ for the simple $U(l_I)$-module of highest weight $\lambda$. This is a finite dimensional $L$-vector space and we define the generalized Verma module of highest weight $\lambda$ as

$$M_I(\lambda) := U(\mathfrak{g}) \otimes_{U(p_I)} L_I(\lambda).$$

The generalized Verma module is an object of $O_{\mathrm{alg}}^I$ and has a unique simple quotient $L(\lambda)$. When $I = \emptyset$, we simply write $M(\lambda) = M_0(\lambda)$ and say that $M(\lambda)$ is a Verma module. We also denote by $P(\lambda)$ the projective cover of the simple module $L(\lambda)$. If $\lambda$ is dominant with respect to $\mathcal{B}$, we call $P(w_0 \cdot \lambda)$ the antidominant projective (with respect to $\lambda$).

## 2.2 Nilpotent action of $U(t)$

Given $I \subset \Delta$ we denote by $\mathfrak{m}_I$ the augmentation ideal of $U(\mathfrak{z}_I)$ and set

$$A_I := U(\mathfrak{z}_I)_{\mathfrak{m}_I} \\ A := A_0 := U(t)_{\mathfrak{m}}.$$

The canonical Lie algebra decomposition $l_I = \mathfrak{z}_I \oplus l_I^{\mathrm{ss}}$ defines a canonical morphism of Lie algebras $p_I : l_I \to \mathfrak{z}_I$ which extends to a morphism $U(l_I) \to U(\mathfrak{z}_I)$ of $L$-algebras also denoted by $p_I$. This morphism induces a surjective morphism $A \to A_I$ of $A_I$-algebras.

We show that the category $O^{I,\infty}$ naturally embeds into the category $U(\mathfrak{g})_{A_I}$-mod, where $U(\mathfrak{g})_{A_I} := U(\mathfrak{g}) \otimes_L A_I$.
For $\mu \in \operatorname{Hom}_L(t, L)$, we write $M^\mu \subset M$ for the $L$-subspace of those $v \in M$ such that, for any $h \in t$, $(h - \mu(h))^n \cdot v = 0$ for some $n \ge 1$. We have

$$M = \bigoplus_{\mu \in \operatorname{Hom}_L(t, L)} M^{\mu}.$$

We write $O_{\operatorname{alg}}^{I,\infty}$ for the full subcategory of $O^{I,\infty}$ whose objects $M$ satisfy $M^\mu = 0$ for $\mu \notin X^*(\mathcal{T})$.

Moreover, we write $O_{\operatorname{alg}}^I \subset O_{\operatorname{alg}}^{I,\infty}$ for the full subcategory whose objects are direct sums of finitely generated semisimple $U(l_I)$-modules (when seen as $U(l_I)$-modules). This coincides with the usual parabolic (algebraic) category $O$, which is denoted $O_{\operatorname{alg}}^{p_I}$ in [OS15]). When $I = \emptyset$ we simply use the notations $O_{\operatorname{alg}}^\infty$ and $O_{\operatorname{alg}}$ for $O_{\operatorname{alg}}^{\emptyset,\infty}$ and $O_{\operatorname{alg}}^\emptyset$. Note that $O_{\operatorname{alg}}^{I,\infty} \subset O_{\operatorname{alg}}^\infty$ for any $I \subset \Delta$. As these categories depend on the choices of $\mathfrak{g}$ and $\mathfrak{b}$ we write $O_{\operatorname{alg}}^{\mathfrak{g},\mathfrak{b}}$ (with additional decorations) instead of $O$, when the context is unclear.

These categories are stable by subobject and quotients in the category of $U(\mathfrak{g})$-modules. Moreover the category $O_{\operatorname{alg}}^{I,\infty}$ is stable under extensions.

For any character $\lambda \in X^*(\mathcal{T})_I^+$, we write $L_I(\lambda)$ for the simple $U(l_I)$-module of highest weight $\lambda$. This is a finite dimensional $L$-vector space and we define the generalized Verma module of highest weight $\lambda$ as

$$M_I(\lambda) := U(\mathfrak{g}) \otimes_{U(p_I)} L_I(\lambda).$$

The generalized Verma module is an object of $O_{\operatorname{alg}}^I$ and has a unique simple quotient $L(\lambda)$. When $I = \emptyset$, we simply write $M(\lambda) = M_0(\lambda)$ and say that $M(\lambda)$ is a Verma module. We also denote by $P(\lambda)$ the projective cover of the simple module $L(\lambda)$. If $\lambda$ is dominant with respect to $\mathcal{B}$, we call $P(w_0 \cdot \lambda)$ the antidominant projective (with respect to $\lambda$).

## 2.2 Nilpotent action of $U(t)$

Given $I \subset \Delta$ we denote by $\mathfrak{m}_I$ the augmentation ideal of $U(\mathfrak{z}_I)$ and set

$$A_I := U(\mathfrak{z}_I)_{\mathfrak{m}_I} \\ A := A_0 := U(t)_{\mathfrak{m}}.$$

The canonical Lie algebra decomposition $l_I = \mathfrak{z}_I \oplus l_I^{ss}$ defines a canonical morphism of Lie algebras $p_I : l_I \to \mathfrak{z}_I$ which extends to a morphism $U(l_I) \to U(\mathfrak{z}_I)$ of $L$-algebras also denoted by $p_I$. This morphism induces a surjective morphism $A \to A_I$ of $A_I$-algebras.

We show that the category $O^{I,\infty}$ naturally embeds into the category $U(\mathfrak{g})_{A_I}$-mod, where $U(\mathfrak{g})_{A_I} := U(\mathfrak{g}) \otimes_L A_I$.
Let $M$ be an object of the category $\mathcal{O}^{I,\infty}$. Let $h \in t$. For $v \in M$ the element $h$ defines an $L$-linear endomorphism of the finite dimensional $L$-vector space $U(t)v$ and we write $h = D_{h,v} + N_{h,v}$ for its Jordan decomposition with semisimple part $D_{h,v}$ and nilpotent part $N_{h,v}$. As $M$ is locally $U(t)$-finite, uniqueness of the Jordan decomposition implies that these endomorphisms “glue” into an endomorphism $D_h$ and a locally nilpotent endomorphism $N_h$ of $M$ such that $D_{h,v}$ respectively $N_{h,v}$ is the restriction of $D_h$ respectively $N_h$ to $U(t)v$ for any $v \in M$.

**Lemma 2.1.** The endomorphism $N_h$ is $U(\mathfrak{g})$-equivariant.

*Proof.* By construction $N_h$ and $D_h$ commute with the action of $t$ and stabilize each $M^\mu$. Let $\alpha \in \Phi$ and $x \in \mathfrak{g}_\alpha$. For $v \in M^\mu$, we have $x \cdot v \in M^{\mu+\alpha}$ and $[h, x] = \alpha(h)x$ so that

$$D_h x \cdot v + N_h x \cdot v = x D_h \cdot v + x N_h \cdot v + \alpha(h) xv.$$

By definition of $M^\mu$, we have $D_h \cdot v = \mu(h)v$ for any $v \in M^\mu$. This implies $D_h x \cdot v = (\mu(h) + \alpha(h))x \cdot v$ and $xD_h \cdot v = \mu(h)x \cdot v$. Therefore $N_h x \cdot v = xN_h \cdot v$. We conclude that $N_h$ commutes with the endomorphism of $M$ induced by $x$. Therefore $N_h$ is $U(\mathfrak{g})$-equivariant. $\square$

Given $M \in \mathcal{O}^{I,\infty}$ Lemma 2.1 implies that we can define an $U(t)$-module structure on $M$ by letting $h \in t \subset U(t)$ act via $N_h$. As the action of each $h$ on $M$ is locally nilpotent, this action extends to an $A$-module structure.

**Lemma 2.2.** Let $M$ be an object of $\mathcal{O}^{I,\infty}$, then the $A$-action on $M$ factors through $A_I$. Moreover, this $A_I$-module structure makes $\mathcal{O}^{I,\infty}$ into a full subcategory of $U(\mathfrak{g})_{A_I}$-mod.

*Proof.* In order to prove that the $A$-action factors through $A \to A_I$ it is enough to prove that for $h \in t \cap \mathfrak{f}^{ss}$ the endomorphism $N_h$ is zero. This is a direct consequence of the fact that $\mathfrak{f}^{ss}$ is a semi-simple Lie algebra and that the $L$-vector space $U(\mathfrak{f}^{ss})v$ is finite dimensional for any $v \in M$ (by definition of $\mathcal{O}^{I,\infty}$). As the $U(\mathfrak{g})$-action commutes with the $A$-action by Lemma 2.1 the module $M$ is an $U(\mathfrak{g})_{A_I}$-module. Finally we note that, given $h \in t$, the construction of $N_h$ is functorial in $M$. $\square$

*Remark 2.3.* Let $M \in \mathcal{O}^{I,\infty}$ and $\mu \in \operatorname{Hom}_L(t, L)$ then the above construction implies that

$$M^{\mu} = \{v \in M \mid hv = (\mu(h)v) + p_I(v) \forall h \in t\}$$

Let $M \in \mathcal{O}_{\text{alg}}^{\infty}$. Lemma 2.1 also implies that we can define another structure of an $U(\mathfrak{g})$-module on $M$ where an element $h \in t$ acts through the semisimple part $D_h$ and the action of an element $x \in \mathfrak{g}_\alpha$ for $\alpha \in \Phi$ is not modified. We denote this $U(\mathfrak{g})$-module structure by $M^{\text{ss}}$. Then $M^{\text{ss}}$ is an object of $\mathcal{O}_{\text{alg}}$ and [OS15, Lemm. 3.2] implies that there is a unique structure of algebraic $\mathcal{B}$-module on $M$ lifting the structure of $U(\mathfrak{b})$-module on $M^{\text{ss}}$. This $\mathcal{B}$-action is compatible with the original $U(\mathfrak{g})$-module structure on $M$ in the following sense:
Let $M$ be an object of the category $\mathcal{O}^{I,\infty}$. Let $h \in t$. For $v \in M$ the element $h$ defines an $L$-linear endomorphism of the finite dimensional $L$-vector space $U(t)v$ and we write $h = D_{h,v} + N_{h,v}$ for its Jordan decomposition with semisimple part $D_{h,v}$ and nilpotent part $N_{h,v}$. As $M$ is locally $U(t)$-finite, uniqueness of the Jordan decomposition implies that these endomorphisms “glue” into an endomorphism $D_h$ and a locally nilpotent endomorphism $N_h$ of $M$ such that $D_{h,v}$ respectively $N_{h,v}$ is the restriction of $D_h$ respectively $N_h$ to $U(t)v$ for any $v \in M$.

**Lemma 2.1.** The endomorphism $N_h$ is $U(\mathfrak{g})$-equivariant.

*Proof.* By construction $N_h$ and $D_h$ commute with the action of $t$ and stabilize each $M^\mu$. Let $\alpha \in \Phi$ and $x \in \mathfrak{g}_\alpha$. For $v \in M^\mu$, we have $x \cdot v \in M^{\mu+\alpha}$ and $[h, x] = \alpha(h)x$ so that

$$D_h x \cdot v + N_h x \cdot v = x D_h \cdot v + x N_h \cdot v + \alpha(h) x v.$$

By definition of $M^\mu$, we have $D_h \cdot v = \mu(h)v$ for any $v \in M^\mu$. This implies $D_h x \cdot v = (\mu(h) + \alpha(h))x \cdot v$ and $xD_h \cdot v = \mu(h)x \cdot v$. Therefore $N_h x \cdot v = xN_h \cdot v$. We conclude that $N_h$ commutes with the endomorphism of $M$ induced by $x$. Therefore $N_h$ is $U(\mathfrak{g})$-equivariant. $\square$

Given $M \in \mathcal{O}^{I,\infty}$ Lemma 2.1 implies that we can define an $U(t)$-module structure on $M$ by letting $h \in t \subset U(t)$ act via $N_h$. As the action of each $h$ on $M$ is locally nilpotent, this action extends to an $A$-module structure.

**Lemma 2.2.** Let $M$ be an object of $\mathcal{O}^{I,\infty}$, then the $A$-action on $M$ factors through $A_I$. Moreover, this $A_I$-module structure makes $\mathcal{O}^{I,\infty}$ into a full subcategory of $U(\mathfrak{g})_{A_I}$-mod.

*Proof.* In order to prove that the $A$-action factors through $A \to A_I$ it is enough to prove that for $h \in t \cap \mathfrak{f}^{ss}$ the endomorphism $N_h$ is zero. This is a direct consequence of the fact that $\mathfrak{f}^{ss}$ is a semi-simple Lie algebra and that the $L$-vector space $U(\mathfrak{f}^{ss})v$ is finite dimensional for any $v \in M$ (by definition of $\mathcal{O}^{I,\infty}$). As the $U(\mathfrak{g})$-action commutes with the $A$-action by Lemma 2.1 the module $M$ is an $U(\mathfrak{g})_{A_I}$-module. Finally we note that, given $h \in t$, the construction of $N_h$ is functorial in $M$. $\square$

**Remark 2.3.** Let $M \in \mathcal{O}^{I,\infty}$ and $\mu \in \operatorname{Hom}_L(t, L)$ then the above construction implies that

$$M^{\mu} = \{v \in M \mid hv = (\mu(h)v) + p_I(v) \forall h \in t\}$$

Let $M \in \mathcal{O}_{\text{alg}}^{\infty}$. Lemma 2.1 also implies that we can define another structure of an $U(\mathfrak{g})$-module on $M$ where an element $h \in t$ acts through the semisimple part $D_h$ and the action of an element $x \in \mathfrak{g}_\alpha$ for $\alpha \in \Phi$ is not modified. We denote this $U(\mathfrak{g})$-module structure by $M^{\text{ss}}$. Then $M^{\text{ss}}$ is an object of $\mathcal{O}_{\text{alg}}$ and [OS15, Lemm. 3.2] implies that there is a unique structure of algebraic $\mathcal{B}$-module on $M$ lifting the structure of $U(\mathfrak{b})$-module on $M^{\text{ss}}$. This $\mathcal{B}$-action is compatible with the original $U(\mathfrak{g})$-module structure on $M$ in the following sense:
**Lemma 2.4.** Let $M$ be an object of $\mathcal{O}_{\text{alg}}^{\infty}$ endowed with the $\mathcal{B}$-module structure defined above. Then

$$b \cdot (X \cdot (b^{-1} \cdot v)) = (\text{Ad}(b)X) \cdot v$$

for any $b \in \mathcal{B}(L)$, $X \in \mathfrak{g}$ and $v \in M$.

*Proof.* It is sufficient to prove the formula for $b \in \mathcal{N}(L)$ and for $b \in \mathcal{T}(L)$. If $b \in \mathcal{N}(L)$, then $b = \exp(n)$ for some $n \in \mathfrak{n}$. It follows that $\text{Ad}(b)X$ is equal to the finite sum $\sum_{k \ge 0} \frac{1}{k!} \text{ad}(n)^k X$ and that the action of $b$ on $M$ is given by the series $\sum_{k \ge 0} \frac{1}{k!} n^k$ (which is locally finite). Therefore we have,

$$ \begin{aligned} b \cdot (X \cdot (b^{-1} \cdot v)) &= \sum_{k \ge 0, \ell \ge 0} (-1)^{\ell} \frac{1}{k! \ell!} n^k X n^{\ell} \cdot v \\ &= \sum_{m \ge 0} \frac{1}{m!} \sum_{k+\ell=m} (-1)^{k-m} \binom{m}{k} n^k X n^{\ell} \cdot v \\ &= \sum_{m \ge 0} \frac{1}{m!} (\text{ad}(n)^m X) \cdot v = \text{Ad}(b)X \cdot v. \end{aligned} $$

If $b \in \mathcal{T}(L)$, then if $\alpha \in \Phi \cup \{0\}$ and $X \in \mathfrak{g}_{\alpha}$, and if $v \in M^{\mu}$, we have

$$ \begin{aligned} b \cdot (X \cdot (b^{-1} \cdot v)) &= b \cdot (X \cdot (\mu(b^{-1})v)) = (\mu + \alpha)(b)\mu(b^{-1})X \cdot v \\ &= \alpha(b)X \cdot v = \text{Ad}(b)X \cdot v \end{aligned} $$

as $\text{Ad}(b)X = \alpha(b)X$.

□

For later use, we note that we can resolve objects in $\mathcal{O}_{\text{alg}}^{\infty}$ as follows:

**Lemma 2.5.** Let $M$ be an object of $\mathcal{O}_{\text{alg}}^{\infty}$. Then there exist finite dimensional $U(\mathfrak{b})$-modules $V_0$ and $V_1$ and an exact sequence of $U(\mathfrak{g})$-modules

$$ U(\mathfrak{g}) \otimes_{U(\mathfrak{b})} V_1 \to U(\mathfrak{g}) \otimes_{U(\mathfrak{b})} V_0 \to M \to 0. \qquad (2) $$

Moreover, this exact sequence is $\mathcal{B}$-equivariant for the $\mathcal{B}$-actions (on each of the three terms) defined just before Lemma 2.4.

*Proof.* The existence of a finite dimensional $U(\mathfrak{b})$-module $V_0$ and a surjective map $U(\mathfrak{g}) \otimes_{U(\mathfrak{b})} V_0 \to M$ is a consequence of Proposition 2.14. The existence of $V_1$ and of the map $U(\mathfrak{g}) \otimes_{U(\mathfrak{b})} V_1 \to U(\mathfrak{g}) \otimes_{U(\mathfrak{b})} V_0$ follows again from Proposition 2.14 applied to the kernel of $U(\mathfrak{g}) \otimes_{U(\mathfrak{b})} V_0 \to M$. The $\mathcal{B}$-equivariance is a direct consequence of the definition of the algebraic action of $\mathcal{B}$-action on each term of the sequence (2). □
**Lemma 2.4.** Let $M$ be an object of $\mathcal{O}_{\text{alg}}^{\infty}$ endowed with the $\mathcal{B}$-module structure defined above. Then

$$b \cdot (X \cdot (b^{-1} \cdot v)) = (\text{Ad}(b)X) \cdot v$$

for any $b \in \mathcal{B}(L)$, $X \in \mathfrak{g}$ and $v \in M$.

*Proof.* It is sufficient to prove the formula for $b \in \mathcal{N}(L)$ and for $b \in \mathcal{T}(L)$. If $b \in \mathcal{N}(L)$, then $b = \exp(n)$ for some $n \in \mathfrak{n}$. It follows that $\text{Ad}(b)X$ is equal to the finite sum $\sum_{k \ge 0} \frac{1}{k!} \text{ad}(n)^k X$ and that the action of $b$ on $M$ is given by the series $\sum_{k \ge 0} \frac{1}{k!} n^k$ (which is locally finite). Therefore we have,

$$\begin{align*}
b \cdot (X \cdot (b^{-1} \cdot v)) &= \sum_{k \ge 0, \ell \ge 0} (-1)^{\ell} \frac{1}{k! \ell!} n^k X n^{\ell} \cdot v \\
&= \sum_{m \ge 0} \frac{1}{m!} \sum_{k+\ell=m} (-1)^{k-m} \binom{m}{k} n^k X n^{\ell} \cdot v \\
&= \sum_{m \ge 0} \frac{1}{m!} (\text{ad}(n)^m X) \cdot v = \text{Ad}(b)X \cdot v.
\end{align*}$$

If $b \in \mathcal{T}(L)$, then if $\alpha \in \Phi \cup \{0\}$ and $X \in \mathfrak{g}_{\alpha}$, and if $v \in M^{\mu}$, we have

$$\begin{align*}
b \cdot (X \cdot (b^{-1} \cdot v)) &= b \cdot (X \cdot (\mu(b^{-1})v)) = (\mu + \alpha)(b)\mu(b^{-1})X \cdot v \\
&= \alpha(b)X \cdot v = \text{Ad}(b)X \cdot v
\end{align*}$$

as $\text{Ad}(b)X = \alpha(b)X$.

□

For later use, we note that we can resolve objects in $\mathcal{O}_{\text{alg}}^{\infty}$ as follows:

**Lemma 2.5.** Let $M$ be an object of $\mathcal{O}_{\text{alg}}^{\infty}$. Then there exist finite dimensional $U(\mathfrak{b})$-modules $V_0$ and $V_1$ and an exact sequence of $U(\mathfrak{g})$-modules

$$U(\mathfrak{g}) \otimes_{U(\mathfrak{b})} V_1 \to U(\mathfrak{g}) \otimes_{U(\mathfrak{b})} V_0 \to M \to 0. \qquad (2)$$

Moreover, this exact sequence is $\mathcal{B}$-equivariant for the $\mathcal{B}$-actions (on each of the three terms) defined just before Lemma 2.4.

*Proof.* The existence of a finite dimensional $U(\mathfrak{b})$-module $V_0$ and a surjective map $U(\mathfrak{g}) \otimes_{U(\mathfrak{b})} V_0 \to M$ is a consequence of Proposition 2.14. The existence of $V_1$ and of the map $U(\mathfrak{g}) \otimes_{U(\mathfrak{b})} V_1 \to U(\mathfrak{g}) \otimes_{U(\mathfrak{b})} V_0$ follows again from Proposition 2.14 applied to the kernel of $U(\mathfrak{g}) \otimes_{U(\mathfrak{b})} V_0 \to M$. The $\mathcal{B}$-equivariance is a direct consequence of the definition of the algebraic action of $\mathcal{B}$-action on each term of the sequence (2). □
## 2.3 Deformations of the category $\mathcal{O}$

Fix $I \subset \Delta$ and let $M$ be some $U(\mathfrak{g})_{A_I}$-module. For $\mu \in X^*(\mathcal{T})$, we define the $A_I$-submodule

$$M^{\mu} := \{v \in M \mid \forall h \in \mathfrak{t}, h \cdot v = (p_I(h) + \mu(h))v\}.$$

We note that for $M \in \mathcal{O}^{I,\infty}$ this coincides with the generalized eigenspace for $\mu$ by Remark 2.3. Inspired by the construction of [Soe92, §3.1], we define $\tilde{\mathcal{O}}_{\text{alg}}^I$ as the category of $U(\mathfrak{g})_{A_I}$-modules $M$ such that

* $M$ is finitely generated over $U(\mathfrak{g})_{A_I}$;

* $M = \bigoplus_{\mu \in X^*(\mathcal{T})} M^{\mu}$ and each $M^{\mu}$ is a finite free $A_I$-module;

* for any $m \in M$ the $A_I$-submodule $(U(\mathfrak{p}_I) \otimes_L A_I)m$ is finitely generated.

