Andrés Sarrazola

G-equivariance of formal models of flag varieties

Keywords: Flag varieties, formal models, Beilinson-Bernstein correspondence, admissible locally analytic representations, localization

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Introduction

Let |ℚ be a finite extension and its ring of integers. Throughout this work will always denote a split connected reductive group scheme over . We will fix a Borel subgroup ⊂ which contain a split maximal torus ⊂ of . We will also denote by ∶= ∕ the smooth flag -scheme associated to and by the smooth formal scheme. In [START_REF] Huyghe | D † -affinity of formal models of flag varieties[END_REF] the authors have introduced certain sheaves of differential operators (with congruence level ∈ ℕ) D † , on a family of formal models of rig the rigid analytic flag variety. They study their cohomological properties and show that is D † , -affine, which is a fundamental result. Moreover, it is proved in [START_REF] Huyghe | D † -affinity of formal models of flag varieties[END_REF]Theorem 5.3.12] that the category of admissible locally analytic representations of the -analytic group ( ) can be described in terms of ( )-equivariant families (M , ) of modules over D † , on the projective system of all formal models of rig . Our motivation is to study the preceding localization theorem in the twisted situation. In this work we will treat the algebraic case. This is, we will only consider characters of the Lie algebra = Lie( ) arriving from characters ∈ ( ) = Hom( , ) via differentiation. In this situation induces an invertible sheaf L ( ) on and we define D † , ( ) as the sheaf of differential operators (with conguence level ) acting on L ( ). We will follow the philosophy described in [START_REF] Huyghe | D † -affinity of formal models of flag varieties[END_REF] introducing sheaves of differential operators on each admisisble blow-up of . Let pr ∶ → be an admissible blow-up, then for >> 01

D † , ( ) ∶= pr * D † , ( ) = O ⊗ pr -1 O pr -1 D † , ( )
is a sheaf of rings on . Let us denote by the so-called Weyl character and let us assume that + ∈ * = Hom ( ⊗ , ) is a dominant and regular character of ∶= ⊗ . In this situation, we will show that the direct image functor pr * induces an equivalence of categories between the category of coherent D † , ( )-modules and the category of coherent D † , ( )modules. In addition, we have pr * D † , ( ) = D † , ( ), which implies that 0 ( , D † , ( )) = 0 ( , D † , ( )).

It is shown that 0 ( , D † , ( )) can be identified with the central redaction D an ( ( ) • )2 of Emerton's analytic distribution algebra D an ( ( ) • ) of the wide open rigid-analytic k-th congruence group ( ) • . Our first result is Theorem 1. Let pr ∶ → be an admissible blow-up. Suppose that ∈ Hom( , ) is an algebraic character such that + ∈ * is a dominant and regular character of . Then 0 ( , •) induces an equivalence between the categories of coherent D † , ( )-modules and finitely presented D an ( ( ) • ) -modules. As in the classical case, the inverse functor is determined by the localization functor

L † , ( )(•) ∶= D † , ( ) ⊗ D an ( ( ) • ) (•).
Let us now describe the most important tools in our localization theorem. On the algebraic side, we will first assume that 0 = ( ) and that ( 0 , ) is the algebra of locally analytic distributions of the compact analytic group 0 (in the sense of [START_REF] Schneider | Continuous and locally analytic representation theory[END_REF]). The key point will be to construct a structure of weak Fréchet-Stein algebra on ( 0 , ) (in the sense of [START_REF] Emerton | Locally Analytic Vectors in Representations of Locally -adic Analytic Groups[END_REF]Definition 1.2.6]) that will allow us to localize the coadmissible ( 0 , )-modules (relative to this weak Fréchet-Stein structure). In fact, if C cts ( 0 , ) ( ) • -an is the vector space of locally analytic vectors of the space of continuous -valued functions and ( ( ) • , 0 ) ∶= (C cts ( 0 , ) ( ) • -an ) ′ is its strong dual, then we have an isomorphism