**Lemma 2.6.** Let $M$ be an object of $\tilde{\mathcal{O}}_{\text{alg}}^I$. Then for any $n \ge 0$, the $U(\mathfrak{g})$-module $M/\mathfrak{m}_I^n M$ is an object of $\mathcal{O}_{\text{alg}}^{I,\infty}$ and $M/\mathfrak{m}_I M$ is in $\mathcal{O}_{\text{alg}}^I$.

*Proof.* This is a direct consequence of the definitions. $\square$

<figure><img src="image_2.png" /></figure>

For $\lambda \in X^*(\mathcal{T})_I^+$ we define the *deformed generalized Verma module* of weight $\lambda$ as

$$\tilde{M}_I(\lambda) := U(\mathfrak{g}) \otimes_{U(\mathfrak{p}_I)} (L_I(\lambda) \otimes_L A_I)$$

where $U(\mathfrak{p}_I)$ acts on $A_I$ via the composition $U(\mathfrak{p}_I) \to U(\mathfrak{l}_I) \xrightarrow{p_I} U(\mathfrak{z}_I) \to A_I$. The module $\tilde{M}_I(V)$ is an object of $\tilde{\mathcal{O}}_{\text{alg}}^I$ and we have an isomorphism of $U(\mathfrak{g})_{A_I}$-modules

$$\tilde{M}_I(\lambda) \otimes_{A_I} A_I/\mathfrak{m}_I = M_I(\lambda).$$

### 2.3.1 Duality

Recall that there exist an internal duality functor $M \mapsto M^{\vee}$ on the category $\mathcal{O}_{\text{alg}}$ (see [Hum08, §3.2]). We will define an analogue on $\tilde{\mathcal{O}}_{\text{alg}}$. Let $M$ be an object of the category $\tilde{\mathcal{O}}_{\text{alg}}^I$. We define an action of $U(\mathfrak{g})$ on $M^* := \text{Hom}_{A_I}(M, A_I)$ by $x \cdot f(m) = f(\tau(x)m)$ where $\tau$ is the anti-involution of $U(\mathfrak{g})$ defined in [Hum08, §0.5]. We then define $M^{\vee}$ to be the sub-$U(\mathfrak{g})$-module of $M^*$ given by

$$M^{\vee} := \bigoplus_{\mu \in X^*(\mathcal{T})} (M^{*})^{\mu}.$$

**Lemma 2.7.** If $M$ is an object of the category $\tilde{\mathcal{O}}_{\text{alg}}^I$, then so is $M^{\vee}$. Moreover there is a canonical isomorphism $M \xrightarrow{\sim} (M^{\vee})^{\vee}$. Consequently $M^{\vee}/\mathfrak{m}_I M^{\vee} \simeq (M/\mathfrak{m}_I M)^{\vee}$ is in the category $\mathcal{O}_{\text{alg}}^I$.
## 2.3 Deformations of the category $\mathcal{O}$

Fix $I \subset \Delta$ and let $M$ be some $U(\mathfrak{g})_{A_I}$-module. For $\mu \in X^*(\mathcal{T})$, we define the $A_I$-submodule

$$M^{\mu} := \{v \in M \mid \forall h \in \mathfrak{t}, h \cdot v = (p_I(h) + \mu(h))v\}.$$

We note that for $M \in \mathcal{O}^{I,\infty}$ this coincides with the generalized eigenspace for $\mu$ by Remark 2.3. Inspired by the construction of [Soe92, §3.1], we define $\tilde{\mathcal{O}}_{\text{alg}}^I$ as the category of $U(\mathfrak{g})_{A_I}$-modules $M$ such that

* $M$ is finitely generated over $U(\mathfrak{g})_{A_I}$;

* $M = \bigoplus_{\mu \in X^*(\mathcal{T})} M^{\mu}$ and each $M^{\mu}$ is a finite free $A_I$-module;

* for any $m \in M$ the $A_I$-submodule $(U(\mathfrak{p}_I) \otimes_L A_I)m$ is finitely generated.

**Lemma 2.6.** Let $M$ be an object of $\tilde{\mathcal{O}}_{\text{alg}}^I$. Then for any $n \ge 0$, the $U(\mathfrak{g})$-module $M/\mathfrak{m}_I^n M$ is an object of $\mathcal{O}_{\text{alg}}^{I,\infty}$ and $M/\mathfrak{m}_I M$ is in $\mathcal{O}_{\text{alg}}^I$.

*Proof.* This is a direct consequence of the definitions. $\square$

<figure><img src="image_2.png" /></figure>

For $\lambda \in X^*(\mathcal{T})_I^+$ we define the *deformed generalized Verma module* of weight $\lambda$ as

$$\tilde{M}_I(\lambda) := U(\mathfrak{g}) \otimes_{U(\mathfrak{p}_I)} (L_I(\lambda) \otimes_L A_I)$$

where $U(\mathfrak{p}_I)$ acts on $A_I$ via the composition $U(\mathfrak{p}_I) \to U(\mathfrak{l}_I) \xrightarrow{p_I} U(\mathfrak{z}_I) \to A_I$. The module $\tilde{M}_I(V)$ is an object of $\tilde{\mathcal{O}}_{\text{alg}}^I$ and we have an isomorphism of $U(\mathfrak{g})_{A_I}$-modules

$$\tilde{M}_I(\lambda) \otimes_{A_I} A_I/\mathfrak{m}_I = M_I(\lambda).$$

### 2.3.1 Duality

Recall that there exist an internal duality functor $M \mapsto M^{\vee}$ on the category $\mathcal{O}_{\text{alg}}$ (see [Hum08, §3.2]). We will define an analogue on $\tilde{\mathcal{O}}_{\text{alg}}$. Let $M$ be an object of the category $\tilde{\mathcal{O}}_{\text{alg}}^I$. We define an action of $U(\mathfrak{g})$ on $M^* := \text{Hom}_{A_I}(M, A_I)$ by $x \cdot f(m) = f(\tau(x)m)$ where $\tau$ is the anti-involution of $U(\mathfrak{g})$ defined in [Hum08, §0.5]. We then define $M^{\vee}$ to be the sub-$U(\mathfrak{g})$-module of $M^*$ given by

$$M^{\vee} := \bigoplus_{\mu \in X^*(\mathcal{T})} (M^{*})^{\mu}.$$

**Lemma 2.7.** If $M$ is an object of the category $\tilde{\mathcal{O}}_{\text{alg}}^I$, then so is $M^{\vee}$. Moreover there is a canonical isomorphism $M \xrightarrow{\sim} (M^{\vee})^{\vee}$. Consequently $M^{\vee}/\mathfrak{m}_I M^{\vee} \simeq (M/\mathfrak{m}_I M)^{\vee}$ is in the category $\mathcal{O}_{\text{alg}}^I$.
*Proof.* We have a canonical isomorphism of $A_I$-modules

$$M^* \simeq \prod_{\mu \in X^*(\mathcal{T})} \mathrm{Hom}_{A_I}(M^\mu, A_I)$$

and we easily check that $(M^*)^\mu = \mathrm{Hom}_{A_I}(M^\mu, A_I)$ for $\mu \in X^*(\mathcal{T})$. As any $M^\mu$ is a finite free $A_I$-module, so is $(M^*)^\mu = (M^\vee)^\mu$.

Let $\mathfrak{n}_I^- := \bigoplus_{\alpha \in -\Phi^+ \setminus \Phi_I} \mathfrak{g}_\alpha$ denote the nilpotent radical of the parabolic Lie subalgebra opposite to $\mathfrak{p}_I$. Note that a $U(\mathfrak{g})_{A_I}$-module $M$ such that $M = \bigoplus_\mu M^\mu$ with $M^\mu$ finite free over $A_I$ is in $\tilde{\mathcal{O}}_{\mathrm{alg}}^I$ if and only if we can write $M = U(\mathfrak{n}_I^-) \cdot (\bigoplus_{\mu \in S} M_\mu)$ for some finite set $S \subset X^*(\mathcal{T})$. By Lemma 2.6, the object $M/\mathfrak{m}_I M$ lies in the category $\mathcal{O}_{\mathrm{alg}}^I$ and it follows from [Hum08, §9.3] that $(M/\mathfrak{m}_I M)^\vee$ lies in $\mathcal{O}_{\mathrm{alg}}^I$. This implies that there exists a finite set $S \subset X^*(\mathcal{T})$ such that

$$(M/\mathfrak{m}_I M)^\vee = U(\mathfrak{n}_I^-) \cdot (\bigoplus_{\mu \in S} (M/\mathfrak{m}_I M)^{\vee, \mu}).$$

It follows that for any $\mu$ such that $(M/\mathfrak{m}_I M)^{\vee, \mu} \neq 0$, the map

$$\bigoplus_{\substack{\nu \in \sum_{\alpha \in -\Phi \setminus \Phi_I} \mathbb{N}\alpha \\ \mu' \in S \\ \mu' + \nu = \mu}} (M/\mathfrak{m}_I M)^{\vee, \mu'} \to (M/\mathfrak{m}_I M)^{\vee, \mu}$$

given by the action of the corresponding element of $U(\mathfrak{n}_I^-)$ on each summand, is surjective. As $M^\mu$ is a finite free $A_I$-module and $A_I$ is a local ring, we deduce from Nakayama's Lemma that the map

$$\bigoplus_{\substack{\nu \in \sum_{\alpha \in -\Phi \setminus \Phi_I} \mathbb{N}\alpha \\ \mu' \in S \\ \mu' + \nu = \mu}} M^{\vee, \mu'} \to M^{\vee, \mu}$$

is surjective and thus that $M^\vee = U(\mathfrak{n}_I^-) \cdot (\bigoplus_{\mu' \in S} M^{\vee, \mu'})$. This implies that $M^\vee$ is a finitely generated $U(\mathfrak{g})_{A_I}$-module and we also deduce from this equality that $M^\vee$ is locally $U(\mathfrak{p}_I)_{A_I}$-finite.

In order to prove that $M \xrightarrow{\sim} (M^\vee)^\vee$ we note that the natural map $M \to (M^\vee)^*$ of $U(\mathfrak{g})_{A_I}$-modules factors through $(M^\vee)^\vee$ and respects the weight decomposition. Moreover as $M^\mu$ is free over $A_I$ for all $\mu$, the induced bi-duality $M^\mu \xrightarrow{\sim} (M^{\mu, *})^*$ morphism is an isomorphism. $\square$

### 2.3.2 Blocks

Let $Z(\mathfrak{g})$ denote the center of $U(\mathfrak{g})$ and let $\chi : Z(\mathfrak{g}) \to L$ be a character of $Z(\mathfrak{g})$. Let $\mathcal{O}_\chi$ be the subcategory of objects $M$ of $\mathcal{O}_{\mathrm{alg}}$ such that $z - \chi(z)$ acts nilpotently on $M$ for any $z \in Z(\mathfrak{g})$. For $I \subset \Delta$, we denote by $\mathcal{O}_\chi^I$ the full subcategory of objects of $\mathcal{O}_{\mathrm{alg}}^I$ which
Proof. We have a canonical isomorphism of $A_I$-modules

$$M^* \simeq \prod_{\mu \in X^*(\mathcal{T})} \mathrm{Hom}_{A_I}(M^\mu, A_I)$$

and we easily check that $(M^*)^\mu = \mathrm{Hom}_{A_I}(M^\mu, A_I)$ for $\mu \in X^*(\mathcal{T})$. As any $M^\mu$ is a finite free $A_I$-module, so is $(M^*)^\mu = (M^\vee)^\mu$.

Let $\mathfrak{n}_I^- := \bigoplus_{\alpha \in -\Phi^+ \setminus \Phi_I} \mathfrak{g}_\alpha$ denote the nilpotent radical of the parabolic Lie subalgebra opposite to $\mathfrak{p}_I$. Note that a $U(\mathfrak{g})_{A_I}$-module $M$ such that $M = \bigoplus_\mu M^\mu$ with $M^\mu$ finite free over $A_I$ is in $\tilde{\mathcal{O}}_{\mathrm{alg}}^I$ if and only if we can write $M = U(\mathfrak{n}_I^-) \cdot (\bigoplus_{\mu \in S} M_\mu)$ for some finite set $S \subset X^*(\mathcal{T})$. By Lemma 2.6, the object $M/\mathfrak{m}_I M$ lies in the category $\mathcal{O}_{\mathrm{alg}}^I$ and it follows from [Hum08, §9.3] that $(M/\mathfrak{m}_I M)^\vee$ lies in $\mathcal{O}_{\mathrm{alg}}^I$. This implies that there exists a finite set $S \subset X^*(\mathcal{T})$ such that

$$(M/\mathfrak{m}_I M)^\vee = U(\mathfrak{n}_I^-) \cdot (\bigoplus_{\mu \in S} (M/\mathfrak{m}_I M)^{\vee, \mu}).$$

It follows that for any $\mu$ such that $(M/\mathfrak{m}_I M)^{\vee, \mu} \neq 0$, the map

$$\bigoplus_{\substack{\nu \in \sum_{\alpha \in -\Phi \setminus \Phi_I} \mathbb{N}\alpha \\ \mu' \in S \\ \mu' + \nu = \mu}} (M/\mathfrak{m}_I M)^{\vee, \mu'} \to (M/\mathfrak{m}_I M)^{\vee, \mu}$$

given by the action of the corresponding element of $U(\mathfrak{n}_I^-)$ on each summand, is surjective. As $M^\mu$ is a finite free $A_I$-module and $A_I$ is a local ring, we deduce from Nakayama's Lemma that the map

$$\bigoplus_{\substack{\nu \in \sum_{\alpha \in -\Phi \setminus \Phi_I} \mathbb{N}\alpha \\ \mu' \in S \\ \mu' + \nu = \mu}} M^{\vee, \mu'} \to M^{\vee, \mu}$$

is surjective and thus that $M^\vee = U(\mathfrak{n}_I^-) \cdot (\bigoplus_{\mu' \in S} M^{\vee, \mu'})$. This implies that $M^\vee$ is a finitely generated $U(\mathfrak{g})_{A_I}$-module and we also deduce from this equality that $M^\vee$ is locally $U(\mathfrak{p}_I)_{A_I}$-finite.

In order to prove that $M \xrightarrow{\sim} (M^\vee)^\vee$ we note that the natural map $M \to (M^\vee)^*$ of $U(\mathfrak{g})_{A_I}$-modules factors through $(M^\vee)^\vee$ and respects the weight decomposition. Moreover as $M^\mu$ is free over $A_I$ for all $\mu$, the induced bi-duality $M^\mu \xrightarrow{\sim} (M^{\mu, *})^*$ morphism is an isomorphism. $\square$

### 2.3.2 Blocks

Let $Z(\mathfrak{g})$ denote the center of $U(\mathfrak{g})$ and let $\chi: Z(\mathfrak{g}) \to L$ be a character of $Z(\mathfrak{g})$. Let $\mathcal{O}_\chi$ be the subcategory of objects $M$ of $\mathcal{O}_{\mathrm{alg}}$ such that $z - \chi(z)$ acts nilpotently on $M$ for any $z \in Z(\mathfrak{g})$. For $I \subset \Delta$, we denote by $\mathcal{O}_\chi^I$ the full subcategory of objects of $\mathcal{O}_{\mathrm{alg}}^I$ which
are also in $\mathcal{O}_\chi$. We deduce from [Hum08, Prop. 1.12] that there is a decomposition into blocks

$$ \mathcal{O}_{\text{alg}}^{I} = \bigoplus_{\chi} \mathcal{O}_{\chi}^{I}. $$

We write $\tilde{\mathcal{O}}_{\chi}^{I}$ for the subcategory of objects $M$ of $\tilde{\mathcal{O}}_{\text{alg}}^{I}$ such that $M/\mathfrak{m}_{I}M$ lies in $\mathcal{O}_{\chi}^{I}$, and similarly $\mathcal{O}_{\chi}^{I,\infty}$.

*Remark 2.8.* For $\lambda \in X^{*}(\mathbb{T})$, let $\chi$ be the character $\chi_{\lambda}$ defined in [Hum08, §1.7]. Then *loc. cit.* implies that $\tilde{M}_{I}(\lambda)$ is in $\tilde{\mathcal{O}}_{\chi\lambda}^{I}$.

**Lemma 2.9.** We have decompositions $\tilde{\mathcal{O}}_{\text{alg}}^{I} = \bigoplus_{\chi} \tilde{\mathcal{O}}_{\chi}^{I}$ and $\mathcal{O}_{\text{alg}}^{I,\infty} = \bigoplus_{\chi} \mathcal{O}_{\chi}^{I,\infty}$.

*Proof.* Let $M$ be an object of $\tilde{\mathcal{O}}_{\text{alg}}^{I}$. For a character $\chi : Z(\mathfrak{g}) \to L$ and $\mu \in X^{*}(\mathbb{T})$, let $M^{\mu,\chi}$ denote the subset of elements $x \in M^{\mu}$ such that $(z - \chi(z))^{n}x \to 0$ for the $\mathfrak{m}_{I}$-adic topology on the finite free $A_{I}$-module $M^{\mu}$. We easily check that $M^{\chi} := \bigoplus_{\mu \in X^{*}(\mathbb{T})} M^{\mu,\chi}$ is an $U(\mathfrak{g})_{A_{I}}$-submodule of $M$ which lies in $\tilde{\mathcal{O}}_{\chi}^{I}$ and that $M = \bigoplus_{\chi} M^{\chi}$. The case of $\mathcal{O}_{\text{alg}}^{I,\infty}$ is similar. $\square$

**Lemma 2.10.** Let $\lambda_{1}, \lambda_{2} \in X^{*}(\mathbb{T})$. Assume that $\tilde{M}_{I}(\lambda_{1})$ and $\tilde{M}_{I}(\lambda_{2})$ are in the same block $\tilde{\mathcal{O}}_{\chi}^{I}$ for a character $\chi : Z(\mathfrak{g}) \to L$. Then there exists $w \in W$ such that $w \cdot \lambda_{1} = \lambda_{2}$.

*Proof.* By Remark 2.8, the claim follows from the same claim in the category $\mathcal{O}_{\chi}^{I}$. As $M_{I}(\lambda_{1})$ and $M_{I}(\lambda_{2})$ are quotients of $M(\lambda_{1})$ and $M(\lambda_{2})$, this is a consequence of [Hum08, Thm. 1.10]. $\square$

When $\lambda$ is a character of $\mathfrak{t}$, we often write by abuse of notation $\mathcal{O}_{\lambda}$ (resp. $\mathcal{O}_{\lambda}^{I,\infty}$, $\tilde{\mathcal{O}}_{\lambda}^{I}$) for the block $\mathcal{O}_{\chi_{\lambda}}$ (resp. $\mathcal{O}_{\chi_{\lambda}}^{I,\infty}$, $\tilde{\mathcal{O}}_{\chi_{\lambda}}^{I}$) where $\chi_{\lambda}$ is the character of $Z(\mathfrak{g})$ giving the action of the center on $M(\lambda)$ (see [Hum08, §1.7]). In particular, $\chi_{\lambda} = \chi_{\mu}$ if, and only if, there is $w \in W$ such that $w \cdot \lambda = \mu$.

**Corollary 2.11.** Let $\lambda \in X^{*}(\mathbb{T})$ be a dominant weight and let $\chi_{\lambda}$ be the associated character of $Z(\mathfrak{g})$. If $M$ is an object of $\tilde{\mathcal{O}}_{\chi_{\lambda}}^{I}$ (resp. $\mathcal{O}_{\chi_{\lambda}}^{I,\infty}$), then $M^{\lambda} = (M^{\lambda})^{\mathbb{n}}$.

*Proof.* Assume that this is false. Then there exists $\alpha \in \Phi^{+}$ and $x \in \mathfrak{g}_{\alpha}$ such that $xM^{\lambda} \neq 0$. Thus there exists $\mu > \lambda$ such that $M^{\mu} \neq 0$. As $M$ lies in the category $\tilde{\mathcal{O}}_{\text{alg}}^{I}$ (resp. $\mathcal{O}_{\chi_{\lambda}}^{I,\infty}$), we can choose $\mu$ to be maximal which then implies $\mathfrak{n}M^{\mu} = 0$. As $M^{\mu} \neq 0$ Nakayama's lemma implies that there exists $v \in M^{\mu}$ which is non zero in $M^{\mu}/\mathfrak{m}M^{\mu}$. Then $v$ defines a map $\tilde{M}_{I}(\mu) \to M$ with $\mu > \lambda$, which is non-zero after reduction by $\mathfrak{m}$. Thus it induces a non-zero map $M_{I}(\mu) \to M/\mathfrak{m}M \in \mathcal{O}_{\chi_{\lambda}}$. It follows that $\mu = w \cdot \lambda$ which is a contradiction. $\square$
are also in $\mathcal{O}_\chi$. We deduce from [Hum08, Prop. 1.12] that there is a decomposition into blocks

$$ \mathcal{O}_{\text{alg}}^{I} = \bigoplus_{\chi} \mathcal{O}_{\chi}^{I}. $$

We write $\tilde{\mathcal{O}}_{\chi}^{I}$ for the subcategory of objects $M$ of $\tilde{\mathcal{O}}_{\text{alg}}^{I}$ such that $M/\mathfrak{m}_{I}M$ lies in $\mathcal{O}_{\chi}^{I}$, and similarly $\mathcal{O}_{\chi}^{I,\infty}$.

*Remark 2.8.* For $\lambda \in X^{*}(\mathbb{T})$, let $\chi$ be the character $\chi_{\lambda}$ defined in [Hum08, §1.7]. Then loc. cit. implies that $\tilde{M}_{I}(\lambda)$ is in $\tilde{\mathcal{O}}_{\chi\lambda}^{I}$.

**Lemma 2.9.** We have decompositions $\tilde{\mathcal{O}}_{\text{alg}}^{I} = \bigoplus_{\chi} \tilde{\mathcal{O}}_{\chi}^{I}$ and $\mathcal{O}_{\text{alg}}^{I,\infty} = \bigoplus_{\chi} \mathcal{O}_{\chi}^{I,\infty}$.

*Proof.* Let $M$ be an object of $\tilde{\mathcal{O}}_{\text{alg}}^{I}$. For a character $\chi : Z(\mathfrak{g}) \to L$ and $\mu \in X^{*}(\mathbb{T})$, let $M^{\mu,\chi}$ denote the subset of elements $x \in M^{\mu}$ such that $(z - \chi(z))^{n}x \to 0$ for the $\mathfrak{m}_{I}$-adic topology on the finite free $A_{I}$-module $M^{\mu}$. We easily check that $M^{\chi} := \bigoplus_{\mu \in X^{*}(\mathbb{T})} M^{\mu,\chi}$ is an $U(\mathfrak{g})_{A_{I}}$-submodule of $M$ which lies in $\tilde{\mathcal{O}}_{\chi}^{I}$ and that $M = \bigoplus_{\chi} M^{\chi}$. The case of $\mathcal{O}_{\text{alg}}^{I,\infty}$ is similar. $\square$

**Lemma 2.10.** Let $\lambda_{1}, \lambda_{2} \in X^{*}(\mathbb{T})$. Assume that $\tilde{M}_{I}(\lambda_{1})$ and $\tilde{M}_{I}(\lambda_{2})$ are in the same block $\tilde{\mathcal{O}}_{\chi}^{I}$ for a character $\chi : Z(\mathfrak{g}) \to L$. Then there exists $w \in W$ such that $w \cdot \lambda_{1} = \lambda_{2}$.

*Proof.* By Remark 2.8, the claim follows from the same claim in the category $\mathcal{O}_{\chi}^{I}$. As $M_{I}(\lambda_{1})$ and $M_{I}(\lambda_{2})$ are quotients of $M(\lambda_{1})$ and $M(\lambda_{2})$, this is a consequence of [Hum08, Thm. 1.10]. $\square$

When $\lambda$ is a character of $\mathfrak{t}$, we often write by abuse of notation $\mathcal{O}_{\lambda}$ (resp. $\mathcal{O}_{\lambda}^{I,\infty}$, $\tilde{\mathcal{O}}_{\lambda}^{I}$) for the block $\mathcal{O}_{\chi_{\lambda}}$ (resp. $\mathcal{O}_{\chi_{\lambda}}^{I,\infty}$, $\tilde{\mathcal{O}}_{\chi_{\lambda}}^{I}$) where $\chi_{\lambda}$ is the character of $Z(\mathfrak{g})$ giving the action of the center on $M(\lambda)$ (see [Hum08, §1.7]). In particular, $\chi_{\lambda} = \chi_{\mu}$ if, and only if, there is $w \in W$ such that $w \cdot \lambda = \mu$.

**Corollary 2.11.** Let $\lambda \in X^{*}(\mathbb{T})$ be a dominant weight and let $\chi_{\lambda}$ be the associated character of $Z(\mathfrak{g})$. If $M$ is an object of $\tilde{\mathcal{O}}_{\chi_{\lambda}}^{I}$ (resp. $\mathcal{O}_{\chi_{\lambda}}^{I,\infty}$), then $M^{\lambda} = (M^{\lambda})^{\mathbb{n}}$.

*Proof.* Assume that this is false. Then there exists $\alpha \in \Phi^{+}$ and $x \in \mathfrak{g}_{\alpha}$ such that $xM^{\lambda} \neq 0$. Thus there exists $\mu > \lambda$ such that $M^{\mu} \neq 0$. As $M$ lies in the category $\tilde{\mathcal{O}}_{\text{alg}}^{I}$ (resp. $\mathcal{O}_{\chi_{\lambda}}^{I,\infty}$), we can choose $\mu$ to be maximal which then implies $\mathfrak{n}M^{\mu} = 0$. As $M^{\mu} \neq 0$ Nakayama's lemma implies that there exists $v \in M^{\mu}$ which is non zero in $M^{\mu}/\mathfrak{m}M^{\mu}$. Then $v$ defines a map $\tilde{M}_{I}(\mu) \to M$ with $\mu > \lambda$, which is non-zero after reduction by $\mathfrak{m}$. Thus it induces a non-zero map $M_{I}(\mu) \to M/\mathfrak{m}M \in \mathcal{O}_{\chi_{\lambda}}$. It follows that $\mu = w \cdot \lambda$ which is a contradiction. $\square$
### 2.3.3 Deformed Verma modules

Let $\lambda \in X^*(\mathcal{T})$ and let $V$ be a finite dimensional $U(\mathfrak{g})$-module. Then we have an isomorphism of $U(\mathfrak{g})_A$-modules

$$ \widetilde{M}(\lambda) \otimes_L V \simeq U(\mathfrak{g})_A \otimes_{U(\mathfrak{b})_A} (V_{|\mathfrak{b}} \otimes_L A(\lambda)). $$

Indeed there is a canonical map from the left to the right, which then is easily checked to be an isomorphism. As $V_{|\mathfrak{b}}$ is a successive extension of one dimensional $U(\mathfrak{b})$-modules, and as $U(\mathfrak{g})_A \otimes_{U(\mathfrak{b})_A} (-)$ is an exact functor (as follows from the PBW Theorem), we have a filtration $(\mathrm{Fil}_i)$ of $\widetilde{M}(\lambda) \otimes_L V$ such that each subquotient $\mathrm{Fil}_i / \mathrm{Fil}_{i-1}$ is isomorphic to $\widetilde{M}(\lambda + \nu_i)$ for $\nu_i$ a weight of $V$. Moreover the family $(\nu_i)$ is the family of weights of $V$ (counted with multiplicity).

**Proposition 2.12.** Let $K$ denote the fraction field of $A$. Then the filtration $(\mathrm{Fil}_i \otimes_A K)$ of $(\widetilde{M}(\lambda) \otimes_L V) \otimes_A K$ splits in the category of $U(\mathfrak{g})_K$-modules, i.e. there exists an isomorphism of $U(\mathfrak{g})_K$-modules

$$ (\widetilde{M}(\lambda) \otimes_L V) \otimes_A K \simeq \bigoplus_i (\widetilde{M}(\lambda + \nu_i) \otimes_A K) $$

compatible with the filtration $(\mathrm{Fil}_i \otimes_A K)$.

*Proof.* This is a consequence of the paragraph preceding [Soe92, Thm. 8]. □

**Lemma 2.13.** Let $\lambda \in X^*(\mathcal{T})_I^+$ be a dominant weight (with respect to $\mathcal{P}_I$) and let $V$ be a finite dimensional $U(\mathfrak{g})$-module. Let $M$ be an object of $\mathcal{O}_{\mathrm{alg}}^{I,\infty}$. Then the map

$$ \mathrm{Hom}_{U(\mathfrak{g})_{A_I}} (\widetilde{M}_I(\lambda) \otimes_L V, M) \to \mathrm{Hom}_{U(\mathfrak{g})} (M_I(\lambda) \otimes_L V, M/\mathfrak{m}_I M) $$

given by reduction modulo $\mathfrak{m}_I$ is surjective.

*Proof.* The $L$-vector space $\mathrm{Hom}_L(V, L)$ has the structure of an $U(\mathfrak{g})$-module induced by $\mathfrak{g}$-action defined by $x \cdot \phi = -\phi(x)$ for $x \in \mathfrak{g}$ and $\phi \in \mathrm{Hom}_L(V, L)$. For any $U(\mathfrak{g})$-modules $M_1$ and $M_2$, the adjunction isomorphism $\mathrm{Hom}_L(M_1 \otimes_L V, M_2) \simeq \mathrm{Hom}_L(M_1, M_2 \otimes_L \mathrm{Hom}_L(V, L))$ is $\mathfrak{g}$-equivariant and hence induces an isomorphism,

$$ \mathrm{Hom}_{U(\mathfrak{g})}(M_1 \otimes_L V, M_2) \simeq \mathrm{Hom}_{U(\mathfrak{g})}(M_1, M_2 \otimes_L \mathrm{Hom}_L(V, L)). $$

Thus, as $M \otimes_L \mathrm{Hom}_L(V, L)$ lies in $\mathcal{O}_{\mathrm{alg}}^{I,\infty}$ we can assume that $V = L$. Using Lemma 2.9, we can assume that $M$ is in $\mathcal{O}_{\chi}^{I,\infty}$ for some character $\chi$ and by Remark 2.8, it is sufficient to consider the case where $\chi = \chi_\lambda$. By construction of the deformed generalized Verma modules we have $\mathrm{Hom}_{U(\mathfrak{g})_{A_I}}(\widetilde{M}_I(\lambda), M) = (M^\lambda)^{n_I}$ and $\mathrm{Hom}_{U(\mathfrak{g})}(M_I(\lambda), M/\mathfrak{m}_I M) = ((M/\mathfrak{m}_I M)^\lambda)^{n_I}$. However it follows from Corollary 2.11 that $(M^\lambda)^{n_I} = M^\lambda$ and $((M/\mathfrak{m}_I M)^\lambda)^{n_I} = (M/\mathfrak{m}_I)^\lambda$. It is thus sufficient to prove that the map $M^\lambda \to (M/\mathfrak{m}_I M)^\lambda$ is surjective, which is obvious. □
### 2.3.3 Deformed Verma modules

Let $\lambda \in X^*(\mathcal{T})$ and let $V$ be a finite dimensional $U(\mathfrak{g})$-module. Then we have an isomorphism of $U(\mathfrak{g})_A$-modules

$$ \widetilde{M}(\lambda) \otimes_L V \simeq U(\mathfrak{g})_A \otimes_{U(\mathfrak{b})_A} (V_{|\mathfrak{b}} \otimes_L A(\lambda)). $$

Indeed there is a canonical map from the left to the right, which then is easily checked to be an isomorphism. As $V_{|\mathfrak{b}}$ is a successive extension of one dimensional $U(\mathfrak{b})$-modules, and as $U(\mathfrak{g})_A \otimes_{U(\mathfrak{b})_A} (-)$ is an exact functor (as follows from the PBW Theorem), we have a filtration $(\mathrm{Fil}_i)$ of $\widetilde{M}(\lambda) \otimes_L V$ such that each subquotient $\mathrm{Fil}_i / \mathrm{Fil}_{i-1}$ is isomorphic to $\widetilde{M}(\lambda + \nu_i)$ for $\nu_i$ a weight of $V$. Moreover the family $(\nu_i)$ is the family of weights of $V$ (counted with multiplicity).

**Proposition 2.12.** Let $K$ denote the fraction field of $A$. Then the filtration $(\mathrm{Fil}_i \otimes_A K)$ of $(\widetilde{M}(\lambda) \otimes_L V) \otimes_A K$ splits in the category of $U(\mathfrak{g})_K$-modules, i.e. there exists an isomorphism of $U(\mathfrak{g})_K$-modules

$$ (\widetilde{M}(\lambda) \otimes_L V) \otimes_A K \simeq \bigoplus_i (\widetilde{M}(\lambda + \nu_i) \otimes_A K) $$

compatible with the filtration $(\mathrm{Fil}_i \otimes_A K)$.

*Proof.* This is a consequence of the paragraph preceding [Soe92, Thm. 8]. □

**Lemma 2.13.** Let $\lambda \in X^*(\mathcal{T})_I^+$ be a dominant weight (with respect to $\mathcal{P}_I$) and let $V$ be a finite dimensional $U(\mathfrak{g})$-module. Let $M$ be an object of $\mathcal{O}_{\mathrm{alg}}^{I,\infty}$. Then the map

$$ \mathrm{Hom}_{U(\mathfrak{g})_{A_I}}(\widetilde{M}_I(\lambda) \otimes_L V, M) \to \mathrm{Hom}_{U(\mathfrak{g})}(M_I(\lambda) \otimes_L V, M/\mathfrak{m}_I M) $$

given by reduction modulo $\mathfrak{m}_I$ is surjective.

*Proof.* The $L$-vector space $\mathrm{Hom}_L(V, L)$ has the structure of an $U(\mathfrak{g})$-module induced by $\mathfrak{g}$-action defined by $x \cdot \phi = -\phi(x)$ for $x \in \mathfrak{g}$ and $\phi \in \mathrm{Hom}_L(V, L)$. For any $U(\mathfrak{g})$-modules $M_1$ and $M_2$, the adjunction isomorphism $\mathrm{Hom}_L(M_1 \otimes_L V, M_2) \simeq \mathrm{Hom}_L(M_1, M_2 \otimes_L \mathrm{Hom}_L(V, L))$ is $\mathfrak{g}$-equivariant and hence induces an isomorphism,

$$ \mathrm{Hom}_{U(\mathfrak{g})}(M_1 \otimes_L V, M_2) \simeq \mathrm{Hom}_{U(\mathfrak{g})}(M_1, M_2 \otimes_L \mathrm{Hom}_L(V, L)). $$

Thus, as $M \otimes_L \mathrm{Hom}_L(V, L)$ lies in $\mathcal{O}_{\mathrm{alg}}^{I,\infty}$ we can assume that $V = L$. Using Lemma 2.9, we can assume that $M$ is in $\mathcal{O}_{\chi}^{I,\infty}$ for some character $\chi$ and by Remark 2.8, it is sufficient to consider the case where $\chi = \chi_\lambda$. By construction of the deformed generalized Verma modules we have $\mathrm{Hom}_{U(\mathfrak{g})_{A_I}}(\widetilde{M}_I(\lambda), M) = (M^\lambda)^{n_I}$ and $\mathrm{Hom}_{U(\mathfrak{g})}(M_I(\lambda), M/\mathfrak{m}_I M) = ((M/\mathfrak{m}_I M)^\lambda)^{n_I}$. However it follows from Corollary 2.11 that $(M^\lambda)^{n_I} = M^\lambda$ and $((M/\mathfrak{m}_I M)^\lambda)^{n_I} = (M/\mathfrak{m}_I)^\lambda$. It is thus sufficient to prove that the map $M^\lambda \to (M/\mathfrak{m}_I M)^\lambda$ is surjective, which is obvious. □
**Proposition 2.14.** Let $M$ be an object of the category $\mathcal{O}_{\text{alg}}^{I,\infty}$. Then there exist weights $\lambda_1, \dots, \lambda_r \in X^*(\bar{T})_I^+$ and finite dimensional $U(\mathfrak{g})$-modules $W_1, \dots, W_r$ and a surjective map of $U(\mathfrak{g})_{A_I}$-modules

$$ (\widetilde{M}_I(\lambda_1) \otimes_L W_1) \oplus \cdots \oplus (\widetilde{M}_I(\lambda_r) \otimes_L W_r) \to M. \qquad (3) $$

In particular $M$ is a quotient of an object of the category $\tilde{\mathcal{O}}_{\text{alg}}^I$. Moreover there exists an integer $N \ge 0$ such that the map (3) factors through

$$ ((\widetilde{M}_I(\lambda_1) \otimes_L W_1) \oplus \cdots \oplus (\widetilde{M}_I(\lambda_r) \otimes_L W_r)) \otimes_{A_I} A_I / \mathfrak{m}_I^N. $$

*Proof.* By [Hum08, Thm. 9.8] (and its proof), there exist dominant weights $\lambda_1, \dots, \lambda_r$, finite dimensional $U(\mathfrak{g})$-modules $W_1, \dots, W_r$ and a surjective map

$$ (M_I(\lambda_1) \otimes_L W_1) \oplus \cdots (M_I(\lambda_r) \otimes_L W_r) \to M / \mathfrak{m}_I M. $$

By Lemma 2.13, this map can be lifted into a $U(\mathfrak{g})_{A_I}$-equivariant map

$$ \widetilde{M}_I(\lambda_1) \otimes_L W_1 \oplus \cdots \widetilde{M}_I(\lambda_r) \otimes_L W_r \to M $$

which is surjective by Nakayama's Lemma. The last assertion is a consequence of the fact that $M$ is finitely generated as a $U(\mathfrak{g})$-module and all its elements are killed by some power of $\mathfrak{m}_I$ so that $M$ is killed by $\mathfrak{m}_I^N$ for some $N \ge 0$. $\square$

## 2.4 Bimodule structure

Let $\xi : Z(\mathfrak{g}) \to U(\mathfrak{t})$ be the Harish-Chandra map. Recall that it is defined as follows: for $x \in Z(\mathfrak{g})$ there exists a unique element $\xi(x) \in U(\mathfrak{t})$ such that $x \in \xi(x) + U(\mathfrak{g})\mathfrak{n}$ (see [Kna01, Lem. 8.17]). For any $\nu \in X^*(\bar{T})$ we denote by $t_\nu$ the unique endomorphism of $U(\mathfrak{t})$ such that $t_\nu(x) = x + \nu(x)$ for $x \in \mathfrak{t}$. Note that $t_{-\delta_G} \circ \xi$ induces an isomorphism from $Z(\mathfrak{g})$ on to $U(\mathfrak{t})^W$ (see [Kna01, Thm. 6.18]). For a dominant weight $\lambda \in X^*(\bar{T})$ we define a map

$$ h_\lambda : A \otimes_L Z(\mathfrak{g}) \xrightarrow{\text{Id} \otimes \xi} A \otimes_L \otimes U(\mathfrak{t}) \xrightarrow{\text{Id} \otimes t_\lambda} A \otimes_{A^W} A $$

following [Soe92, §3.2], It follows from [Soe92, Thm. 9] that $h_\lambda$ is surjective (note that $W_\lambda$ in loc. cit. is trivial in our situation). If $I \subset \Delta$ is a finite subset, tensorization on the left with $p_I : A \to A_I$ yields a map $h_\lambda : A_I \otimes_L Z(\mathfrak{g}) \to A_I \otimes_{A^W} A$.

For $w \in W$, let $I_w \subset A_I \otimes_L Z(\mathfrak{g})$ denote the kernel of the map

$$ h_{\lambda,w} : A_I \otimes_L Z(\mathfrak{g}) \xrightarrow{\text{Id} \otimes h_\lambda} A_I \otimes_{A^W} A \xrightarrow{x \otimes y \to (x p_I(\text{Ad}(w)y))} A_I. $$

It is not hard to see that this kernel only depends on the choice of $\bar{w} \in W_I \setminus W$.

**Proposition 2.15.** For $w \in I^W$, the $A_I \otimes_L Z(\mathfrak{g})$-modules $\widetilde{M}_I(w \cdot \lambda)$ and $\widetilde{M}_I(w \cdot \lambda)^\vee$ are annihilated by $I_w$.
**Proposition 2.14.** Let $M$ be an object of the category $\mathcal{O}_{\text{alg}}^{I,\infty}$. Then there exist weights $\lambda_1, \dots, \lambda_r \in X^*(\bar{T})_I^+$ and finite dimensional $U(\mathfrak{g})$-modules $W_1, \dots, W_r$ and a surjective map of $U(\mathfrak{g})_{A_I}$-modules

$$ (\widetilde{M}_I(\lambda_1) \otimes_L W_1) \oplus \cdots \oplus (\widetilde{M}_I(\lambda_r) \otimes_L W_r) \to M. \qquad (3) $$

In particular $M$ is a quotient of an object of the category $\tilde{\mathcal{O}}_{\text{alg}}^I$. Moreover there exists an integer $N \ge 0$ such that the map (3) factors through

$$ ((\widetilde{M}_I(\lambda_1) \otimes_L W_1) \oplus \cdots \oplus (\widetilde{M}_I(\lambda_r) \otimes_L W_r)) \otimes_{A_I} A_I / \mathfrak{m}_I^N. $$

*Proof.* By [Hum08, Thm. 9.8] (and its proof), there exist dominant weights $\lambda_1, \dots, \lambda_r$, finite dimensional $U(\mathfrak{g})$-modules $W_1, \dots, W_r$ and a surjective map

$$ (M_I(\lambda_1) \otimes_L W_1) \oplus \cdots (M_I(\lambda_r) \otimes_L W_r) \to M / \mathfrak{m}_I M. $$

By Lemma 2.13, this map can be lifted into a $U(\mathfrak{g})_{A_I}$-equivariant map

$$ \widetilde{M}_I(\lambda_1) \otimes_L W_1 \oplus \cdots \widetilde{M}_I(\lambda_r) \otimes_L W_r \to M $$

which is surjective by Nakayama's Lemma. The last assertion is a consequence of the fact that $M$ is finitely generated as a $U(\mathfrak{g})$-module and all its elements are killed by some power of $\mathfrak{m}_I$ so that $M$ is killed by $\mathfrak{m}_I^N$ for some $N \ge 0$. $\square$

## 2.4 Bimodule structure

Let $\xi : Z(\mathfrak{g}) \to U(\mathfrak{t})$ be the Harish-Chandra map. Recall that it is defined as follows: for $x \in Z(\mathfrak{g})$ there exists a unique element $\xi(x) \in U(\mathfrak{t})$ such that $x \in \xi(x) + U(\mathfrak{g})\mathfrak{n}$ (see [Kna01, Lem. 8.17]). For any $\nu \in X^*(\bar{T})$ we denote by $t_\nu$ the unique endomorphism of $U(\mathfrak{t})$ such that $t_\nu(x) = x + \nu(x)$ for $x \in \mathfrak{t}$. Note that $t_{-\delta_G} \circ \xi$ induces an isomorphism from $Z(\mathfrak{g})$ on to $U(\mathfrak{t})^W$ (see [Kna01, Thm. 6.18]). For a dominant weight $\lambda \in X^*(\bar{T})$ we define a map

$$ h_\lambda : A \otimes_L Z(\mathfrak{g}) \xrightarrow{\text{Id}\otimes\xi} A \otimes_L \otimes U(\mathfrak{t}) \xrightarrow{\text{Id}\otimes t_\lambda} A \otimes_{A^W} A $$

following [Soe92, §3.2], It follows from [Soe92, Thm. 9] that $h_\lambda$ is surjective (note that $W_\lambda$ in loc. cit. is trivial in our situation). If $I \subset \Delta$ is a finite subset, tensorization on the left with $p_I : A \to A_I$ yields a map $h_\lambda : A_I \otimes_L Z(\mathfrak{g}) \to A_I \otimes_{A^W} A$.