( 0 , ) ≃ ← ← ← ← ← ← ← → lim ← ← ← ← ← ← ← ← ← ← ←
in the sense of the Abelian groups. Now, let be a coadmissible ( 0 , ) -module and ∶= ′ its associated locally analytic representation. The vector space of ( ) • -analytic vectors ( ) • -an ⊆ is stable under the action of 0 and its dual ∶= ( ( ) • -an ) ′ is a finitely presented ( ( ) • , 0 )-module. In this situation, theorem 2 produces a coherent D † , ( )-module

L † , ( )( ) ∶= D † , ( ) ⊗ D an ( ( ) • )
for each element ( , ) ∈ F . We will note this family L 0 ( ) ∶= L † , ( )( ) ( , )∈F . Theorem 3. Let ∈ Hom( , ) be an algebraic character such that + ∈ * is a dominant and regular character of . The functors L 0 (•) and Γ(•) induce equivalences of categories between the category C 0 , (of coadmissible ( 0 , ) -modules) and the category C 0 , .

Finally, the last part of this work is devoted to the study of coadmissible ( , ) -modules, where ∶= ( ) 6 . To do this, we will consider the Bruhat-Tits building B of ( [START_REF] Bruhat | Groupes réductifs sur un corps local I[END_REF] and [START_REF] Bruhat | Groupes réductifs sur un corps local II[END_REF]). It is a simplicial complex equipped with a -action. For any special vertex ∈ B, the theory of Bruhat and Tits associates a reductive group whose generic fiber is canonically isomorphic to × Spec( ) Spec( ). Let be the flag scheme of , and its formal completion along its special fiber. We consider the set F composed of triples ( , , ) such that is a special vertex, → is an admissible blow-up of and ≥ . According to (7.0.2) F is partially ordered. In addition, for each special vertex ∈ B, each element ∈ induces an isomorphism ∶ → , such that if ( ) ♮ ∶ O → ( ) * O is the comorphism map and ∶ → is an admissible blow-up along (I ), then the (admissible) blow-up along (( ) -1 ( ) * I ) produces a scheme with an isomorphism ∶ → , such that = and for every , ℎ ∈ we have ℎ • = ℎ .

A coadmissible -equivariant arithmetic D( )-module on F , consists of a family (M ( , , ) ) ( , , )∈F of coherent D † , ( )modules satisfying the condition ( †) plus some compatibility properties (definition 7.0.4) that allow us to form the projective limit

Γ(M ) ∶= lim ← ← ← ← ← ← ← ← ← ← ← ( , , )∈F 0 ( , M ( , , ) ),
which, as we will show, has a structure of coadmissible ( , ) -module. On the other hand, given a coadmissible ( , ) -module , we consider ∶= ′ its continuous dual, which is a locally analytic representation of . Then let , be the dual space of the subspace ( ) • -an ⊆ of ( ) • -analytic vectors. For every ( , , ) ∈ F , we have a coherent D † , ( )-module

L † , ( )( , ) = D † , ( ) ⊗ D an ( ( ) • ) , .
We note this family L ( ). We will show the following result (theorem 7.0.6).

Theorem 4. Let ∈ Hom( , ) be an algebraic character such that + ∈ * is a dominant and regular character of . The functors L (•) and Γ(•) give an equivalence between the categories of coadmissible ( , ) -modules and coadmissible -equivariant arithmetic D( )-modules.

The last task will be to study the projective limit

∞ ∶= lim ← ← ← ← ← ← ← ← ← ← ← ( , , )
. This is the Zariski-Riemann space associated to the rigid flag variety rig . We can also form the projective limit D( ) of the sheaves D † , ( ) which is a sheaf of -equivariant differential operators on ∞ . Similarly, if (M ( , , ) ) ( , , )∈F is a coadmissible -equivariant arithmetic D( )-module, then we can form the projective limit M ∞ . The data M ( , , )∈F ⇝ M ∞ induces a faithful functor from the category of coadmissible -equivariant arithmetic D( )-modules on F to the category of -equivariant D( )-modules on ∞ (theorem 7.0.8). In fact, this is a fully faithful functor as we will briefly explain in our Final remark (7.0.9).