For $w \in W$, let $I_w \subset A_I \otimes_L Z(\mathfrak{g})$ denote the kernel of the map

$$ h_{\lambda,w} : A_I \otimes_L Z(\mathfrak{g}) \xrightarrow{\text{Id}\otimes h_\lambda} A_I \otimes_{A^W} A \xrightarrow{x\otimes y \to (x p_I(\text{Ad}(w)y))} A_I. $$

It is not hard to see that this kernel only depends on the choice of $\bar{w} \in W_I \setminus W$.

**Proposition 2.15.** For $w \in I^W$, the $A_I \otimes_L Z(\mathfrak{g})$-modules $\widetilde{M}_I(w \cdot \lambda)$ and $\widetilde{M}_I(w \cdot \lambda)^\vee$ are annihilated by $I_w$.
Proof. The result for $\tilde{M}_I(w \cdot \lambda)^\vee$ follows from the result for $\tilde{M}_I(w \cdot \lambda)^\vee$ and the inclusion

$$ \tilde{M}_I(w \cdot \lambda)^\vee \subset \operatorname{Hom}_A(\tilde{M}_I(w \cdot \lambda), A). $$

Hence it is enough to check that the action of $A_I \otimes_L Z(\mathfrak{g})$ on $\tilde{M}_I(w \cdot \lambda)$ factors through $h_{\lambda,w}$. As this action is central and $\tilde{M}_I(w \cdot \lambda)$ is generated by $\tilde{M}_I(w \cdot \lambda)^{w \cdot \lambda}$ as an $U(\mathfrak{g})_{A_I}$-module, it is sufficient to check that the action $A_I \otimes_L Z(\mathfrak{g})$ on $\tilde{M}_I(w \cdot \lambda)^{w \cdot \lambda}$ factors through $h_{\lambda,w}$. Using the fact that $\mathfrak{n}$ acts trivially on $\tilde{M}_I(w \cdot \lambda)^{w \cdot \lambda}$, an element $x \in Z(\mathfrak{g})$ acts on this space via $\xi$. For the clarity of the computation let us write $\varepsilon_\nu : U(\mathfrak{t}) \to A_I$ for the $L$-algebra homomorphism associated to an $L$-linear map $\nu : \mathfrak{t} \to A_I$ and let $\iota : \mathfrak{t} \hookrightarrow A \to A_I$. Then for $x \in Z(\mathfrak{g})$ and $v \in \tilde{M}_I(w \cdot \lambda)$, we have

$$ \begin{aligned} \varepsilon_{w \cdot \lambda + \iota}(\xi(x)) &= \varepsilon_{w(\lambda + \delta_G + w^{-1}(\iota))(\mathfrak{t} - \delta_G)(\xi(x))} \\ &= \varepsilon_{\lambda + \delta_G + w^{-1}(\iota)(\mathfrak{t} - \delta_G)(\xi(x))} \\ &= \varepsilon_{w^{-1}(\iota)(h_\lambda(x))} = p_I(\operatorname{Ad}(w)(h_\lambda(x))) \end{aligned} $$

(where we use that the image of $t_{-\delta_G} \circ \xi$ lies in $U(\mathfrak{t})^W$). As an element $y \in U(\mathfrak{t})$ acts by multiplication by $\varepsilon_{w \cdot \lambda + \iota}(y)$ on $\tilde{M}_I(w \cdot \lambda)^{w \cdot \lambda}$, we conclude that an element $x \otimes z \in A_I \otimes_L Z(\mathfrak{g})$ acts by multiplication by $x p_I(\operatorname{Ad}(w)(h_\lambda(x)))$ on $\tilde{M}_I(w \cdot \lambda)^{w \cdot \lambda}$, which is the desired formula. $\square$

Remark 2.16. The ring $U(\mathfrak{t})$ (resp. $U(\mathfrak{z}_I)$) is the affine coordinate ring of the (affine) $L$-scheme associated to the dual $\mathfrak{t}^*$ of $\mathfrak{t}$ (resp. to the dual $\mathfrak{z}_I^*$ of $\mathfrak{z}_I$) so that $A$ (resp. $A_I$) is the stalk of the structure sheaf of $\mathfrak{t}^*$ (resp. of $\mathfrak{z}_I^*$) at the origin. The ideal $I_w$ is the ideal defining the irreducible component $T_{I,w}$ of $(\mathfrak{z}_I^* \times_{\mathfrak{t}^*/W} \mathfrak{t}^*)_{(0,0)}$ consisting of pairs $(\lambda, \mu) \in \mathfrak{z}_I^* \times \mathfrak{t}^*$ of characters such that $\mu = w(\lambda)$.

Later in the paper we will view the $L$-scheme $\mathfrak{t}^*$ as the Lie algebra $\mathfrak{t}^\vee$ of the dual torus $\underline{T}^\vee_L$ of the Langlands dual group $\underline{G}^\vee_L$, that we consider as an algebraic group over $L$. As we will later specialize to the case where $\underline{G}$ is isomorphic to a product of $r$ copies of $\mathrm{GL}_n$ the reductive group $\underline{G}$ is self dual and we will identify $\mathfrak{t}^* = \mathfrak{t}^\vee$ with $\mathfrak{t}$ in order to avoid the additional $(-)^\vee$ in the notation. In particular we will consider $U(\mathfrak{t})$ as the affine coordinate ring of $\mathfrak{t}$. The inclusion $\mathfrak{z}_I^* \hookrightarrow \mathfrak{t}_I^*$ induced by the projection $p_I : \mathfrak{t}_I \to \mathfrak{z}_I$ is then identified with the inclusion $\mathfrak{z}_I^\vee \hookrightarrow \mathfrak{t}_I^\vee$ of the center of the Lie algebra of the Langlands dual group of $\underline{L}$ and again we use self duality (in the case of products of copies of $\mathrm{GL}_n$) to identify this map with $\mathfrak{z}_I \hookrightarrow \mathfrak{t}_I$. Hence we obtain a canonical map $\mathfrak{z}_I \hookrightarrow \mathfrak{t}$ of $L$-schemes corresponding to the morphism $U(\mathfrak{t}) \to U(\mathfrak{z}_I)$. With this identification the ideal $I_w$ defines the irreducible component $T_{I,w}$ of $(\mathfrak{z}_I \times_{\mathfrak{t}/W} \mathfrak{t})_{(0,0)}$ whose points are the pairs $(x, y) \in \mathfrak{t}^2$ such that $y = w^{-1}(x)$.

We finally recall the following result of Soergel (Endomorphismensatz 7 [Soe90]).

**Proposition 2.17.** The action of $Z(\mathfrak{g})$ on $P(w_0 \cdot \lambda)$ factors through the map $t_\lambda \circ \xi : Z(\mathfrak{g}) \to L \otimes_{A^W} A$ and induces an isomorphism $L \otimes_{A^W} A \simeq \mathrm{End}_\mathcal{O}(P(w_0 \cdot \lambda))$.
Proof. The result for $\tilde{M}_I(w \cdot \lambda)^\vee$ follows from the result for $\tilde{M}_I(w \cdot \lambda)^\vee$ and the inclusion

$$ \tilde{M}_I(w \cdot \lambda)^\vee \subset \operatorname{Hom}_A(\tilde{M}_I(w \cdot \lambda), A). $$

Hence it is enough to check that the action of $A_I \otimes_L Z(\mathfrak{g})$ on $\tilde{M}_I(w \cdot \lambda)$ factors through $h_{\lambda,w}$. As this action is central and $\tilde{M}_I(w \cdot \lambda)$ is generated by $\tilde{M}_I(w \cdot \lambda)^{w \cdot \lambda}$ as an $U(\mathfrak{g})_{A_I}$-module, it is sufficient to check that the action $A_I \otimes_L Z(\mathfrak{g})$ on $\tilde{M}_I(w \cdot \lambda)^{w \cdot \lambda}$ factors through $h_{\lambda,w}$. Using the fact that $\mathfrak{n}$ acts trivially on $\tilde{M}_I(w \cdot \lambda)^{w \cdot \lambda}$, an element $x \in Z(\mathfrak{g})$ acts on this space via $\xi$. For the clarity of the computation let us write $\varepsilon_\nu : U(\mathfrak{t}) \to A_I$ for the $L$-algebra homomorphism associated to an $L$-linear map $\nu : \mathfrak{t} \to A_I$ and let $\iota : \mathfrak{t} \hookrightarrow A \to A_I$. Then for $x \in Z(\mathfrak{g})$ and $v \in \tilde{M}_I(w \cdot \lambda)$, we have

$$ \begin{aligned} \varepsilon_{w \cdot \lambda + \iota}(\xi(x)) &= \varepsilon_{w(\lambda + \delta_G + w^{-1}(\iota))(\mathfrak{t} - \delta_G)(\xi(x))} \\ &= \varepsilon_{\lambda + \delta_G + w^{-1}(\iota)(\mathfrak{t} - \delta_G)(\xi(x))} \\ &= \varepsilon_{w^{-1}(\iota)(h_\lambda(x))} = p_I(\operatorname{Ad}(w)(h_\lambda(x))) \end{aligned} $$

(where we use that the image of $t_{-\delta_G} \circ \xi$ lies in $U(\mathfrak{t})^W$). As an element $y \in U(\mathfrak{t})$ acts by multiplication by $\varepsilon_{w \cdot \lambda + \iota}(y)$ on $\tilde{M}_I(w \cdot \lambda)^{w \cdot \lambda}$, we conclude that an element $x \otimes z \in A_I \otimes_L Z(\mathfrak{g})$ acts by multiplication by $xp_I(\operatorname{Ad}(w)(h_\lambda(x)))$ on $\tilde{M}_I(w \cdot \lambda)^{w \cdot \lambda}$, which is the desired formula. $\square$

Remark 2.16. The ring $U(\mathfrak{t})$ (resp. $U(\mathfrak{z}_I)$) is the affine coordinate ring of the (affine) $L$-scheme associated to the dual $\mathfrak{t}^*$ of $\mathfrak{t}$ (resp. to the dual $\mathfrak{z}_I^*$ of $\mathfrak{z}_I$) so that $A$ (resp. $A_I$) is the stalk of the structure sheaf of $\mathfrak{t}^*$ (resp. of $\mathfrak{z}_I^*$) at the origin. The ideal $I_w$ is the ideal defining the irreducible component $T_{I,w}$ of $(\mathfrak{z}_I^* \times_{\mathfrak{t}^*/W} \mathfrak{t}^*)_{(0,0)}$ consisting of pairs $(\lambda, \mu) \in \mathfrak{z}_I^* \times \mathfrak{t}^*$ of characters such that $\mu = w(\lambda)$.

Later in the paper we will view the $L$-scheme $\mathfrak{t}^*$ as the Lie algebra $\mathfrak{t}^\vee$ of the dual torus $\underline{T}_L^\vee$ of the Langlands dual group $\underline{G}_L^\vee$, that we consider as an algebraic group over $L$. As we will later specialize to the case where $\underline{G}$ is isomorphic to a product of $r$ copies of $\mathrm{GL}_n$ the reductive group $\underline{G}$ is self dual and we will identify $\mathfrak{t}^* = \mathfrak{t}^\vee$ with $\mathfrak{t}$ in order to avoid the additional $(-)^\vee$ in the notation. In particular we will consider $U(\mathfrak{t})$ as the affine coordinate ring of $\mathfrak{t}$. The inclusion $\mathfrak{z}_I^* \hookrightarrow \mathfrak{t}_I^*$ induced by the projection $p_I : \mathfrak{t}_I \to \mathfrak{z}_I$ is then identified with the inclusion $\mathfrak{z}_I^\vee \hookrightarrow \mathfrak{t}_I^\vee$ of the center of the Lie algebra of the Langlands dual group of $\underline{L}$ and again we use self duality (in the case of products of copies of $\mathrm{GL}_n$) to identify this map with $\mathfrak{z}_I \hookrightarrow \mathfrak{t}_I$. Hence we obtain a canonical map $\mathfrak{z}_I \hookrightarrow \mathfrak{t}$ of $L$-schemes corresponding to the morphism $U(\mathfrak{t}) \to U(\mathfrak{z}_I)$. With this identification the ideal $I_w$ defines the irreducible component $T_{I,w}$ of $(\mathfrak{z}_I \times_{\mathfrak{t}/W} \mathfrak{t})_{(0,0)}$ whose points are the pairs $(x, y) \in \mathfrak{t}^2$ such that $y = w^{-1}(x)$.

We finally recall the following result of Soergel (Endomorphismensatz 7 [Soe90]).

**Proposition 2.17.** The action of $Z(\mathfrak{g})$ on $P(w_0 \cdot \lambda)$ factors through the map $t_\lambda \circ \xi : Z(\mathfrak{g}) \to L \otimes_{A^W} A$ and induces an isomorphism $L \otimes_{A^W} A \simeq \mathrm{End}_\mathcal{O}(P(w_0 \cdot \lambda))$.
## 3 The Emerton–Jacquet functor

Let $\underline{G}$ be a quasi-split reductive group defined over $\mathbb{Q}_p$. Let $\underline{B}$ be a Borel subgroup and $\underline{T}$ be a maximal torus of $\underline{G}$ contained in $\underline{B}$. We set $G := \underline{G}(\mathbb{Q}_p)$, $B := \underline{B}(\mathbb{Q}_p)$, $T := \underline{T}(\mathbb{Q}_p)$. We also fix $L$ a finite extension of $\mathbb{Q}_p$ which will be the coefficient field of our representations. We assume that $L$ is big enough so that the torus $\underline{T} \times_{\mathbb{Q}_p} L$ is split (and then $\underline{G} \times_{\mathbb{Q}_p} L$ is split). We denote $\mathfrak{g}$, $\mathfrak{b}$ etc. the Lie algebras of $\underline{G} \times_{\mathbb{Q}_p} L$, $\underline{B} \times_{\mathbb{Q}_p} L$ etc. In the following we will consider the category $\text{Rep}_L^{\text{la}}G$ of locally analytic $G$-representations on locally convex $L$-vector spaces, as well as the corresponding variants for the $(\mathbb{Q}_p$-analytic) groups $B, T$, etc. In [Eme06a, Def. 3.4.5] Emerton constructs a functor

$$J_B : \text{Rep}_L^{\text{la}}G \to \text{Rep}_L^{\text{la}}T$$

that we refer to as the Emerton–Jacquet functor. It is defined as follows: Let $N_0$ be a compact open subgroup of $N$ and let $T^+ := \{t \in T \mid tN_0t^{-1} \subset N_0\}$. If $V$ is a $L$-linear representation of $B$, we endow the $L$-vector space $V^{N_0}$ with the action of the monoid $T^+$ defined by

$$[t]v := [N_0 : tN_0t^{-1}]^{-1} \sum_{u \in N_0/tN_0t^{-1}} ut(v).$$

Then $J_B(V)$ is the finite slope space $(V^{N_0})_{\text{fs}}$ of $V^{N_0}$ with respect to the action of $T^+$ on which the $T^+$-action extends to a locally analytic representation of $T$.

### 3.1 Families of locally analytic representations of the Borel subgroup

Let $s \in \mathbb{Z}_{\ge 0}$ be an integer and let $\Pi$ be a locally analytic $L$-representation of $\mathbb{Z}_p^s \times B$. We consider the following hypothesis on $\Pi$:

**Hypothesis 3.1.** There exists a locally analytic representation of $N_0$ on a locally convex $L$-vector space of compact type $V$ such that

$$\Pi_{|\mathbb{Z}_p^s \times N_0} \simeq C^{\text{la}}(\mathbb{Z}_p^s, L) \hat{\otimes}_L V.$$

Given $s$, we set $S := \mathcal{O}_L[\mathbb{Z}_p^s]$ and write $\text{Spf}(S)^{\text{rig}}$ for the rigid analytic generic fiber of $\text{Spf}(S)$. This space is a rigid analytic open polydisc and we write

$$S^{\text{rig}} = \Gamma(\text{Spf}(S)^{\text{rig}}, \mathcal{O}_{\text{Spf} S^{\text{rig}}})$$

for its ring of rigid analytic functions, which is a Fréchet $L$-algebra (when endowed with its natural topology). We note that a finitely generated, projective $S^{\text{rig}}$-module $C$ defines a vector bundle on $\text{Spf}(S)^{\text{rig}}$. As every vector bundle on a rigid analytic polydisc is free, it follows that $C$ is free as well, i.e. every finitely generated projective $S^{\text{rig}}$-module is finite free. Moreover, finite dimensional quotients of $S^{\text{rig}}$ admit resolutions by a perfect complexes:
### 3 The Emerton–Jacquet functor

Let $\underline{G}$ be a quasi-split reductive group defined over $\mathbb{Q}_p$. Let $\underline{B}$ be a Borel subgroup and $\underline{T}$ be a maximal torus of $\underline{G}$ contained in $\underline{B}$. We set $G := \underline{G}(\mathbb{Q}_p)$, $B := \underline{B}(\mathbb{Q}_p)$, $T := \underline{T}(\mathbb{Q}_p)$. We also fix $L$ a finite extension of $\mathbb{Q}_p$ which will be the coefficient field of our representations. We assume that $L$ is big enough so that the torus $\underline{T} \times_{\mathbb{Q}_p} L$ is split (and then $\underline{G} \times_{\mathbb{Q}_p} L$ is split). We denote $\mathfrak{g}$, $\mathfrak{b}$ etc. the Lie algebras of $\underline{G} \times_{\mathbb{Q}_p} L$, $\underline{B} \times_{\mathbb{Q}_p} L$ etc. In the following we will consider the category $\text{Rep}_L^{\text{la}}G$ of locally analytic $G$-representations on locally convex $L$-vector spaces, as well as the corresponding variants for the $(\mathbb{Q}_p$-analytic) groups $B, T$, etc. In [Eme06a, Def. 3.4.5] Emerton constructs a functor

$$J_B : \text{Rep}_L^{\text{la}}G \to \text{Rep}_L^{\text{la}}T$$

that we refer to as the Emerton–Jacquet functor. It is defined as follows: Let $N_0$ be a compact open subgroup of $N$ and let $T^+ := \{t \in T \mid tN_0t^{-1} \subset N_0\}$. If $V$ is a $L$-linear representation of $B$, we endow the $L$-vector space $V^{N_0}$ with the action of the monoid $T^+$ defined by

$$[t]v := [N_0 : tN_0t^{-1}]^{-1} \sum_{u \in N_0/tN_0t^{-1}} ut(v).$$

Then $J_B(V)$ is the finite slope space $(V^{N_0})_{\text{fs}}$ of $V^{N_0}$ with respect to the action of $T^+$ on which the $T^+$-action extends to a locally analytic representation of $T$.

#### 3.1 Families of locally analytic representations of the Borel subgroup

Let $s \in \mathbb{Z}_{\ge 0}$ be an integer and let $\Pi$ be a locally analytic $L$-representation of $\mathbb{Z}_p^s \times B$. We consider the following hypothesis on $\Pi$:

**Hypothesis 3.1.** There exists a locally analytic representation of $N_0$ on a locally convex $L$-vector space of compact type $V$ such that

$$\Pi_{|\mathbb{Z}_p^s \times N_0} \simeq C^{\text{la}}(\mathbb{Z}_p^s, L) \hat{\otimes}_L V.$$

Given $s$, we set $S := \mathcal{O}_L[\mathbb{Z}_p^s]$ and write $\text{Spf}(S)^{\text{rig}}$ for the rigid analytic generic fiber of $\text{Spf}(S)$. This space is a rigid analytic open polydisc and we write

$$S^{\text{rig}} = \Gamma(\text{Spf}(S)^{\text{rig}}, \mathcal{O}_{\text{Spf} S^{\text{rig}}})$$

for its ring of rigid analytic functions, which is a Fréchet $L$-algebra (when endowed with its natural topology). We note that a finitely generated, projective $S^{\text{rig}}$-module $C$ defines a vector bundle on $\text{Spf}(S)^{\text{rig}}$. As every vector bundle on a rigid analytic polydisc is free, it follows that $C$ is free as well, i.e. every finitely generated projective $S^{\text{rig}}$-module is finite free. Moreover, finite dimensional quotients of $S^{\text{rig}}$ admit resolutions by a perfect complexes:
**Lemma 3.2.** Let $\mathfrak{a} \subset S^{\text{rig}}$ be a closed strict ideal such that $\dim_L S^{\text{rig}}/\mathfrak{a} < \infty$. Then there exists perfect complex $C_\bullet$ of $S^{\text{rig}}$-modules which is a resolution of $S^{\text{rig}}/\mathfrak{a}$ and such that $C_0 = S^{\text{rig}}$.

*Proof.* As $S[1/p]$ is dense in $S^{\text{rig}}$, its image in $S^{\text{rig}}/\mathfrak{a}$ is dense $L$-vector space and, as $S^{\text{rig}}/\mathfrak{a}$ is finite dimensional, is in fact equal to $S^{\text{rig}}/\mathfrak{a}$. Setting $\mathfrak{a}_0 := \mathfrak{a} \cap S[1/p]$, we have $S[1/p]/\mathfrak{a}_0 \simeq S^{\text{rig}}/\mathfrak{a}$. As $S^{\text{rig}}$ is a flat $S[1/p]$-module, it is sufficient to prove that $S[1/p]/\mathfrak{a}_0$ has a finite resolution by finite projective $S[1/p]$-modules, which is a consequence of the fact that $S[1/p]$ is a regular noetherian ring. $\square$

Let $C_\bullet$ be a complex of finite free $S^{\text{rig}}$-modules. For each $n \ge 0$, $C_n$ is endowed with its canonical topology induced by the topology of $S^{\text{rig}}$, then the differentials in the complex $C_\bullet$ are continuous. The complex $\Pi^\bullet := \text{Hom}_{S^{\text{rig}}}(C_\bullet, \Pi)$ is then a complex of locally analytic $L$-representations of $\mathbb{Z}_p^s \times B$. We also set $\Pi^{N_0, \bullet} := \text{Hom}_{S^{\text{rig}}}(C_\bullet, \Pi^{N_0})$ and $J_B(\Pi)^\bullet := \text{Hom}_{S^{\text{rig}}}(C_\bullet, J_B(\Pi))$.

**Lemma 3.3.** Let $0 \to U \to V \to W \to 0$ be a short exact sequence of topological $L$-vector spaces of compact type (resp. nuclear Fréchet spaces) and let $X$ be a topological $L$-vector space of compact type (resp. nuclear Fréchet space). Then the following sequence is exact

$$0 \to U \hat{\otimes}_L X \to V \hat{\otimes}_L X \to W \hat{\otimes}_L X \to 0.$$

*Proof.* The claim follows from [Sch11, Lemm. 4.13], [ST02, Cor. 1.4] and from [Eme17, Prop. 1.1.32]. $\square$

**Lemma 3.4.** Let $\Pi$ be a locally analytic representation of $\mathbb{Z}_p^s \times B$ satisfying Hypothesis 3.1. Then the two complexes $\Pi^\bullet$ and $\Pi^{N_0, \bullet}$ are complexes of $L$-vector spaces of compact type with strict continuous transition maps. Moreover for any integer $n \ge 0$, we have an isomorphism of topological $T^+$-modules

$$H^n(\Pi^{N_0, \bullet}) \simeq H^n(\Pi^\bullet)^{N_0}.$$

*Proof.* Fix an isomorphism $\Pi_{|\mathbb{Z}_p^s \times N_0} \simeq C^\text{la}(\mathbb{Z}_p^s, L) \hat{\otimes}_L V$ whose existence comes from hypothesis 3.1. As any $C_m$ is a finite free $S^{\text{rig}}$-module and as the completed tensor product $-\hat{\otimes}_L -$ commutes with finite direct sums ([Koh07, Lem. 1.2.13]), we have an isomorphism of complexes of topological representations of $\mathbb{Z}_p^s \times N_0$:

$$\Pi^\bullet \simeq \text{Hom}_{S^{\text{rig}}}(C_\bullet, C^\text{la}(\mathbb{Z}_p^s, L)) \hat{\otimes}_L V.$$

As $C^\text{la}(\mathbb{Z}_p^s, L)$ is an admissible locally analytic representation of $\mathbb{Z}_p^s$, the complex $\text{Hom}_{S^{\text{rig}}}(C_\bullet, C^\text{la}(\mathbb{Z}_p^s, L))$ has strict transition maps with closed images ([ST03, Prop. 6.4]). We deduce from this fact and from Lemma 3.3 that the complex $\Pi^\bullet$ has strict transition maps and that we have topological isomorphisms $H^n(\Pi^\bullet) \simeq H^n(\text{Hom}_{S^{\text{rig}}}(C_\bullet, C^\text{la}(\mathbb{Z}_p^s, L))) \hat{\otimes}_L V$ for any
**Lemma 3.2.** Let $\mathfrak{a} \subset S^{\text{rig}}$ be a closed strict ideal such that $\dim_L S^{\text{rig}}/\mathfrak{a} < \infty$. Then there exists perfect complex $C_\bullet$ of $S^{\text{rig}}$-modules which is a resolution of $S^{\text{rig}}/\mathfrak{a}$ and such that $C_0 = S^{\text{rig}}$.

*Proof.* As $S[1/p]$ is dense in $S^{\text{rig}}$, its image in $S^{\text{rig}}/\mathfrak{a}$ is dense $L$-vector space and, as $S^{\text{rig}}/\mathfrak{a}$ is finite dimensional, is in fact equal to $S^{\text{rig}}/\mathfrak{a}$. Setting $\mathfrak{a}_0 := \mathfrak{a} \cap S[1/p]$, we have $S[1/p]/\mathfrak{a}_0 \simeq S^{\text{rig}}/\mathfrak{a}$. As $S^{\text{rig}}$ is a flat $S[1/p]$-module, it is sufficient to prove that $S[1/p]/\mathfrak{a}_0$ has a finite resolution by finite projective $S[1/p]$-modules, which is a consequence of the fact that $S[1/p]$ is a regular noetherian ring. $\square$

Let $C_\bullet$ be a complex of finite free $S^{\text{rig}}$-modules. For each $n \ge 0$, $C_n$ is endowed with its canonical topology induced by the topology of $S^{\text{rig}}$, then the differentials in the complex $C_\bullet$ are continuous. The complex $\Pi^\bullet := \text{Hom}_{S^{\text{rig}}}(C_\bullet, \Pi)$ is then a complex of locally analytic $L$-representations of $\mathbb{Z}_p^s \times B$. We also set $\Pi^{N_0, \bullet} := \text{Hom}_{S^{\text{rig}}}(C_\bullet, \Pi^{N_0})$ and $J_B(\Pi)^\bullet := \text{Hom}_{S^{\text{rig}}}(C_\bullet, J_B(\Pi))$.

**Lemma 3.3.** Let $0 \to U \to V \to W \to 0$ be a short exact sequence of topological $L$-vector spaces of compact type (resp. nuclear Fréchet spaces) and let $X$ be a topological $L$-vector space of compact type (resp. nuclear Fréchet space). Then the following sequence is exact

$$0 \to U \hat{\otimes}_L X \to V \hat{\otimes}_L X \to W \hat{\otimes}_L X \to 0.$$

*Proof.* The claim follows from [Sch11, Lemm. 4.13], [ST02, Cor. 1.4] and from [Eme17, Prop. 1.1.32]. $\square$

**Lemma 3.4.** Let $\Pi$ be a locally analytic representation of $\mathbb{Z}_p^s \times B$ satisfying Hypothesis 3.1. Then the two complexes $\Pi^\bullet$ and $\Pi^{N_0, \bullet}$ are complexes of $L$-vector spaces of compact type with strict continuous transition maps. Moreover for any integer $n \ge 0$, we have an isomorphism of topological $T^+$-modules

$$H^n(\Pi^{N_0, \bullet}) \simeq H^n(\Pi^\bullet)^{N_0}.$$

*Proof.* Fix an isomorphism $\Pi_{|\mathbb{Z}_p^s \times N_0} \simeq C^\text{la}(\mathbb{Z}_p^s, L) \hat{\otimes}_L V$ whose existence comes from hypothesis 3.1. As any $C_m$ is a finite free $S^{\text{rig}}$-module and as the completed tensor product $-\hat{\otimes}_L -$ commutes with finite direct sums ([Koh07, Lem. 1.2.13]), we have an isomorphism of complexes of topological representations of $\mathbb{Z}_p^s \times N_0$:

$$\Pi^\bullet \simeq \text{Hom}_{S^{\text{rig}}}(C_\bullet, C^\text{la}(\mathbb{Z}_p^s, L)) \hat{\otimes}_L V.$$

As $C^\text{la}(\mathbb{Z}_p^s, L)$ is an admissible locally analytic representation of $\mathbb{Z}_p^s$, the complex $\text{Hom}_{S^{\text{rig}}}(C_\bullet, C^\text{la}(\mathbb{Z}_p^s, L))$ has strict transition maps with closed images ([ST03, Prop. 6.4]). We deduce from this fact and from Lemma 3.3 that the complex $\Pi^\bullet$ has strict transition maps and that we have topological isomorphisms $H^n(\Pi^\bullet) \simeq H^n(\text{Hom}_{S^{\text{rig}}}(C_\bullet, C^\text{la}(\mathbb{Z}_p^s, L))) \hat{\otimes}_L V$ for any
$n \ge 0$. The commutation of $\hat{\otimes}_L$ with finite direct sum implies that we have a topological isomorphism of $L$-vector spaces for any $m \ge 0$:

$$(\mathrm{Hom}_{\mathrm{Srig}}(C_m, \Pi^{N_0}) \simeq \mathrm{Hom}_{\mathrm{Srig}}(C_m, C^{\mathrm{la}}(\mathbb{Z}_p^s, L)) \hat{\otimes}_L V^{N_0}).$$

We deduce as before that the complex $\Pi^{\bullet, N_0}$ has strict transition maps and that we have isomorphisms

$$H^n(\Pi^{N_0, \bullet}) \simeq H^n(\Pi^{\bullet})^{N_0}$$

for any $n \ge 0$.

□

**Proposition 3.5.** For any integer $n \ge 0$, there is an isomorphism

$$H^n(J_B(\Pi^{\bullet}) \simeq J_B(H^n(\Pi^{\bullet}))$$

of locally analytic $L$-representations of $\mathbb{Z}_p^s \times T$.

*Proof.* It follows from [Eme06a, Prop. 3.2.4.(ii)] that there is a natural continuous $T^+$-equivariant map of complexes $(\Pi^{N_0, \bullet})_{\mathrm{fs}} \to \Pi^{N_0, \bullet}$ inducing a continuous $T^+$-equivariant morphism $H^n(\Pi_{\mathrm{fs}}^{N_0, \bullet}) \to H^n(\Pi^{N_0, \bullet})$. By loc. cit., the universal property of the functor $(-)_{\mathrm{fs}}$ provides a $T$-equivariant map $H^n(\Pi_{\mathrm{fs}}^{N_0, \bullet}) \to H^n(\Pi^{N_0, \bullet})_{\mathrm{fs}}$. It follows from Lemma 3.4 that it is sufficient to prove that this map is a topological isomorphism.

We now deduce from [Eme06a, Prop. 3.2.27] and [Fu, Thm. 4.5] that given an exact sequence $0 \to U \to V \to W \to 0$ of spaces of compact type with continuous action of $T^+$, then $0 \to U_{\mathrm{fs}} \to V_{\mathrm{fs}} \to W_{\mathrm{fs}} \to 0$ is exact, the image of $U_{\mathrm{fs}}$ is closed in $V_{\mathrm{fs}}$ and the map $V_{\mathrm{fs}} \to W_{\mathrm{fs}}$ is strict. The open mapping theorem then implies that the sequence is strict exact. As the complex $\Pi^{N_0, \bullet}$ has strict transition maps by Lemma 3.4, we conclude that the map $H^n(\Pi_{\mathrm{fs}}^{N_0, \bullet}) \to H^n(\Pi^{N_0, \bullet})_{\mathrm{fs}}$ is a topological isomorphism. □

**Proposition 3.6.** Let $\Pi$ be a locally analytic $L$-representation of $\mathbb{Z}_p^s \times B$ satisfying the hypothesis 3.1. Let $\mathfrak{a}$ be a closed strict ideal of $S^{\mathrm{rig}}$ such that $\dim_L S^{\mathrm{rig}}/\mathfrak{a} < +\infty$. Then the map

$$\mathfrak{a} \otimes_{S^{\mathrm{rig}}} J_B(\Pi)' \to J_B(\Pi)'$$

is injective.

*Proof.* By Lemma 3.2, there exists a perfect complex $C_{\bullet}$ of $S^{\mathrm{rig}}$-modules such that, $C_0 = S^{\mathrm{rig}}$, $H_0(C_{\bullet}) \simeq S^{\mathrm{rig}}/\mathfrak{a}$ and $H_i(C_{\bullet}) = 0$ for $i > 0$. By Hypothesis 3.1, we have $\Pi|_{\mathbb{Z}_p^s \times N_0} \simeq C^{\mathrm{la}}(\mathbb{Z}_p^s, L) \hat{\otimes}_L V$ for some topological $L$-vector space of compact type $V$. As $C_{\bullet}$ has strict transition maps, it follows from Lemma 3.3 that the complex $C_{\bullet} \otimes_{S^{\mathrm{rig}}} \Pi' \simeq C_{\bullet} \hat{\otimes}_L V'$ is a resolution of $(S^{\mathrm{rig}}/\mathfrak{a}) \hat{\otimes}_L V'$. We then deduce from $\mathrm{Hom}_{S^{\mathrm{rig}}}(C_i, \Pi)' \simeq C_i \otimes_{S^{\mathrm{rig}}} \Pi'$ for any $i \ge 0$, that $H^i(\mathrm{Hom}_{S^{\mathrm{rig}}}(C_{\bullet}, \Pi)) = 0$ for $i > 0$. Therefore Proposition 3.5 implies that $H^i(\mathrm{Hom}_{S^{\mathrm{rig}}}(C_{\bullet}, J_B(\Pi))) = 0$ for $i > 0$. We denote by $(-)'$ the duality between spaces of compact type and Fréchet spaces. This duality implies that $H_i(C_{\bullet} \otimes_{S^{\mathrm{rig}}} J_B(\Pi)') = 0$ for $i > 0$. As $\mathfrak{a} = \mathrm{Coker}(C_2 \to C_1)$, we deduce that

$$\begin{aligned} \mathfrak{a} \otimes_{S^{\mathrm{rig}}} J_B(\Pi)' &= \mathrm{Coker}(C_2 \otimes_{S^{\mathrm{rig}}} J_B(\Pi)' \to C_1 \otimes_{S^{\mathrm{rig}}} J_B(\Pi)') \\ &\subset C_0 \otimes_{S^{\mathrm{rig}}} J_B(\Pi)' = J_B(\Pi)'. \end{aligned} \quad \square$$
$n \ge 0$. The commutation of $\hat{\otimes}_L$ with finite direct sum implies that we have a topological isomorphism of $L$-vector spaces for any $m \ge 0$:

$$(\mathrm{Hom}_{S\mathrm{rig}}(C_m, \Pi^{N_0}) \simeq \mathrm{Hom}_{S\mathrm{rig}}(C_m, C^{\mathrm{la}}(\mathbb{Z}_p^s, L))\hat{\otimes}_L V^{N_0}).$$

We deduce as before that the complex $\Pi^{\bullet, N_0}$ has strict transition maps and that we have isomorphisms

$$H^n(\Pi^{N_0, \bullet}) \simeq H^n(\Pi^{\bullet})^{N_0}$$

for any $n \ge 0$.

□

**Proposition 3.5.** For any integer $n \ge 0$, there is an isomorphism

$$H^n(J_B(\Pi)^{\bullet}) \simeq J_B(H^n(\Pi^{\bullet}))$$

of locally analytic $L$-representations of $\mathbb{Z}_p^s \times T$.

*Proof.* It follows from [Eme06a, Prop. 3.2.4.(ii)] that there is a natural continuous $T^+$-equivariant map of complexes $(\Pi^{N_0, \bullet})_{\mathrm{fs}} \to \Pi^{N_0, \bullet}$ inducing a continuous $T^+$-equivariant morphism $H^n(\Pi_{\mathrm{fs}}^{N_0, \bullet}) \to H^n(\Pi^{N_0, \bullet})$. By *loc. cit.*, the universal property of the functor $(-)_{\mathrm{fs}}$ provides a $T$-equivariant map $H^n(\Pi_{\mathrm{fs}}^{N_0, \bullet}) \to H^n(\Pi^{N_0, \bullet})_{\mathrm{fs}}$. It follows from Lemma 3.4 that it is sufficient to prove that this map is a topological isomorphism.

We now deduce from [Eme06a, Prop. 3.2.27] and [Fu, Thm. 4.5] that given an exact sequence $0 \to U \to V \to W \to 0$ of spaces of compact type with continuous action of $T^+$, then $0 \to U_{\mathrm{fs}} \to V_{\mathrm{fs}} \to W_{\mathrm{fs}} \to 0$ is exact, the image of $U_{\mathrm{fs}}$ is closed in $V_{\mathrm{fs}}$ and the map $V_{\mathrm{fs}} \to W_{\mathrm{fs}}$ is strict. The open mapping theorem then implies that the sequence is strict exact. As the complex $\Pi^{N_0, \bullet}$ has strict transition maps by Lemma 3.4, we conclude that the map $H^n(\Pi_{\mathrm{fs}}^{N_0, \bullet}) \to H^n(\Pi^{N_0, \bullet})_{\mathrm{fs}}$ is a topological isomorphism. □

**Proposition 3.6.** Let $\Pi$ be a locally analytic $L$-representation of $\mathbb{Z}_p^s \times B$ satisfying the hypothesis 3.1. Let $\mathfrak{a}$ be a closed strict ideal of $S^{\mathrm{rig}}$ such that $\dim_L S^{\mathrm{rig}}/\mathfrak{a} < +\infty$. Then the map

$$\mathfrak{a} \otimes_{S^{\mathrm{rig}}} J_B(\Pi)' \to J_B(\Pi)'$$

is injective.

*Proof.* By Lemma 3.2, there exists a perfect complex $C_\bullet$ of $S^{\mathrm{rig}}$-modules such that, $C_0 = S^{\mathrm{rig}}$, $H_0(C_\bullet) \simeq S^{\mathrm{rig}}/\mathfrak{a}$ and $H_i(C_\bullet) = 0$ for $i > 0$. By Hypothesis 3.1, we have $\Pi|_{\mathbb{Z}_p^s \times N_0} \simeq C^{\mathrm{la}}(\mathbb{Z}_p^s, L)\hat{\otimes}_L V$ for some topological $L$-vector space of compact type $V$. As $C_\bullet$ has strict transition maps, it follows from Lemma 3.3 that the complex $C_\bullet \otimes_{S^{\mathrm{rig}}} \Pi' \simeq C_\bullet \hat{\otimes}_L V'$ is a resolution of $(S^{\mathrm{rig}}/\mathfrak{a})\hat{\otimes}_L V'$. We then deduce from $\mathrm{Hom}_{S^{\mathrm{rig}}}(C_i, \Pi)' \simeq C_i \otimes_{S^{\mathrm{rig}}} \Pi'$ for any $i \ge 0$, that $H^i(\mathrm{Hom}_{S^{\mathrm{rig}}}(C_\bullet, \Pi)) = 0$ for $i > 0$. Therefore Proposition 3.5 implies that $H^i(\mathrm{Hom}_{S^{\mathrm{rig}}}(C_\bullet, J_B(\Pi))) = 0$ for $i > 0$. We denote by $(-)'$ the duality between spaces of compact type and Fréchet spaces. This duality implies that $H_i(C_\bullet \otimes_{S^{\mathrm{rig}}} J_B(\Pi)) = 0$ for $i > 0$. As $\mathfrak{a} = \mathrm{Coker}(C_2 \to C_1)$, we deduce that

$$\begin{aligned} \mathfrak{a} \otimes_{S^{\mathrm{rig}}} J_B(\Pi)' &= \mathrm{Coker}(C_2 \otimes_{S^{\mathrm{rig}}} J_B(\Pi)' \to C_1 \otimes_{S^{\mathrm{rig}}} J_B(\Pi)') \\ &\subset C_0 \otimes_{S^{\mathrm{rig}}} J_B(\Pi)' = J_B(\Pi)'. \end{aligned}$$

□
## 3.2 Families of locally analytic representations of G

Let $\Pi$ be an admissible locally analytic $L$-representation of $\mathbb{Z}_p^s \times G$. The aim of this section is to use $\Pi$ in order to construct a functor

$$M \mapsto \mathrm{Hom}_{U(\mathfrak{g})}(M, \Pi)$$

from the category $\mathcal{O}_{\mathrm{alg}}^{\infty}$ to the category of locally analytic $\mathbb{Z}_p^s \times B$-representations, and then, by composing with $J_B$, to locally analytic $\mathbb{Z}_p^s \times T$-representations. We will usually assume that we are in the following situation:

**Hypothesis 3.7.** There exists a uniform open pro-$p$-subgroup $H$ of $G$, an integer $m \ge 0$ and a topological $\mathbb{Z}_p^s \times H$-equivariant isomorphism

$$\Pi_{|\mathbb{Z}_p^s \times H} \simeq \mathcal{C}^{\mathrm{la}}(\mathbb{Z}_p^s \times H, L)^m.$$

Recall from section 2.2 that if $M$ is an object of $\mathcal{O}_{\mathrm{alg}}^{\infty}$, there is a unique algebraic action of $\mathcal{B}(L)$ on $M$ which lifts the structure of $U(\mathfrak{b})$-module on $M^{\mathrm{ss}}$. We endow $M$ with the action of $B = \mathcal{B}(\mathbb{Q}_p)$ obtained by restriction to $B$.