We summarize the main results of this work with the following commutative diagrams of functors (cf. [ Notation: Throughout this work will denote the uniformizer of . Furthermore, if is an arbitrary noetherian scheme over , then for every ∈ ℕ we will denote by ∶= × Spec( ) Spec( ∕ +1 ) the reduction modulo +1 , and by

= lim ← ← ← ← ← ← ← ← ← ← →
the formal completion of along the special fiber. Moreover, if E is a sheaf of -modules on then its -completion

E ∶= lim ← ← ← ← ← ← ← ← ← ← ← E ∕ +1
E will be considered as a sheaf on . Finally, the base change of a sheaf of -modules on (resp. on ) to will always be denoted by the subscript ℚ. For instance

E ℚ ∶= E ⊗ (resp. E ℚ ∶= E ⊗ ).
2 Arithmetic definitions

p-adic coefficients and divided powers

Let be a prime number and let us fix a positive integer . Throughout this work, we will denote by ℤ the ring of -adic integers and by ℤ ( ) the localization of ℤ with respect to the prime ideal ( ). Moreover, if ∈ ℕ, we will note the quotient of the euclidean division of by . Berthelot has introduced in [START_REF] Berthelot | D-modules arithmétiques I. Opérateurs différentiels de niveau fini[END_REF] the following coefficients for any two integers

, ′ with ≥ ′ ′ ∶= ! ′ ! ′′ ! , ′′ ∶= -′ .
In fact, we can generalize these coefficients for multi-indexes = ( 1 , ...,

) ∈ ℕ by defining ! ∶= 1 ! ... ! and ′ ∶= ! ′ ! ′′ ! ∈ ℕ and ⟨ ′ ⟩ ∶= ′ ′ -1 ∈ ℤ .
Now, let A be a ℤ ( ) algebra. We say that a triple ( , , ) is an -PD ideal of , if defines a structure of divided powers on (a PD-structure in the sense of [START_REF] Berthelot | Notes on crystalline cohomology[END_REF]) and is endowed of a system of partial divided powers, meaning that for any integer , which decomposes as = + (with < ), there exists an operation defined for every ∈ by = .

Arithmetic differential operators

Let us suppose that is endowed with the m-PD-structure ( , , [ ]) defined in example 2.1.1. Let be a smooth -scheme, and I ⊂ O a quasi-coherent ideal. Let us consider the sheaf of principal parties P ( ) ( ) [3, Section 2.1],which contains an m-PD structure ( Ī, Ĩ, [ ]) and the sequence of ideals ( Ī{ } ) ∈ℕ defining the m-PD-filtration [4, 1.3]. For every ∈ ℕ, the algebra P ,( ) ∶= P ( ) (I)∕ Ī{ } is quasi-coherent and can be considered as a sheaf on . Moreover, the projections 1 , 2 ∶ × → induce two morphisms 1 , 2 ∶ O → P ,( ) endowing P ,( ) of a left and a right structure of O -algebra, respectively. Definition 2.2.1. Let , be positive integers. The sheaf of differential operators of level and order less or equal to n on is defined by D ( ) , ∶= H O (P ,( ) , O ).

If ≤ ′ then [3, Proposition 1.4.1] gives us a canonical surjection P ′ ,( ) → P ,( ) which induces the injection D ( ) , → D ( ) , ′ and the sheaf of differential operators of level is defined by

D ( ) ∶= ⋃ ∈ℕ D ( ) , .
We remark for the reader that by definition D ( ) is endowed with a natural filtration called the order filtration, and like the sheaves P ,( ) , the sheaves D ( ) , are endowed with two natural structures of O -modules. Moreover, the sheaf D ( ) acts on O : if ∈ D ( ) , , then this action is given by the composition O

1 ← ← ← ← ← ← ← ← ← ← → P ,( ) ← ← ← ← ← ← ← ← → O .
D ( ) , ( ) = ∑ | |≤ < > | ∈ O ( ) and ∈ ℕ . ( 3
)

Symmetric algebra of finite level

In this subsection we will focus on introducing the constructions in [START_REF] Huyghe | D † -affinité de l'espace projectif[END_REF]. As before, let denote a smooth -scheme and let us consider L a locally free O -module of finite rank, S (L) the symmetric algebra associated to L and I the ideal of homogeneous elements of degree 1. If P S (L),( ) (I) denotes the -divided power enveloping of (S (L), I) ([3, Proposition 1.4.1]) then we can consider the coherent sheaves on Γ ,( ) (L) ∶= P S (L),( ) (I) and Γ ,(

) (L) ∶= Γ ,( ) (L)∕ Ī{ +1} . ( 4
)
Those algebras are graded [START_REF] Huyghe | D † -affinité de l'espace projectif[END_REF]Proposition 1.3.3] and if 1 , ..., is a local basis of L, we have

Γ ,( ) (L) = ⨁ | |≤ O { } .
As before { } = ∏ =1 { } and ! { } = . We define by duality

Sym ( ) (L) ∶= ⋃ ∈ℕ H O Γ ,( ) (L ∨ ),
( ) (L) = ⨁ | |= O < > , where ! ! < > = .
Remark 2.3.1. By [5, A.10] we have that S (0) (L) is the symmetric algebra of L, which justifies the terminology.

We end this subsection by remarking the following results from [START_REF] Huyghe | D † -affinité de l'espace projectif[END_REF]. Let I be the kernel of the comorphism Δ ♯ of the diagonal embedding Δ ∶ → × Spec( ) . In [START_REF] Huyghe | D † -affinité de l'espace projectif[END_REF]Proposition 1.3.7.3] Huyghe shows that the graded algebra associated to the m-PD-adic filtration of P ,( ) it is identified with the graded m-PD-algebra Γ ,( ) (I∕I 2 ) = Γ ,( ) (Ω 1 ). More exactly, we dispose of a canonical morphism of O -algebras

S (Ω ) → • P ,( ) which extends, via universal property [3, Proposition 1.4.1], to a morphism Γ ,( ) (Ω 1 ) ≃ ← ← ← ← ← ← ← → • (P ,( ) )
. By definition, it induces a graded morphism

Sym ( ) (T ) → • D ( ) (5)
which is in fact an isomorphism of O -algebras.

Arithmetic distribution algebra of finite level

As in the introduction, let us consider a split connected reductive group scheme over and ∈ ℕ fixed. We propose to give a description of the algebra of distributions of level introduced in [START_REF] Huyghe | D-modules arithmétiques, distributions et localisation[END_REF]. Let denote the kernel of the surjective morphism of -algebras ∶ [ ] → , given by the identity element of . We know that ∕ 2 is a free = [ ]∕module of finite rank. Let 1 , . . . , ∈ such that modulo 2 these elements form a basis of ∕ 2 . The -divided power enveloping algebra of ( [ ], ), denoted by ( ) ( ), it is a free -module with basis the elements

{ } = { 1 } 1 . . . { } ,
where ! { } = , for every = + and 0 ≤ < . These algebras are endowed with a decreasing filtration by ideals { } (the -PD filtration), such that and for any two integers , ′ such that ≤ ′ we have a canonical surjection ′ , ∶ ′ ( ) ( ) → ( ) ( ). The module of distributions of level and order is ( ) ( ) ∶= Hom( ( ) ( ), ). The algebra of distributions of level m is

{ } = ⊕ | |≥ { } . The quotients ( ) ( ) ∶= ( ) ( )∕ { +1}
( ) ( ) ∶= lim ← ← ← ← ← ← ← ← ← ← → ( ) ( ),
where the limit is formed respect to the maps Hom ( ′ , , ). The multiplication is defined as follows. By universal property ([3, Proposition 1.4.1]) there exists a canonical application ( ) ′ ( ), we define . as the composition