Let $M$ be an object of $\mathcal{O}_{\mathrm{alg}}^{\infty}$ with its semi-simplified $B$-action. We define an action of $B$ on $\mathrm{Hom}_L(M, \Pi)$ by

$$b \cdot f = bf(b^{-1}-)$$

for $f \in \mathrm{Hom}_L(M, \Pi)$ and $b \in B$. It follows from Lemma 2.4 that this action preserves the subspace $\mathrm{Hom}_{U(\mathfrak{g})}(M, \Pi)$. We moreover endow $\mathrm{Hom}_{U(\mathfrak{g})}(M, \Pi)$ with the left $\mathbb{Z}_p^s$-action inherited from the one on $\Pi$. While the definition of the $B$-action using the semi-simplified action on $M$ might not seem very natural at a first glance, the following lemma says that this definition applied to deformed Verma modules allows us to compute generalized eigenspaces. Given an $U(\mathfrak{t})$-module $X$ we write

$$X[(\mathfrak{t} - \lambda)^k] = \{x \in X | \forall t \in \mathfrak{t}, (t - \lambda(t))^k x = 0\}.$$

With this notation we have the following result:

**Lemma 3.8.** Let $\lambda \in X^*(\underline{T})_I^+$ and $M = \widetilde{M}_I(\lambda) \otimes_{A_I} A_I/\mathfrak{m}_I^k$. Then there is an isomorphism

$$\mathrm{Hom}_{U(\mathfrak{g})}(M, \Pi) \simeq (\Pi^{n_I} \otimes_L L_I(\lambda)')[\mathfrak{m}_I^k]$$

of $B$-representations, where $(-)'$ denote the dual (algebraic) representation. In particular, when $I = \emptyset$,

$$\mathrm{Hom}_{U(\mathfrak{g})}(\widetilde{M}(\lambda), \Pi) \simeq (\Pi^n(\lambda^{-1})) [\mathfrak{m}^k] \simeq (\Pi^n[(\mathfrak{t} - \lambda)^k])(\lambda^{-1}).$$

*Proof.* We compute using the $U(\mathfrak{g})$-structure

$$\begin{align*}
\mathrm{Hom}_{U(\mathfrak{g})}(M, \Pi) &= \mathrm{Hom}_{U(\mathfrak{g})}(U(\mathfrak{g}) \otimes_{U(\mathfrak{b})} (L_I(\lambda) \otimes_L A_I/\mathfrak{m}_I^k), \Pi) \\
&= \mathrm{Hom}_{U(\mathfrak{b})}(L_I(\lambda) \otimes_L A_I/\mathfrak{m}_I^k, \Pi^{n_I}) \\
&= \mathrm{Hom}_{U(\mathfrak{t})}(A_I/\mathfrak{m}_I^k, \Pi^{n_I} \otimes L_I(\lambda)') \\
&= (\Pi^{n_I} \otimes L_I(\lambda)')(\mathfrak{m}_I^k].
\end{align*}$$
## 3.2 Families of locally analytic representations of G

Let $\Pi$ be an admissible locally analytic $L$-representation of $\mathbb{Z}_p^s \times G$. The aim of this section is to use $\Pi$ in order to construct a functor

$$M \mapsto \mathrm{Hom}_{U(\mathfrak{g})}(M, \Pi)$$

from the category $\mathcal{O}_{\mathrm{alg}}^{\infty}$ to the category of locally analytic $\mathbb{Z}_p^s \times B$-representations, and then, by composing with $J_B$, to locally analytic $\mathbb{Z}_p^s \times T$-representations. We will usually assume that we are in the following situation:

**Hypothesis 3.7.** There exists a uniform open pro-$p$-subgroup $H$ of $G$, an integer $m \ge 0$ and a topological $\mathbb{Z}_p^s \times H$-equivariant isomorphism

$$\Pi_{|\mathbb{Z}_p^s \times H} \simeq \mathcal{C}^{\mathrm{la}}(\mathbb{Z}_p^s \times H, L)^m.$$

Recall from section 2.2 that if $M$ is an object of $\mathcal{O}_{\mathrm{alg}}^{\infty}$, there is a unique algebraic action of $\mathcal{B}(L)$ on $M$ which lifts the structure of $U(\mathfrak{b})$-module on $M^{\mathrm{ss}}$. We endow $M$ with the action of $B = \mathcal{B}(\mathbb{Q}_p)$ obtained by restriction to $B$.

Let $M$ be an object of $\mathcal{O}_{\mathrm{alg}}^{\infty}$ with its semi-simplified $B$-action. We define an action of $B$ on $\mathrm{Hom}_L(M, \Pi)$ by

$$b \cdot f = bf(b^{-1}-)$$

for $f \in \mathrm{Hom}_L(M, \Pi)$ and $b \in B$. It follows from Lemma 2.4 that this action preserves the subspace $\mathrm{Hom}_{U(\mathfrak{g})}(M, \Pi)$. We moreover endow $\mathrm{Hom}_{U(\mathfrak{g})}(M, \Pi)$ with the left $\mathbb{Z}_p^s$-action inherited from the one on $\Pi$. While the definition of the $B$-action using the semi-simplified action on $M$ might not seem very natural at a first glance, the following lemma says that this definition applied to deformed Verma modules allows us to compute generalized eigenspaces. Given an $U(\mathfrak{t})$-module $X$ we write

$$X[(\mathfrak{t} - \lambda)^k] = \{x \in X | \forall t \in \mathfrak{t}, (\mathfrak{t} - \lambda(t))^k x = 0\}.$$

With this notation we have the following result:

**Lemma 3.8.** Let $\lambda \in X^*(\mathcal{T})_I^+$ and $M = \widetilde{M}_I(\lambda) \otimes_{A_I} A_I/\mathfrak{m}_I^k$. Then there is an isomorphism

$$\mathrm{Hom}_{U(\mathfrak{g})}(M, \Pi) \simeq (\Pi^{n_I} \otimes_L L_I(\lambda)')[\mathfrak{m}_I^k]$$

of $B$-representations, where $(-)'$ denote the dual (algebraic) representation. In particular, when $I = \emptyset$,

$$\mathrm{Hom}_{U(\mathfrak{g})}(\widetilde{M}(\lambda), \Pi) \simeq (\Pi^n(\lambda^{-1})) [\mathfrak{m}^k] \simeq (\Pi^n[(\mathfrak{t} - \lambda)^k])(\lambda^{-1}).$$

*Proof.* We compute using the $U(\mathfrak{g})$-structure

$$
\begin{align*}
\mathrm{Hom}_{U(\mathfrak{g})}(M, \Pi) &= \mathrm{Hom}_{U(\mathfrak{g})}(U(\mathfrak{g}) \otimes_{U(\mathfrak{b})} (L_I(\lambda) \otimes_L A_I/\mathfrak{m}_I^k), \Pi) \\
&= \mathrm{Hom}_{U(\mathfrak{b})}(L_I(\lambda) \otimes_L A_I/\mathfrak{m}_I^k, \Pi^{n_I}) \\
&= \mathrm{Hom}_{U(\mathfrak{t})}(A_I/\mathfrak{m}_I^k, \Pi^{n_I} \otimes L_I(\lambda)') \\
&= (\Pi^{n_I} \otimes L_I(\lambda)') [\mathfrak{m}_I^k].
\end{align*}
$$
Moreover each equality is compatible with the semi-simplified B-actions. □

**Lemma 3.9.** Let $\Pi$ be a locally analytic representation of $\mathbb{Z}_p^s \times G$ and let $M$ be an object of $\mathcal{O}_{\text{alg}}^{\infty}$. Then the $\mathbb{Z}_p^s \times B$-representation $\text{Hom}_{U(\mathfrak{g})}(M, \Pi)$ is locally analytic.

*Proof.* Let $U(\mathfrak{g}) \otimes_{U(\mathfrak{b})} V_1 \to U(\mathfrak{g}) \otimes_{U(\mathfrak{b})} V_0 \to M \to 0$ be a resolution as in Lemma 2.5. Then $\text{Hom}_{U(\mathfrak{g})}(M, \Pi)$ is the kernel of the map

$$ \text{Hom}_{U(\mathfrak{g})}(U(\mathfrak{g}) \otimes_{U(\mathfrak{b})} V_0, \Pi) \simeq (V'_0 \otimes_L \Pi)^b \to \text{Hom}_{U(\mathfrak{g})}(U(\mathfrak{g}) \otimes_{U(\mathfrak{b})} V_1, \Pi) \simeq (V'_1 \otimes_L \Pi)^b $$
which is continuous and B-equivariant. Therefore $\text{Hom}_{U(\mathfrak{g})}(M, \Pi)$ is isomorphic to a closed B-stable subspace of $V'_0 \otimes_L \Pi$. As $V_0$ is an algebraic finite dimensional representation of B, the representation $V'_0 \otimes_L \Pi$ is locally analytic and hence so is $\text{Hom}_{U(\mathfrak{g})}(M, \Pi)$. □

As $\text{Hom}_{U(\mathfrak{g})}(M, \Pi)$ is a locally analytic representation of B this action may be derived and induces the structure of an $U(\mathfrak{b})$-module on $\text{Hom}_{U(\mathfrak{g})}(M, \Pi)$. Via restriction to $U(\mathfrak{t}) \subset U(\mathfrak{b})$ we may view $\text{Hom}_{U(\mathfrak{g})}(M, \Pi)$ as an $U(\mathfrak{t})$-module.

**Lemma 3.10.** Let $\Pi$ be a locally analytic representation of $\mathbb{Z}_p^s \times G$ and let $M$ be an object of $\mathcal{O}_{\text{alg}}^{\infty}$. Then the $U(\mathfrak{t})$ action on $\text{Hom}_{U(\mathfrak{g})}(M, \Pi)$ factors through a finite dimensional quotient.

*Proof.* By Proposition 2.14 there exist dominant weights $\lambda_1, \dots, \lambda_r$, finite dimensional $\mathfrak{g}$-modules $V_1, \dots, V_r$ and a surjective map

$$ \widetilde{M}(\lambda_1) \otimes_L V_1 \oplus \dots \oplus \widetilde{M}(\lambda_r) \otimes_L V_r \to M. $$

Moreover by Lemma 2.14 there exists $k \ge 1$ such that this map factors through $\mathfrak{m}^k$ (recall that $A$ is the localization of $U(\mathfrak{t})$ at its augmentation ideal $\mathfrak{m}$). Therefore we have an inclusion of $U(\mathfrak{t})$-modules

$$ \text{Hom}_{U(\mathfrak{g})}(M, \Pi) \to \bigoplus_{i=1}^{r} \text{Hom}_{U(\mathfrak{g})}(\widetilde{M}(\lambda_i) \otimes_A \mathfrak{m}^k \otimes_L V_i, \Pi). $$

By Lemma 3.8, $\text{Hom}_{U(\mathfrak{g})}(\widetilde{M}(\lambda_i)/\mathfrak{m}^k \otimes_L V_i, \Pi) = (\Pi \otimes V_i(\lambda_i))^{\text{n}}[\mathfrak{m}^k]$.

Let $\mu_1, \dots, \mu_s$ be the finitely many characters which appears in the restriction to $U(\mathfrak{t})$ of $V_1(\lambda_1), \dots, V_r(\lambda_r)$. Then the action of $U(\mathfrak{t})$ on $\text{Hom}_{U(\mathfrak{g})}(M, \Pi)$ factors through the quotient of $U(\mathfrak{t})$ by the intersection of the $k$-th powers of the kernels of the $\mu_i$. □

**Lemma 3.11.** Assume that $\Pi$ is an admissible locally analytic L-representation of $\mathbb{Z}_p^s \times G$ satisfying Hypothesis 3.7 and $M \in \mathcal{O}_{\text{alg}}^{\infty}$. Then $\text{Hom}_{U(\mathfrak{g})}(M, \Pi)$ satisfies Hypothesis 3.1
Moreover each equality is compatible with the semi-simplified B-actions. $\square$

**Lemma 3.9.** Let $\Pi$ be a locally analytic representation of $\mathbb{Z}_p^s \times G$ and let $M$ be an object of $\mathcal{O}_{\text{alg}}^{\infty}$. Then the $\mathbb{Z}_p^s \times B$-representation $\text{Hom}_{U(\mathfrak{g})}(M, \Pi)$ is locally analytic.

*Proof.* Let $U(\mathfrak{g}) \otimes_{U(\mathfrak{b})} V_1 \to U(\mathfrak{g}) \otimes_{U(\mathfrak{b})} V_0 \to M \to 0$ be a resolution as in Lemma 2.5. Then $\text{Hom}_{U(\mathfrak{g})}(M, \Pi)$ is the kernel of the map

$$ \text{Hom}_{U(\mathfrak{g})}(U(\mathfrak{g}) \otimes_{U(\mathfrak{b})} V_0, \Pi) \simeq (V'_0 \otimes_L \Pi)^b \to \text{Hom}_{U(\mathfrak{g})}(U(\mathfrak{g}) \otimes_{U(\mathfrak{b})} V_1, \Pi) \simeq (V'_1 \otimes_L \Pi)^b $$
which is continuous and B-equivariant. Therefore $\text{Hom}_{U(\mathfrak{g})}(M, \Pi)$ is isomorphic to a closed B-stable subspace of $V'_0 \otimes_L \Pi$. As $V_0$ is an algebraic finite dimensional representation of B, the representation $V'_0 \otimes_L \Pi$ is locally analytic and hence so is $\text{Hom}_{U(\mathfrak{g})}(M, \Pi)$. $\square$

As $\text{Hom}_{U(\mathfrak{g})}(M, \Pi)$ is a locally analytic representation of B this action may be derived and induces the structure of an $U(\mathfrak{b})$-module on $\text{Hom}_{U(\mathfrak{g})}(M, \Pi)$. Via restriction to $U(\mathfrak{t}) \subset U(\mathfrak{b})$ we may view $\text{Hom}_{U(\mathfrak{g})}(M, \Pi)$ as an $U(\mathfrak{t})$-module.

**Lemma 3.10.** Let $\Pi$ be a locally analytic representation of $\mathbb{Z}_p^s \times G$ and let $M$ be an object of $\mathcal{O}_{\text{alg}}^{\infty}$. Then the $U(\mathfrak{t})$ action on $\text{Hom}_{U(\mathfrak{g})}(M, \Pi)$ factors through a finite dimensional quotient.

*Proof.* By Proposition 2.14 there exist dominant weights $\lambda_1, \dots, \lambda_r$, finite dimensional $\mathfrak{g}$-modules $V_1, \dots, V_r$ and a surjective map

$$ \widetilde{M}(\lambda_1) \otimes_L V_1 \oplus \dots \oplus \widetilde{M}(\lambda_r) \otimes_L V_r \to M. $$

Moreover by Lemma 2.14 there exists $k \ge 1$ such that this map factors through $\mathfrak{m}^k$ (recall that $A$ is the localization of $U(\mathfrak{t})$ at its augmentation ideal $\mathfrak{m}$). Therefore we have an inclusion of $U(\mathfrak{t})$-modules

$$ \text{Hom}_{U(\mathfrak{g})}(M, \Pi) \to \bigoplus_{i=1}^{r} \text{Hom}_{U(\mathfrak{g})}(\widetilde{M}(\lambda_i) \otimes_A \mathfrak{m}^k \otimes_L V_i, \Pi). $$

By Lemma 3.8, $\text{Hom}_{U(\mathfrak{g})}(\widetilde{M}(\lambda_i)/\mathfrak{m}^k \otimes_L V_i, \Pi) = (\Pi \otimes V_i(\lambda_i))^{\text{n}}[\mathfrak{m}^k]$.

Let $\mu_1, \dots, \mu_s$ be the finitely many characters which appears in the restriction to $U(\mathfrak{t})$ of $V_1(\lambda_1), \dots, V_r(\lambda_r)$. Then the action of $U(\mathfrak{t})$ on $\text{Hom}_{U(\mathfrak{g})}(M, \Pi)$ factors through the quotient of $U(\mathfrak{t})$ by the intersection of the $k$-th powers of the kernels of the $\mu_i$. $\square$

**Lemma 3.11.** Assume that $\Pi$ is an admissible locally analytic L-representation of $\mathbb{Z}_p^s \times G$ satisfying Hypothesis 3.7 and $M \in \mathcal{O}_{\text{alg}}^{\infty}$. Then $\text{Hom}_{U(\mathfrak{g})}(M, \Pi)$ satisfies Hypothesis 3.1
Proof. We can assume that $N_0 \subset H$. As we assume Hypothesis 3.7, there is an isomorphism $\Pi \cong C^{\text{la}}(\mathbb{Z}_p^s \times H, L)^m \simeq C^{\text{la}}(\mathbb{Z}_p^s, L) \hat{\otimes}_L C(H, L)^m$ of $\mathbb{Z}_p^s \times H$-representation.

Let $[U(\mathfrak{g}) \otimes_{U(\mathfrak{b})} V_1 \to U(\mathfrak{g}) \otimes_{U(\mathfrak{b})} V_0]$ be a resolution of $M$ as in Lemma 2.5. Then $\text{Hom}_{U(\mathfrak{g})}(M, C^{\text{la}}(H, L)^m)$ is the kernel of the map

$$ (V'_0 \otimes_L C^{\text{la}}(H, L)^m)^{\mathfrak{b}} \to (V'_1 \otimes_L C^{\text{la}}(H, L)^m)^{\mathfrak{b}}. \qquad (4) $$

We claim that this is a strict map, then the lemma follows, as exactness of the functor $C^{\text{la}}(\mathbb{Z}_p^s, L) \hat{\otimes}_L (-)$ implies that we have an isomorphism of locally analytic $\mathbb{Z}_p^s \times N_0$-representation

$$ \text{Hom}_{U(\mathfrak{g})}(M, \Pi) \simeq C^{\text{la}}(\mathbb{Z}_p^s, L) \hat{\otimes}_L \text{Hom}_{U(\mathfrak{g})}(M, C^{\text{la}}(H, L))^m. $$

In order to prove that (4) is strict, we use an additional $H$-action. We let $H$-act on $C^{\text{la}}(H, L)$ by right translation and extend this to $V'_i \otimes_L C^{\text{la}}(H, L)$ by acting trivially on $V'_i$. This action commutes with the (diagonal) action of $U(\mathfrak{b})$, as the $U(\mathfrak{b})$ action on $C^{\text{la}}(H, L)$ is induced by left translations. It follows that $(V'_i \otimes_L C^{\text{la}}(H, L)^m)^{\mathfrak{b}}$ is a closed $H$-stable subspace of an admissible locally analytic $H$-representation, and hence an admissible locally analytic $H$-representation itself. Hence (4) is an $H$-equivariant map between admissible locally analytic $H$-representations and hence a strict map which proves the claim. $\square$

**Proposition 3.12.** Let $\Pi$ be an admissible locally analytic representation of $\mathbb{Z}_p^s \times G$ satisfying the hypothesis 3.7 and let $M$ be an object of $\mathcal{O}_{\text{alg}}^{\infty}$. Then the locally analytic representation $J_B(\text{Hom}_{U(\mathfrak{g})}(M, \Pi))$ of $\mathbb{Z}_p^s \times T$ is essentially admissible.

Proof. Let $U(\mathfrak{g}) \otimes_{U(\mathfrak{b})} V_1 \to U(\mathfrak{g}) \otimes_{U(\mathfrak{b})} V_0 \to M \to 0$ be a resolution of $M$ given by Lemma 2.5. Then we have an exact sequence

$$ 0 \to \text{Hom}_{U(\mathfrak{g})}(M, \Pi) \to \text{Hom}_{U(\mathfrak{g})}(U(\mathfrak{g}) \otimes_{U(\mathfrak{b})} V_0, \Pi) \to \text{Hom}_{U(\mathfrak{g})}(U(\mathfrak{g}) \otimes_{U(\mathfrak{b})} V_1, \Pi) $$

of locally analytic representations of $\mathbb{Z}_p^s \times B$ (see Lemma 3.9). As the functor $J_B$ is left exact ([Eme06a, Lem. 3.4.7.(iii)]), this induces a short exact sequence

$$ \begin{aligned} 0 \to J_B(\text{Hom}_{U(\mathfrak{g})}(M, \Pi)) &\to J_B(\text{Hom}_{U(\mathfrak{g})}(U(\mathfrak{g}) \otimes_{U(\mathfrak{b})} V_0, \Pi)) \\ &\to J_B(\text{Hom}_{U(\mathfrak{g})}(U(\mathfrak{g}) \otimes_{U(\mathfrak{b})} V_1, \Pi)) \end{aligned} $$

of locally analytic representations of $\mathbb{Z}_p^s \times T$. As the kernel of a morphism between essentially admissible representations is essentially admissible ([Eme06a, Thm. 3.1.3]), it is sufficient to prove that $J_B(\text{Hom}_{U(\mathfrak{g})}(U(\mathfrak{g}) \otimes_{U(\mathfrak{b})} V, \Pi))$ is essentially admissible for any finite dimensional algebraic representation $V$ of $B$. As an algebraic representation of $B$ is an extension of rank 1 object, it is sufficient to prove this when $V$ is 1-dimensional and $V^n = V$. The left exactness of $J_B$ implies that

$$ J_B(\text{Hom}_{U(\mathfrak{g})}(U(\mathfrak{g}) \otimes_{U(\mathfrak{b})} V, \Pi)) \simeq J_B(\text{Hom}_{U(\mathfrak{b})}(V, \Pi^n)) \simeq \text{Hom}_{U(\mathfrak{t})}(V, J_B(\Pi)). $$
Proof. We can assume that $N_0 \subset H$. As we assume Hypothesis 3.7, there is an isomorphism $\Pi \cong C^{\text{la}}(\mathbb{Z}_p^s \times H, L)^m \simeq C^{\text{la}}(\mathbb{Z}_p^s, L) \hat{\otimes}_L C(H, L)^m$ of $\mathbb{Z}_p^s \times H$-representation.

Let $[U(\mathfrak{g}) \otimes_{U(\mathfrak{b})} V_1 \to U(\mathfrak{g}) \otimes_{U(\mathfrak{b})} V_0]$ be a resolution of $M$ as in Lemma 2.5. Then $\text{Hom}_{U(\mathfrak{g})}(M, C^{\text{la}}(H, L)^m)$ is the kernel of the map

$$ (V'_0 \otimes_L C^{\text{la}}(H, L)^m)^{\mathfrak{b}} \to (V'_1 \otimes_L C^{\text{la}}(H, L)^m)^{\mathfrak{b}}. \qquad (4) $$

We claim that this is a strict map, then the lemma follows, as exactness of the functor $C^{\text{la}}(\mathbb{Z}_p^s, L) \hat{\otimes}_L (-)$ implies that we have an isomorphism of locally analytic $\mathbb{Z}_p^s \times N_0$-representation

$$ \text{Hom}_{U(\mathfrak{g})}(M, \Pi) \simeq C^{\text{la}}(\mathbb{Z}_p^s, L) \hat{\otimes}_L \text{Hom}_{U(\mathfrak{g})}(M, C^{\text{la}}(H, L))^m. $$

In order to prove that (4) is strict, we use an additional $H$-action. We let $H$-act on $C^{\text{la}}(H, L)$ by right translation and extend this to $V'_i \otimes_L C^{\text{la}}(H, L)$ by acting trivially on $V'_i$. This action commutes with the (diagonal) action of $U(\mathfrak{b})$, as the $U(\mathfrak{b})$ action on $C^{\text{la}}(H, L)$ is induced by left translations. It follows that $(V'_i \otimes_L C^{\text{la}}(H, L)^m)^{\mathfrak{b}}$ is a closed $H$-stable subspace of an admissible locally analytic $H$-representation, and hence an admissible locally analytic $H$-representation itself. Hence (4) is an $H$-equivariant map between admissible locally analytic $H$-representations and hence a strict map which proves the claim. $\square$

**Proposition 3.12.** Let $\Pi$ be an admissible locally analytic representation of $\mathbb{Z}_p^s \times G$ satisfying the hypothesis 3.7 and let $M$ be an object of $\mathcal{O}_{\text{alg}}^{\infty}$. Then the locally analytic representation $J_B(\text{Hom}_{U(\mathfrak{g})}(M, \Pi))$ of $\mathbb{Z}_p^s \times T$ is essentially admissible.

Proof. Let $U(\mathfrak{g}) \otimes_{U(\mathfrak{b})} V_1 \to U(\mathfrak{g}) \otimes_{U(\mathfrak{b})} V_0 \to M \to 0$ be a resolution of $M$ given by Lemma 2.5. Then we have an exact sequence

$$ 0 \to \text{Hom}_{U(\mathfrak{g})}(M, \Pi) \to \text{Hom}_{U(\mathfrak{g})}(U(\mathfrak{g}) \otimes_{U(\mathfrak{b})} V_0, \Pi) \to \text{Hom}_{U(\mathfrak{g})}(U(\mathfrak{g}) \otimes_{U(\mathfrak{b})} V_1, \Pi) $$

of locally analytic representations of $\mathbb{Z}_p^s \times B$ (see Lemma 3.9). As the functor $J_B$ is left exact ([Eme06a, Lem. 3.4.7.(iii)]), this induces a short exact sequence

$$ \begin{aligned} 0 \to J_B(\text{Hom}_{U(\mathfrak{g})}(M, \Pi)) &\to J_B(\text{Hom}_{U(\mathfrak{g})}(U(\mathfrak{g}) \otimes_{U(\mathfrak{b})} V_0, \Pi)) \\ &\to J_B(\text{Hom}_{U(\mathfrak{g})}(U(\mathfrak{g}) \otimes_{U(\mathfrak{b})} V_1, \Pi)) \end{aligned} $$

of locally analytic representations of $\mathbb{Z}_p^s \times T$. As the kernel of a morphism between essentially admissible representations is essentially admissible ([Eme06a, Thm. 3.1.3]), it is sufficient to prove that $J_B(\text{Hom}_{U(\mathfrak{g})}(U(\mathfrak{g}) \otimes_{U(\mathfrak{b})} V, \Pi))$ is essentially admissible for any finite dimensional algebraic representation $V$ of $B$. As an algebraic representation of $B$ is an extension of rank 1 object, it is sufficient to prove this when $V$ is 1-dimensional and $V^n = V$. The left exactness of $J_B$ implies that

$$ J_B(\text{Hom}_{U(\mathfrak{g})}(U(\mathfrak{g}) \otimes_{U(\mathfrak{b})} V, \Pi)) \simeq J_B(\text{Hom}_{U(\mathfrak{b})}(V, \Pi^n)) \simeq \text{Hom}_{U(\mathfrak{t})}(V, J_B(\Pi)). $$
By [BHS17b, Prop. 3.4] (whose proof follows [Eme06a, Thm. 0.5]), the locally analytic representation $J_B(\Pi)$ of $\mathbb{Z}_p^s \times T$ is essentially admissible. As $U(t)$ is finitely generated, we conclude that $\mathrm{Hom}_{U(t)}(V, J_B(\Pi))$ is essentially admissible. $\square$

**Lemma 3.13.** Let $\Pi$ be a locally analytic representation of $\mathbb{Z}_p^s \times G$ satisfying Hypothesis 3.7.

(i) The functor $M \mapsto \mathrm{Hom}_{U(\mathfrak{g})}(M, \Pi)$ from $\mathcal{O}_{\mathrm{alg}}^{\infty}$ to the category of locally analytic representations of $\mathbb{Z}_p^s \times B$ is exact.

(ii) The functor $M \mapsto \mathrm{Hom}_{U(\mathfrak{g})}(M, \Pi)^{N_0}$ from $\mathcal{O}_{\mathrm{alg}}^{\infty}$ to the category of locally convex $L$-vector spaces sends short exact sequences on short exact sequences with strict maps.

*Proof.* The assertion (i) is [BHS19, Lem. 5.2.5]. We recall the proof as we will need notation for the proof of (ii). Let $M$ be an object of the category $\mathcal{O}_{\mathrm{alg}}^{\infty}$. Let $H \subset G$ be a uniform compact open pro-$p$-subgroup. Recall (see for example the proof of [ST03, Prop. 6.5]) that $\Pi_{|\mathbb{Z}_p^s \times H} = \lim_{\underset{r<1}{\longrightarrow}} \Pi_r$ with

$$ \Pi_r = \mathrm{Hom}_L^{\mathrm{cont}}(D_r(\mathbb{Z}_p^s \times H) \otimes_{D(\mathbb{Z}_p^s \times G, L)} \Pi, L). $$

As $M$ is a finitely presented $U(\mathfrak{g})$-module, we have

$$ \mathrm{Hom}_{U(\mathfrak{g})}(M, \Pi) \simeq \lim_{\underset{r<1}{\longrightarrow}} \mathrm{Hom}_{U(\mathfrak{g})}(M, \Pi_r) = \lim_{\underset{r}{\longrightarrow}} \mathrm{Hom}_{U_r(\mathfrak{g})}(M_r, \Pi_r) $$

with $M_r := U_r(\mathfrak{g}) \otimes_{U(\mathfrak{g})} M$. Note that there exists an integer $m \ge 0$ such that $\Pi_r \simeq \mathrm{Hom}_L^{\mathrm{cont}}(D_r(\mathbb{Z}_p^s \times H), L)^m$. Therefore we have

$$ \mathrm{Hom}_{D_r(H)}(D_r(H) \otimes_{U(\mathfrak{g})} M, \Pi_r) \\ \simeq \mathrm{Hom}_L^{\mathrm{cont}}(D_r(H) \otimes_{U_r(\mathfrak{g})} M_r, \mathrm{Hom}_L^{\mathrm{cont}}(D_r(\mathbb{Z}_p^s, L), L))^m, $$

for $r < 1$. As the functor $M \mapsto M_r$ is exact and $D_r(H)$ is a finite free $U_r(\mathfrak{g})$-module, this proves (i).

Now we prove (ii). As $N_0$ is a compact group and $L$ is of characteristic 0, it is equivalent to prove (ii) after replacing $N_0$ by an open subgroup. Therefore we can assume that $N_0 = H \cap N$ and that $H = (\bar{N} \cap H)(T \cap H)(N \cap H)$ where $\bar{N}$ is the group of $Q_p$-points of the unipotent subgroup of $G$ opposite to $N$. Let $r < 1$. The space $\mathrm{Hom}_{U(\mathfrak{g})}(M, \Pi_r)^{N_0}$ is the space of maps from $M$ to $\Pi_r$ that are equivariant for the actions of $N_0$ and $U(\mathfrak{g})$. Therefore we have

$$ \mathrm{Hom}_{U(\mathfrak{g})}(M, \Pi_r)^{N_0} = \mathrm{Hom}_{U_r(\mathfrak{g}) \otimes_{U_r(n)} D_r(N_0)}(M_r, \Pi_r) \\ \simeq \mathrm{Hom}_L^{\mathrm{cont}}(D_r(H) \otimes_{(U_r(\mathfrak{g}) \otimes_{U_r(n)} D_r(N_0))} M_r, \mathrm{Hom}_L^{\mathrm{cont}}(D_r(\mathbb{Z}_p^s, L), L))^m. $$

As $D_r(H)$ is a finite free right $U_r(\mathfrak{g}) \otimes_{U_r(n)} D_r(N_0)$-module (see [Koh07, Thm 1.4]), this proves the claim. $\square$
By [BHS17b, Prop. 3.4] (whose proof follows [Eme06a, Thm. 0.5]), the locally analytic representation $J_B(\Pi)$ of $\mathbb{Z}_p^s \times T$ is essentially admissible. As $U(t)$ is finitely generated, we conclude that $\mathrm{Hom}_{U(t)}(V, J_B(\Pi))$ is essentially admissible. $\square$

**Lemma 3.13.** Let $\Pi$ be a locally analytic representation of $\mathbb{Z}_p^s \times G$ satisfying Hypothesis 3.7.

(i) The functor $M \mapsto \mathrm{Hom}_{U(\mathfrak{g})}(M, \Pi)$ from $\mathcal{O}_{\mathrm{alg}}^{\infty}$ to the category of locally analytic representations of $\mathbb{Z}_p^s \times B$ is exact.

(ii) The functor $M \mapsto \mathrm{Hom}_{U(\mathfrak{g})}(M, \Pi)^{N_0}$ from $\mathcal{O}_{\mathrm{alg}}^{\infty}$ to the category of locally convex $L$-vector spaces sends short exact sequences on short exact sequences with strict maps.

*Proof.* The assertion (i) is [BHS19, Lem. 5.2.5]. We recall the proof as we will need notation for the proof of (ii). Let $M$ be an object of the category $\mathcal{O}_{\mathrm{alg}}^{\infty}$. Let $H \subset G$ be a uniform compact open pro-$p$-subgroup. Recall (see for example the proof of [ST03, Prop. 6.5]) that $\Pi_{|\mathbb{Z}_p^s \times H} = \lim_{\underset{r<1}{\longrightarrow}} \Pi_r$ with

$$ \Pi_r = \mathrm{Hom}_L^{\mathrm{cont}}(D_r(\mathbb{Z}_p^s \times H) \otimes_{D(\mathbb{Z}_p^s \times G, L)} \Pi, L). $$

As $M$ is a finitely presented $U(\mathfrak{g})$-module, we have

$$ \mathrm{Hom}_{U(\mathfrak{g})}(M, \Pi) \simeq \lim_{r<1} \mathrm{Hom}_{U(\mathfrak{g})}(M, \Pi_r) = \lim_{r} \mathrm{Hom}_{U_r(\mathfrak{g})}(M_r, \Pi_r) $$

with $M_r := U_r(\mathfrak{g}) \otimes_{U(\mathfrak{g})} M$. Note that there exists an integer $m \ge 0$ such that $\Pi_r \simeq \mathrm{Hom}_L^{\mathrm{cont}}(D_r(\mathbb{Z}_p^s \times H), L)^m$. Therefore we have

$$ \mathrm{Hom}_{D_r(H)}(D_r(H) \otimes_{U(\mathfrak{g})} M, \Pi_r) \simeq \mathrm{Hom}_L^{\mathrm{cont}}(D_r(H) \otimes_{U_r(\mathfrak{g})} M_r, \mathrm{Hom}_L^{\mathrm{cont}}(D_r(\mathbb{Z}_p^s, L), L))^m, $$

for $r < 1$. As the functor $M \mapsto M_r$ is exact and $D_r(H)$ is a finite free $U_r(\mathfrak{g})$-module, this proves (i).

Now we prove (ii). As $N_0$ is a compact group and $L$ is of characteristic 0, it is equivalent to prove (ii) after replacing $N_0$ by an open subgroup. Therefore we can assume that $N_0 = H \cap N$ and that $H = (\bar{N} \cap H)(T \cap H)(N \cap H)$ where $\bar{N}$ is the group of $\mathbb{Q}_p$-points of the unipotent subgroup of $G$ opposite to $N$. Let $r < 1$. The space $\mathrm{Hom}_{U(\mathfrak{g})}(M, \Pi_r)^{N_0}$ is the space of maps from $M$ to $\Pi_r$ that are equivariant for the actions of $N_0$ and $U(\mathfrak{g})$. Therefore we have

$$ \mathrm{Hom}_{U(\mathfrak{g})}(M, \Pi_r)^{N_0} = \mathrm{Hom}_{U_r(\mathfrak{g}) \otimes_{U_r(n)} D_r(N_0)}(M_r, \Pi_r) \simeq \mathrm{Hom}_L^{\mathrm{cont}}(D_r(H) \otimes_{(U_r(\mathfrak{g}) \otimes_{U_r(n)} D_r(N_0))} M_r, \mathrm{Hom}_L^{\mathrm{cont}}(D_r(\mathbb{Z}_p^s, L), L))^m. $$

As $D_r(H)$ is a finite free right $U_r(\mathfrak{g}) \otimes_{U_r(n)} D_r(N_0)$-module (see [Koh07, Thm 1.4]), this proves the claim. $\square$
**Theorem 3.14.** The functor $M \mapsto J_B(\mathrm{Hom}_{U(g)}(M, \Pi))$ from the category $\mathcal{O}_{\mathrm{alg}}^{\infty}$ to the category of essentially admissible representations of $T$ is exact.

*Proof.* This is essentially a consequence of Lemma 3.13 (ii) and we conclude as at the end of the proof Proposition 3.5. □

### 3.3 The case of Banach representations with coefficients

Let $R$ be a complete local noetherian $\mathcal{O}_L$-algebra. As above we will write $R^{\mathrm{rig}}$ for the ring of rigid analytic functions on $(\mathrm{Spf} R)^{\mathrm{rig}}$. Let $\Pi$ be an $R$-admissible $R$-Banach representation of the group $G$ (see [BHS17b, Def. 3.1]). We assume that our representations satisfies the following property:

**Hypothesis 3.15.** there exists an integer $s \ge 0$, a local morphism of $\mathcal{O}_L$-algebras $S := \mathcal{O}_L[[\mathbb{Z}_p^s]] \to R$ such that, for some (resp. any) open pro-$p$-subgroup $G_0 \subset G$, the $S[[G_0]]\backslash[1/p]$-module $\Pi' := \mathrm{Hom}_L^{\mathrm{cont}}(\Pi, L)$ is finite free (as a consequence $\Pi$ is also $S$-admissible).

Using the hypothesis, one shows that the $R$-analytic vectors $\Pi^{R-\mathrm{an}}$ and the $S$-analytic vectors $\Pi^{S-\mathrm{an}}$ of $\Pi$ coincide and they also coincides with the subspace of $\mathbb{Z}_p^s \times G$-locally analytic vectors in $\Pi$ (see [BHS17b, Prop. 3.8]). We will simply denote this subspace by $\Pi^{\mathrm{la}}$ in what follows. This is a locally analytic representation of $\mathbb{Z}_p^s \times G$ with an action of $R^{\mathrm{rig}}$ commuting with $G$. Moreover if we forget the $R^{\mathrm{rig}}$-action, the representation $\Pi^{\mathrm{la}}$ satisfies Hypothesis 3.7.

In the following we will write $\hat{T}$ for the rigid analytic space of continuous characters of $T$ and $\hat{T}_0$ for the space of continuous characters of the maximal compact subgroup $T_0 \subset T$. We recall that the ring of rigid analytic functions on $\hat{T}_0$ is identified with the algebra $D(T_0, L)$ of $L$-valued distributions on $T_0$. Restriction to $T_0$ defines a canonical projection $\hat{T} \to \hat{T}_0$. Moreover, the derivative of a character at 1 defines a weight map

$$ \mathrm{wt} : \hat{T}_0 \to t^*, \qquad (5) $$

where by abuse of notation we write $t^*$ for the rigid analytic space associated to the $L$-vector space $t^*$. The map $\mathrm{wt}$ is étale and locally finite. Moreover, étaleness implies that for any character $\delta_0 : T_0 \to L^\times$ we can identify the tangent space of $\hat{T}_0$ at $\delta_0$ with the $L$-vector space $t^*$.

**Lemma 3.16.** For any object $M$ in $\mathcal{O}_{\mathrm{alg}}^{\infty}$, the dual $J_B(\mathrm{Hom}_{U(g)}(M, \Pi^{\mathrm{la}}))'$ of the Emerton-Jacquet module $J_B(\mathrm{Hom}_{U(g)}(M, \Pi^{\mathrm{la}}))$ is coadmissible as an $R^{\mathrm{rig}}\hat{\otimes}_L \mathcal{O}(\hat{T})$-module.

*Proof.* This is essentially the same proof than for Proposition 3.12 using the fact that $J_B(\Pi^{\mathrm{la}})$ is essentially admissible as a representation of $\mathbb{Z}_p^{s'} \times T$ for any $s'$ and surjection $\mathcal{O}_L[[\mathbb{Z}_p^{s'}]] \to R$ by [BHS17b, Prop. 3.4]. □
**Theorem 3.14.** The functor $M \mapsto J_B(\mathrm{Hom}_{U(g)}(M, \Pi))$ from the category $\mathcal{O}_{\mathrm{alg}}^{\infty}$ to the category of essentially admissible representations of $T$ is exact.

*Proof.* This is essentially a consequence of Lemma 3.13 (ii) and we conclude as at the end of the proof Proposition 3.5. □

### 3.3 The case of Banach representations with coefficients

Let $R$ be a complete local noetherian $\mathcal{O}_L$-algebra. As above we will write $R^{\mathrm{rig}}$ for the ring of rigid analytic functions on $(\mathrm{Spf} R)^{\mathrm{rig}}$. Let $\Pi$ be an $R$-admissible $R$-Banach representation of the group $G$ (see [BHS17b, Def. 3.1]). We assume that our representations satisfies the following property:

**Hypothesis 3.15.** there exists an integer $s \ge 0$, a local morphism of $\mathcal{O}_L$-algebras $S := \mathcal{O}_L[[\mathbb{Z}_p^s]] \to R$ such that, for some (resp. any) open pro-$p$-subgroup $G_0 \subset G$, the $S[[G_0]]\backslash[1/p]$-module $\Pi' := \mathrm{Hom}_L^{\mathrm{cont}}(\Pi, L)$ is finite free (as a consequence $\Pi$ is also $S$-admissible).

Using the hypothesis, one shows that the $R$-analytic vectors $\Pi^{R-\mathrm{an}}$ and the $S$-analytic vectors $\Pi^{S-\mathrm{an}}$ of $\Pi$ coincide and they also coincides with the subspace of $\mathbb{Z}_p^s \times G$-locally analytic vectors in $\Pi$ (see [BHS17b, Prop. 3.8]). We will simply denote this subspace by $\Pi^{\mathrm{la}}$ in what follows. This is a locally analytic representation of $\mathbb{Z}_p^s \times G$ with an action of $R^{\mathrm{rig}}$ commuting with $G$. Moreover if we forget the $R^{\mathrm{rig}}$-action, the representation $\Pi^{\mathrm{la}}$ satisfies Hypothesis 3.7.

In the following we will write $\hat{T}$ for the rigid analytic space of continuous characters of $T$ and $\hat{T}_0$ for the space of continuous characters of the maximal compact subgroup $T_0 \subset T$. We recall that the ring of rigid analytic functions on $\hat{T}_0$ is identified with the algebra $D(T_0, L)$ of $L$-valued distributions on $T_0$. Restriction to $T_0$ defines a canonical projection $\hat{T} \to \hat{T}_0$. Moreover, the derivative of a character at 1 defines a weight map

$$ \mathrm{wt} : \hat{T}_0 \to t^*, \qquad (5) $$

where by abuse of notation we write $t^*$ for the rigid analytic space associated to the $L$-vector space $t^*$. The map $\mathrm{wt}$ is étale and locally finite. Moreover, étaleness implies that for any character $\delta_0 : T_0 \to L^\times$ we can identify the tangent space of $\hat{T}_0$ at $\delta_0$ with the $L$-vector space $t^*$.

**Lemma 3.16.** For any object $M$ in $\mathcal{O}_{\mathrm{alg}}^{\infty}$, the dual $J_B(\mathrm{Hom}_{U(g)}(M, \Pi^{\mathrm{la}}))'$ of the Emerton-Jacquet module $J_B(\mathrm{Hom}_{U(g)}(M, \Pi^{\mathrm{la}}))$ is coadmissible as an $R^{\mathrm{rig}}\hat{\otimes}_L \mathcal{O}(\hat{T})$-module.

*Proof.* This is essentially the same proof than for Proposition 3.12 using the fact that $J_B(\Pi^{\mathrm{la}})$ is essentially admissible as a representation of $\mathbb{Z}_p^{s'} \times T$ for any $s'$ and surjection $\mathcal{O}_L[[\mathbb{Z}_p^{s'}]] \to R$ by [BHS17b, Prop. 3.4]. □
Let $M$ be an object of $\mathcal{O}_{\text{alg}}^{\infty}$. It follows from Lemma 3.16 that there exists a unique
up to unique isomorphism coherent sheaf $\mathcal{M}_{\Pi}(M)$ on $\text{Spf}(R)^{\text{rig}} \times \hat{T}$ such that

$$
\Gamma(\text{Spf}(R)^{\text{rig}} \times \hat{T}, \mathcal{M}_{\Pi}(M)) = J_B(\text{Hom}_{U(\mathfrak{g})}(M, \Pi^{\text{la}}))'.
$$

In particular we obtain a functor from $\mathcal{O}_{\text{alg}}^{\infty}$ to the category of coherent sheaves on
$\text{Spf}(R)^{\text{rig}} \times \hat{T}$.

**Theorem 3.17.** *The coherent sheaf $\mathcal{M}_{\Pi}(M)$ on $\text{Spf}(R)^{\text{rig}} \times \hat{T}$ is, locally on $\text{Spf}(R)^{\text{rig}} \times \hat{T}$, finite free over $\text{Spf}(S)^{\text{rig}}$. In particular, if nonzero, it is Cohen-Macaulay of dimension $s$.*

Proof. Let $T_0$ be the maximal compact subgroup of $T$ and let $\hat{T}_0$ be the rigid analytic
space of characters of $T_0$ over $L$. Set $N := J_B(\text{Hom}_{U(\mathfrak{g})}(M, \Pi^{\text{la}}))'$. It follows from the
proof of [BHS17b, Prop. 3.11] that there exists a family $\mathcal{I}$ of pairs $(U, V)$ where $U$ is a
rational open subset of $\text{Spf}(R)^{\text{rig}} \times \hat{T}$ and $V$ is a rational open subset of $\text{Spf}(S)^{\text{rig}} \times \hat{T}_0$
such that $V$ is the image of $U$ and such that $\text{Supp}(\mathcal{M}_{\Pi}(M)) \subset \bigcup_{(U,V) \in \mathcal{I}} U$. Moreover,
we may assume that $\Gamma(U, \mathcal{M}_{\Pi}(M))$ is a finite projective $\mathcal{O}(V)$-module that is a direct
factor of $\mathcal{O}(V) \hat{\otimes}_{S^{\text{rig}} \hat{\otimes}_L D(T_0, L)} N$.