, ′ ∶ + ′ ( ) ( ) → ( ) ) ⊗ ′ ( ) ( ). If ( , ) ∈ ( ) ( ) ×
. ∶ + ′ ( ) ( ) , ′ ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← → ( ) ( ) ⊗ ′ ( ) ( ) ⊗ ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← → .
Let us denote by ∶= Hom ( ∕ 2 , ) the Lie algebra of . This is a free -module with basis 1 , . . . , defined as the dual basis of the elements 1 , . . . , . If for every multi-index ∈ ℕ , | | ≤ , we denote by < > the dual of the element { } ∈ ( ) ( ), then ( ) ( ) is a free -module of finite rank with a basis given by the elements

< > with | | ≤ [22, proposition 4.1.6].
Remark 2.4.1. 7 Let A be an -algebra and a free -module of finite rank with base ( 1 , ..., ). Let ( 1 , ..., ) be the dual base of ∨ ∶= Hom ( , ). As in the preceding subsection, let S( ∨ ) be the symmetric algebra and I( ∨ ) the augmentation ideal. Let Γ ( ) ( ∨ ) be the -divided power enveloping algebra of (S( ∨ ), I( ∨ )). We put

Γ ( ) ( ∨ ) ∶= Γ ( ) ( ∨ )∕ { +1}
. These are free -modules with base

{ 1 } 1 ... { } with ∑ ≤ [20, 1.1.2]. Let { < > } | |≤
be the dual base of Hom (Γ ( ) ( ∨ ), ). We define

( ) ( ) ∶= ⋃ ∈ℕ Hom Γ ( ) ( ∨ ), .
This is a free -module with a base given by all the < > . The inclusion Sym ( ) ( ) ⊆ Sym ( ) ( ) ⊗ gives the relation

< > = ! ! . ( 6
)
Moreover, it also has a structure of algebra defined as follows. By [START_REF] Huyghe | D † -affinité de l'espace projectif[END_REF]Proposition 1.3.1] there exists an application

Δ , ′ ∶ Γ + ′ ( ) ( ∨ ) → Γ ( ) ( ∨ ) ⊗ Γ ′ ( ) ( ∨ )
, which allows to define the product of ∈ (Γ ( ) ( ∨ ), ) and ∈ (Γ ′ ( ) ( ∨ ), ) by the composition

. ∶ Γ + ′ ( ) ( ∨ ) Δ , ′ ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← → Γ ( ) ( ∨ ) ⊗ Γ ′ ( ) ( ∨ ) ⊗ ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← → .
This maps endows ( ) ( ) of a structure of a graded noetherian -algebra [START_REF] Huyghe | D † -affinité de l'espace projectif[END_REF]Propositions 1.3.1,1.3.3 and 1.3.6].

We have the following important properties [START_REF] Huyghe | D-modules arithmétiques, distributions et localisation[END_REF]Proposition 4.1.15].

Proposition 2.4.2.

(i) There exists a canonical isomorphism of graded -algebras • ( ( ) ( )) ≃ ( ) ( ).

(ii) The -algebras • ( ( ) ( )) and ( ) ( ) are noetherian.

Integral models

In this section we will assume that is a smooth -scheme endowed with a right -action.

Definition 2.5.1. Let be an -algebra (resp. a sheaf of -algebras). We say that an -subalgebra 0 (resp. a subsheaf of -algebras) is an integral model of if 0 ⊗ = .

Remark 2.5.2. Let us recall that throughout this paper denotes the Lie algebra of the split connected reductive group -scheme and U ( ) its universal enveloping algebra. As we have remarked in the introduction, if denotes the -Lie algebra of the algebraic group = × Spec( ) Spec( ) and U ( ) its universal enveloping algebra, then U ( ) is an integral model of U ( ). Moreover, the algebra of distributions of level , introduced in the preceding subsection, it is also an integral model of U ( ) [22, subsection 4.1]. This latest example will be specially important in this work.