After shrinking each $U$ and $V$ if necessary, we may even assume (by the construction
of the family $\mathcal{I}$) that for each $(U, V) \in \mathcal{I}$, the rational open $V$ is of the form $V_1 \times V_2$
with $V_1$ rational open in $\text{Spf}(S)^{\text{rig}}$ and $V_2$ rational open in $\hat{T}_0$. It is sufficient to prove
that, for any pair $(U, V_1 \times V_2) \in \mathcal{I}$, the $\mathcal{O}(V_1)$-module $\Gamma(U, \mathcal{M}(M))$ is finitely generated
and flat.

The map $V_2 \to t^*$ has finite fibers (as the weight map is locally finite), and hence
there are only finitely many points of $V_2$ lying over a given character of $U(t)$. It thus
follows from Lemma 3.10 that the action of $L[T_0]$ on $\Gamma(U, \mathcal{M}(M))$ factors through a finite
dimensional quotient. It follows that $\Gamma(U, \mathcal{M}(M))$ is finitely generated over $\mathcal{O}(V_1)$.

Let $\mathfrak{m} \subset \mathcal{O}(V_1)$ be a maximal ideal. As $\mathcal{O}(V_1)$ is an affinoid $L$-algebra, $\mathfrak{m}$ is closed in $\mathcal{O}(V_1)$ and $\mathcal{O}(V_1)/\mathfrak{m}$ is a finite extension of $L$. As the image of $S^{\text{rig}}$ in $\mathcal{O}(V_1)$ is dense, we have $S^{\text{rig}}/(S^{\text{rig}} \cap \mathfrak{m}) \simeq \mathcal{O}(V_1)/\mathfrak{m}$. The ideal $\mathfrak{a} := S^{\text{rig}} \cap \mathfrak{m}$ of $S^{\text{rig}}$ is finitely generated by Lemma 3.2, so that the sheaf $\mathfrak{a} \otimes_{S^{\text{rig}}} \mathcal{M}_{\Pi}(M)$ is coherent and

$$
\Gamma(\mathrm{Spf}(R)^{\mathrm{rig}} \times \hat{T}, \mathfrak{a} \otimes_{S^{\mathrm{rig}}} \mathcal{M}_{\Pi}(M)) \simeq \mathfrak{a} \otimes_{S^{\mathrm{rig}}} \Gamma(\mathrm{Spf}(R)^{\mathrm{rig}} \times \hat{T}, \mathcal{M}_{\Pi}(M)).
$$

As the functor $\mathcal{M} \mapsto \Gamma(U, \mathcal{M}_{\Pi})$ is exact on the category of coherent sheaves, we have
an isomorphism

$$
\Gamma(U, \mathfrak{a} \otimes_{S^{\text{rig}}} \mathcal{M}_{\Pi}(M)) \simeq \mathfrak{a} \otimes_{S^{\text{rig}}} \Gamma(U, \mathcal{M}_{\Pi}(M)) \simeq \mathfrak{m} \otimes_{\mathcal{O}(V_1)} \Gamma(U, \mathcal{M}_{\Pi}(M)).
$$

Therefore we deduce from Proposition 3.6 that the map

$$
\mathfrak{m} \otimes_{\mathcal{O}(V_1)} \otimes \Gamma(U, \mathcal{M}_{\Pi}(M)) \to \Gamma(U, \mathcal{M}(M))
$$

is injective. This implies that $\Gamma(U, \mathcal{M}_{\Pi}(M))$ is a flat $\mathcal{O}(V_1)$-module. $\square$

<figure><img src="image_3.png" /></figure>
Let $M$ be an object of $\mathcal{O}_{\text{alg}}^{\infty}$. It follows from Lemma 3.16 that there exists a unique up to unique isomorphism coherent sheaf $\mathcal{M}_{\Pi}(M)$ on $\text{Spf}(R)^{\text{rig}} \times \hat{T}$ such that

$$ \Gamma(\text{Spf}(R)^{\text{rig}} \times \hat{T}, \mathcal{M}_{\Pi}(M)) = J_B(\text{Hom}_{U(\mathfrak{g})}(M, \Pi^{\text{la}}))'. $$

In particular we obtain a functor from $\mathcal{O}_{\text{alg}}^{\infty}$ to the category of coherent sheaves on $\text{Spf}(R)^{\text{rig}} \times \hat{T}$.

**Theorem 3.17.** The coherent sheaf $\mathcal{M}_{\Pi}(M)$ on $\text{Spf}(R)^{\text{rig}} \times \hat{T}$ is, locally on $\text{Spf}(R)^{\text{rig}} \times \hat{T}$, finite free over $\text{Spf}(S)^{\text{rig}}$. In particular, if nonzero, it is Cohen-Macaulay of dimension $s$.

*Proof.* Let $T_0$ be the maximal compact subgroup of $T$ and let $\hat{T}_0$ be the rigid analytic space of characters of $T_0$ over $L$. Set $N := J_B(\text{Hom}_{U(\mathfrak{g})}(M, \Pi^{\text{la}}))'$. It follows from the proof of [BHS17b, Prop. 3.11] that there exists a family $\mathcal{I}$ of pairs $(U, V)$ where $U$ is a rational open subset of $\text{Spf}(R)^{\text{rig}} \times \hat{T}$ and $V$ is a rational open subset of $\text{Spf}(S)^{\text{rig}} \times \hat{T}_0$ such that $V$ is the image of $U$ and such that $\text{Supp}(\mathcal{M}_{\Pi}(M)) \subset \bigcup_{(U,V) \in \mathcal{I}} U$. Moreover, we may assume that $\Gamma(U, \mathcal{M}_{\Pi}(M))$ is a finite projective $\mathcal{O}(V)$-module that is a direct factor of $\mathcal{O}(V) \hat{\otimes}_{S^{\text{rig}} \hat{\otimes}_L D(T_0, L)} N$.

After shrinking each $U$ and $V$ if necessary, we may even assume (by the construction of the family $\mathcal{I}$) that for each $(U, V) \in \mathcal{I}$, the rational open $V$ is of the form $V_1 \times V_2$ with $V_1$ rational open in $\text{Spf}(S)^{\text{rig}}$ and $V_2$ rational open in $\hat{T}_0$. It is sufficient to prove that, for any pair $(U, V_1 \times V_2) \in \mathcal{I}$, the $\mathcal{O}(V_1)$-module $\Gamma(U, \mathcal{M}(M))$ is finitely generated and flat.

The map $V_2 \to t^*$ has finite fibers (as the weight map is locally finite), and hence there are only finitely many points of $V_2$ lying over a given character of $U(t)$. It thus follows from Lemma 3.10 that the action of $L[T_0]$ on $\Gamma(U, \mathcal{M}(M))$ factors through a finite dimensional quotient. It follows that $\Gamma(U, \mathcal{M}(M))$ is finitely generated over $\mathcal{O}(V_1)$.

Let $\mathfrak{m} \subset \mathcal{O}(V_1)$ be a maximal ideal. As $\mathcal{O}(V_1)$ is an affinoid $L$-algebra, $\mathfrak{m}$ is closed in $\mathcal{O}(V_1)$ and $\mathcal{O}(V_1)/\mathfrak{m}$ is a finite extension of $L$. As the image of $S^{\text{rig}}$ in $\mathcal{O}(V_1)$ is dense, we have $S^{\text{rig}}/(S^{\text{rig}} \cap \mathfrak{m}) \simeq \mathcal{O}(V_1)/\mathfrak{m}$. The ideal $\mathfrak{a} := S^{\text{rig}} \cap \mathfrak{m}$ of $S^{\text{rig}}$ is finitely generated by Lemma 3.2, so that the sheaf $\mathfrak{a} \otimes_{S^{\text{rig}}} \mathcal{M}_{\Pi}(M)$ is coherent and

$$ \Gamma(\text{Spf}(R)^{\text{rig}} \times \hat{T}, \mathfrak{a} \otimes_{S^{\text{rig}}} \mathcal{M}_{\Pi}(M)) \simeq \mathfrak{a} \otimes_{S^{\text{rig}}} \Gamma(\text{Spf}(R)^{\text{rig}} \times \hat{T}, \mathcal{M}_{\Pi}(M)). $$

As the functor $\mathcal{M} \mapsto \Gamma(U, \mathcal{M}_{\Pi})$ is exact on the category of coherent sheaves, we have an isomorphism

$$ \Gamma(U, \mathfrak{a} \otimes_{S^{\text{rig}}} \mathcal{M}_{\Pi}(M)) \simeq \mathfrak{a} \otimes_{S^{\text{rig}}} \Gamma(U, \mathcal{M}_{\Pi}(M)) \simeq \mathfrak{m} \otimes_{\mathcal{O}(V_1)} \Gamma(U, \mathcal{M}_{\Pi}(M)). $$

Therefore we deduce from Proposition 3.6 that the map

$$ \mathfrak{m} \otimes_{\mathcal{O}(V_1)} \Gamma(U, \mathcal{M}_{\Pi}(M)) \to \Gamma(U, \mathcal{M}(M)) $$

is injective. This implies that $\Gamma(U, \mathcal{M}_{\Pi}(M))$ is a flat $\mathcal{O}(V_1)$-module. $\square$

<figure><img src="image_3.png" /></figure>
**Corollary 3.18.** Assume that the representation Π satisfies Hypothesis 3.15. Then the functor $M \mapsto \mathcal{M}_{\Pi}(M)$ is an exact functor from the category $\mathcal{O}_{\text{alg}}^{\infty}$ to the category of Cohen-Macaulay sheaves on $\text{Spf}(R)^{\text{rig}} \times \hat{T}$. Moreover if $\mathcal{M}_{\Pi}(M)$ is nonzero, its support is s-dimensional, where s is as in Hypothesis 3.15.

## 3.4 Comparison with the parabolic Jacquet functor

Let Π be an R-admissible Banach representation of G satisfying hypothesis 3.15. We end this section by computing the evaluation of $\mathcal{M}_{\Pi}$ on generalized (deformed) Verma modules in terms of Emerton’s parabolic Jacquet-module.

Let $I \subset \Delta$ be a subset of simple roots. Let $\lambda \in X^*(T)_I^+$ be an algebraic character dominant with respect to $\mathfrak{p}_I$. Recall that, by [Eme06a, §3.4], the L-representation $J_{P_I}(\Pi^{\text{la}})$ of $L_I$ is locally analytic. Following [Wu, §5.2], we define

$$J_{P_I}(\Pi^{\text{la}})_{\lambda} := \text{Hom}_{U(\mathfrak{I}^{\text{ss}})}(L_I(\lambda), J_{P_I}(\Pi^{\text{la}})) \otimes_L L_I(\lambda)$$

$$J_{I,\lambda}(\Pi^{\text{la}}) := J_{B \cap L_I}(J_{P_I}(\Pi^{\text{la}})_{\lambda}).$$

Similarly to Lemma 3.16 we have the following finiteness result:

**Proposition 3.19.** The $R^{\text{rig}} \hat{\otimes}_L \mathcal{O}(\hat{T})$-module $J_{I,\lambda}(\Pi^{\text{la}})'$ is coadmissible.

*Proof.* This is a consequence of [Wu, Lemm. 5.1 & 5.2]. □

<figure><img src="image_4.png" /></figure>

By the above proposition there is a coherent sheaf $\mathcal{M}_{\Pi}^{I,\lambda}$ on $\text{Spf}(R)^{\text{rig}} \times \hat{T}$ such that

$$\Gamma(\text{Spf}(R)^{\text{rig}} \times \hat{T}, \mathcal{M}_{\Pi}^{I,\lambda}) = J_{I,\lambda}(\Pi^{\text{la}})'$$

For $k \ge 1$, let $\hat{T}_k^{\text{sm}}$ be the k-th infinitesimal neighborhood of the closed subspace $\hat{T}^{\text{sm}}$ of smooth characters in $\hat{T}$ and let $i_k$ be the closed immersion of $\hat{T}_k^{\text{sm}}$ in $\hat{T}$. Moreover, for $\lambda \in X^*(T) \subset \hat{T}$, we write $t_{\lambda}: \hat{T} \to \hat{T}$ for the map defined by $t_{\lambda}(\delta) = \delta\lambda$.

**Proposition 3.20.** Let $\lambda \in X^*(T)_I^+$ be an algebraic character of $\underline{T}$ dominant with respect to $\underline{P}_I$ and let $M = \tilde{M}_I(\lambda) \otimes_{A_I} A_I/\mathfrak{m}^k \in \mathcal{O}_{\text{alg}}^{\infty}$. Then there is an isomorphism of coherent sheaves on $\text{Spf}(R)^{\text{rig}} \times \hat{T}$:

$$\mathcal{M}_{\Pi}(M) \simeq i_{k,*} i_k^* t_{\lambda}^* \mathcal{M}_{\Pi}^{I,\lambda}.$$

*Proof.* Using the left exactness of the functor $J_{P_I}(-)$, we have an isomorphism an $R^{\text{rig}}$-equivariant morphism of locally analytic representations of $L_I$:

$$J_{P_I}(\text{Hom}_{U(\mathfrak{g})}(\tilde{M}_I(\lambda) \otimes_{A_I} A_I/\mathfrak{m}^k, \Pi^{\text{la}})) \simeq \text{Hom}_{U(\mathfrak{I}_I)}(L_I(\lambda) \otimes_L A_I/\mathfrak{m}^k, J_{P_I}(\Pi^{\text{la}})).$$
**Corollary 3.18.** Assume that the representation Π satisfies Hypothesis 3.15. Then the functor $M \mapsto M_{\Pi}(M)$ is an exact functor from the category $O_{alg}^{\infty}$ to the category of Cohen-Macaulay sheaves on Spf$(R)^{\text{rig}} \times \hat{T}$. Moreover if $M_{\Pi}(M)$ is nonzero, its support is $s$-dimensional, where $s$ is as in Hypothesis 3.15.

## 3.4 Comparison with the parabolic Jacquet functor

Let Π be an $R$-admissible Banach representation of $G$ satisfying hypothesis 3.15. We end this section by computing the evaluation of $M_{\Pi}$ on generalized (deformed) Verma modules in terms of Emerton’s parabolic Jacquet-module.

Let $I \subset \Delta$ be a subset of simple roots. Let $\lambda \in X^*(T)_I^+$ be an algebraic character dominant with respect to $\mathfrak{p}_I$. Recall that, by [Eme06a, §3.4], the $L$-representation $J_{P_I}(\Pi^{\text{la}})$ of $L_I$ is locally analytic. Following [Wu, §5.2], we define

$$J_{P_I}(\Pi^{\text{la}})_{\lambda} := \operatorname{Hom}_{U(\mathfrak{I}^{\text{ss}})}(L_I(\lambda), J_{P_I}(\Pi^{\text{la}})) \otimes_L L_I(\lambda)$$
$$J_{I,\lambda}(\Pi^{\text{la}}) := J_{B \cap L_I}(J_{P_I}(\Pi^{\text{la}})_{\lambda}).$$

Similarly to Lemma 3.16 we have the following finiteness result:

**Proposition 3.19.** The $R^{\text{rig}} \hat{\otimes}_L O(\hat{T})$-module $J_{I,\lambda}(\Pi^{\text{la}})'$ is coadmissible.

*Proof.* This is a consequence of [Wu, Lemm. 5.1 & 5.2]. □

<figure><img src="image_4.png" /></figure>

By the above proposition there is a coherent sheaf $M_{\Pi}^{I,\lambda}$ on Spf$(R)^{\text{rig}} \times \hat{T}$ such that

$$\Gamma(\operatorname{Spf}(R)^{\text{rig}} \times \hat{T}, M_{\Pi}^{I,\lambda}) = J_{I,\lambda}(\Pi^{\text{la}})'$$

For $k \ge 1$, let $\hat{T}_k^{\text{sm}}$ be the $k$-th infinitesimal neighborhood of the closed subspace $\hat{T}^{\text{sm}}$ of smooth characters in $\hat{T}$ and let $i_k$ be the closed immersion of $\hat{T}_k^{\text{sm}}$ in $\hat{T}$. Moreover, for $\lambda \in X^*(T) \subset \hat{T}$, we write $t_{\lambda}: \hat{T} \to \hat{T}$ for the map defined by $t_{\lambda}(\delta) = \delta\lambda$.

**Proposition 3.20.** Let $\lambda \in X^*(T)_I^+$ be an algebraic character of $\underline{T}$ dominant with respect to $\underline{P}_I$ and let $M = \widetilde{M}_I(\lambda) \otimes_{A_I} A_I/\mathfrak{m}^k \in O_{\text{alg}}^{\infty}$. Then there is an isomorphism of coherent sheaves on Spf$(R)^{\text{rig}} \times \hat{T}$:

$$M_{\Pi}(M) \simeq i_{k,*} i_k^* t_{\lambda}^* M_{\Pi}^{I,\lambda}.$$

*Proof.* Using the left exactness of the functor $J_{P_I}(-)$, we have an isomorphism an $R^{\text{rig}}$-equivariant morphism of locally analytic representations of $L_I$:

$$J_{P_I}(\operatorname{Hom}_{U(\mathfrak{g})}(\widetilde{M}_I(\lambda) \otimes_{A_I} A_I/\mathfrak{m}^k, \Pi^{\text{la}})) \simeq \operatorname{Hom}_{U(\mathfrak{I}_I)}(L_I(\lambda) \otimes_L A_I/\mathfrak{m}^k, J_{P_I}(\Pi^{\text{la}})).$$
Therefore

$$
\begin{align*}
\operatorname{Hom}_{U(t)}(\lambda \otimes_L A_I/\mathfrak{m}^k, J_{B \cap L_I}(J_{P_I}(\Pi^{\mathrm{la}})_\lambda)) & \\
&\simeq J_{B \cap L_I}(\operatorname{Hom}_{U(t)}(A_I/\mathfrak{m}^k, \operatorname{Hom}_{U(\mathfrak{I}^{\mathrm{ss}})}(L_I(\lambda), J_{P_I}(\Pi^{\mathrm{la}})))) \\
&= J_{B \cap L_I}(\operatorname{Hom}_{U(\mathfrak{I}_I)}(L_I(\lambda) \otimes_L A_I/\mathfrak{m}^k, J_{P_I}(\Pi^{\mathrm{la}}))) \\
&\simeq J_{B \cap L_I}(J_{P_I}(\operatorname{Hom}_{U(\mathfrak{P}_I)}(L_I(\lambda) \otimes_L A_I/\mathfrak{m}^k, \Pi^{\mathrm{la}}))) \\
&\simeq J_B(\operatorname{Hom}_{U(\mathfrak{g})}(\tilde{M}_I(\lambda) \otimes_{A_I} A_I/\mathfrak{m}^k, \Pi^{\mathrm{la}}))
\end{align*}
$$

where the first isomorphism comes from [Wu, Lemm. 5.3]. The claim now follows from the fact that the source of this chain of isomorphisms is the dual (of the global sections) of $i_{k,*}i_k^*t_\lambda^*\mathcal{M}_{\Pi}^{I,\lambda}$ and the target is the dual of $\mathcal{M}_{\Pi}(M)$. $\square$

# 4 Quasi-trianguline local deformation rings

Let $F$ be a finite extension of $\mathbb{Q}$. We keep the notation of section 3 but we specialize ourselves to the case $\underline{G} = \operatorname{Res}_{(F \otimes_{\mathbb{Q}} \mathbb{Q}_p)/\mathbb{Q}_p}(\operatorname{GL}_{n,F \otimes_{\mathbb{Q}} \mathbb{Q}_p}) \simeq \coprod_{v|p} \operatorname{Res}_{F_v/\mathbb{Q}_p} \operatorname{GL}_{n,F_v}$. We fix $\underline{B}$ the upper triangular Borel subgroup and $\underline{T}$ the diagonal torus. It is therefore sufficient to choose $L$ a finite extension of $\mathbb{Q}_p$ splitting all the $F_v$. We point out that, though the field $L$ of coefficients is the same as in the preceding section, the group $\underline{G}$ in this section should be considered as the Langlands dual group of the group in section 3.

Let $\Sigma_F$ be the set of embeddings of $F$ in $L$. This set can be decomposed as $\Sigma_F = \coprod_{v|p} \Sigma_{F_v}$, where $\Sigma_{F_v}$ is the set of $\mathbb{Q}_p$-linear embeddings of $F_v$ into $L$ and where the index set is the set of places $v$ of $F$ that divide $p$. We have a decomposition

$$
\mathfrak{g} \simeq ( \bigoplus_{\tau \in \Sigma_F} \operatorname{Lie}(\underline{G}) \otimes_{F \otimes_{\mathbb{Q}} \mathbb{Q}_p, \tau} L ) \simeq \bigoplus_{\tau \in \Sigma_F} \operatorname{Lie}(\operatorname{GL}_{n,L}).
$$

Let $\Delta$ be the set of simple roots of $\underline{G}_L$ with respect to $\underline{B}_L$. Then

$$
\Delta = \coprod_{\tau \in \Sigma_F} \Delta_{\tau}, \quad \Delta_{\tau} = \{\alpha_{1,\tau}, \dots, \alpha_{n-1,\tau}\}
$$

where $\alpha_{1,\tau}, \dots, \alpha_{n-1,\tau}$ are the simple roots of the copy of $\operatorname{Lie}(\operatorname{GL}_{n,L})$ corresponding to $\tau$. For $I \subset \Delta$ we denote $\underline{P}_I$ the standard parabolic subgroup of $\underline{G}_L$ corresponding to $I$.

## 4.1 Local models

Let $\tilde{\mathfrak{g}} := \underline{G}_L \times_{\underline{B}_L} \mathfrak{b}$ be the Grothendieck-Springer resolution of $\mathfrak{g}$ (which is considered as a scheme over $L$ not just as a vector space in this section). We have a closed embedding $\tilde{\mathfrak{g}} \hookrightarrow \underline{G}_L/\underline{B} \times_{\mathfrak{g}} \mathfrak{g}$ given by $(g\underline{B}, X) \mapsto (b\underline{B}, \operatorname{Ad}(g)X)$ and set

$$
X := \tilde{\mathfrak{g}} \times_{\mathfrak{g}} \tilde{\mathfrak{g}} \subset \underline{G}_L/\underline{B}_L \times_{\mathfrak{g}} \underline{G}_L/\underline{B}_L.
$$