Proposition 2.5.3. The right -action induces a canonical homomorphism of filtered -algebras

Φ ( ) ∶ ( ) ( ) → 0 ( , D ( ) ).
Proof. The reader can find the proof of this proposition in [22, Proposition 4.4.1 (ii)], we will briefly discuss the construction of Φ ( ) . The central idea in the construction is that if ∶ × → denotes the -action, then the comorphism

♮ ∶ O → O ⊗ [ ] induces a morphism ( ) ∶ P ,( ) → O ⊗ ( ) ( )
for every ∈ ℕ. Those applications are compatible when varying . Let ∈ ( ) ( ) we define Φ ( ) ( ) by

Φ ( ) ( ) ∶ P ,( ) ( ) ← ← ← ← ← ← ← ← ← ← ← ← ← ← → O ⊗ ( ) ( ) ⊗ ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← → O .
Again, those applications are compatible when varying and we get the morphism of the proposition.

Remark 2.5.4.

(i) If is endowed with a left -action, then it turns out that Φ ( ) is an anti-homomorphism.

(ii) In [START_REF] Huyghe | D-modules arithmétiques, distributions et localisation[END_REF]Theorem 4.4.8.3] Huyghe and Schmidt have shown that if = and we consider the right (resp. left) regular action, then the morphism of the preceding proposition is in fact a canonical filtered isomorphism (resp. an anti-isomorphism) between ( ) ( ) and 0 ( , D ( ) ) , the -submodule of (left) -invariant global sections. This isomorphism induces a bijection between ( ) ( ) and 0 ( , D ( ) , ) , and it is compatible when varying .

We will denote by

Φ ( ) ∶ O ⊗ ( ) ( ) → D ( )
the morphism of sheaves (of -modules) defined by: if ⊂ is an open subset and ∈ O ( ), ∈ ( ) ( ), then

Φ ( ) , ( ⊗ ) ∶= ⋅ Φ ( ) ( )| .
Let us define A ( ) = O ⊗ ( ) ( ), and let us remark that we can endow this sheaf with the skew ring multiplication coming from the action of ( ) ( ) on O via the morphism Φ ( ) . This is

( ⊗ ) ⋅ ( ⊗ ) ∶= ⋅ Φ ( ) ( ) ⊗ + ⊗ . ( 7
)
This multiplication defines over A ( ) a structure of a sheaf of associative -algebras, such that it becomes an integral model of the sheaf of -algebras U • ∶= O ⊗ U ( ). To see this, let us recall how the multiplicative structure of the sheaf U • is defined (cf. [30, subsection 5.1] or [29, section 2]).

Differentiating the right action of on we get a morphism of Lie algebras

∶ → 0 ( , T ). (8)
This implies that acts on O by derivations and we can endow U • with the skew ring multiplication

( ⊗ )( ⊗ ) = ( ( )) ⊗ + ⊗ (9)
for ∈ , ∈ U ( ) and , ∈ O . With this product the sheaf U • becomes a sheaf of associative algebras [29, Section 2, page 11].

Remark 2.5.5. As in [START_REF] Bosch | Néron models[END_REF] we can define a morphism (called the operator-representation) of sheaves of -algebras

Ψ ∶ O ⊗ U ( ) → D ; ⊗  → ( ) ( ∈ O , ∈ .).
We get the following commutative diagram

( ) ( ) 0 ( , D ( ) ) U ( ) 0 ( , D ). Φ ( ) Ψ
Given that ( ) ( ) is an integral model of the universal enveloping algebra U ( ), then by [START_REF] Bosch | Néron models[END_REF] and ( 9) we can conclude that A ( ) is also a sheaf of associative -algebras being a subsheaf of U • . Proposition 2.5.6. [START_REF] Huyghe | D-modules arithmétiques, distributions et localisation[END_REF]Corollary 4.4.6] (i) The sheaf A ( ) is a locally free O -module.

(ii) There exists a unique structure over A ( ) of filtered O -ring and there is a canonical isomorphism of graded Oalgebras (A ( ) ) = O ⊗ Sym ( ) ( ).

(iii) The sheaf A ( ) (resp. (A ( ) )) is a coherent sheaf of O -rings (resp. a coherent sheaf of O -algebras), with noetherian sections over open affine subsets of .

Twisted arithmetic differential operators with congruence level

In this chapter we will introduce congruence levels to the constructions given in the preceding sections. This means, deformations of our (integral) differential operators. This notion will be a fundamental tool to define differential operators on an admissible blow-up of the flag -scheme . (i) For any ∈ there exists an isomorphism ∶ -1 ( )

Linearization of group actions

≃ ← ← ← ← ← ← ← → 1 .
(ii) For any , ∈ and for any open affine subset

= Spec( [ ]) ⊆ ∩ the automorphism ∶ • -1 | ∶ 1 → 1 of 1 is given by a linear automorphism ♮ of [ ].
This means, ♮ ( ) = for any ∈ , and

♮ ( ) =
for a suitable ∈ .

In the preceding definition, the scheme L is obtained by glueing the trivial line bundles 1, ∶ × 1 → via the linear transition functions ( ). Thus, each fibre L is a line, in the sense that it has a canonical structure of a 1-dimensional affine space. Definition 3.1.2. Given a line bundle ∶ L → and a morphism ∶ ′ → , the pull-back * (L) is the fiber product L × ′ equipped with its projection to ′ . Now, let ∶ L → be a line bundle over , then a section of over an open subset ⊂ is a morphism ∶ → L such that • = . Moreover the presheaf L defined by

⊆  → { ∶ → L | is a section over }
is a sheaf called the sheaf of sections of the line bundle L. This is an invertible sheaf (i.e., a locally free sheaf of rank 1).

On the other hand, if E is a locally free sheaf of rank 1 on and we let

V(E ) ∶= Spec Sym O (E )
be the line bundle over associated to E [16, 1.7.8], then we have a one-to-one correspondence between isomorphism classes of locally free sheaves of rank 1 on and isomorphic classes of (geometric) line bundles over [17, Chapter II, Exercises 5.1 (a) and 5.

(d)]

{Isomorphism classes of locally free sheaves of rank 1} ↔ {Isomorphic classes of line bundles}

E  → V(E ∨ ) L ↤ L (10)
Let ∶ L → be a line bundle over , let L be its sheaf of sections and ∶ ′ → a morphism of schemes; an easy calculation shows that the sheaf of sections of the pull-back line bundle

* (L) ∶= L × ′ → ′ is equal to * (L).
Let us suppose now that is endowed with a right -action ∶ × Spec( ) → . In particular, for every ∈ ( ) we dispose of a translation morphism

∶ = × Spec( ) Spec( ) × ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← → × Spec( ) ← ← ← ← ← ← →
In the next lines we will study (geometric) line bundles which are endowed with a right -action.

Definition 3.1.3. Let ∶ L → be a line bundle. A -Linearization of L is a right -action ∶ L × Spec( ) → L
satisfying the following two conditions:

(i) The diagram L × Spec( ) L × Spec( ) × is commutative.
(ii) The action on the fibers is -linear.

Let ∈ ( ) and let us suppose that Ψ ∶ * (L) → * 1 (L) is a morphism of line bundles over × Spec( ) . Let us consider the translation morphism

∶ = × Spec( ) Spec( ) × ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← → × Spec( ) ← ← ← ← ← ← → .
We

Ψ ∶ * (L) → * 1 (L) of line bundles over L × Spec( ) , such that Ψ ℎ = Ψ • * (Ψ ℎ ) for all , ℎ ∈ ( ).
Proof. By definition of linearization we have the following commutative diagram

L × Spec( ) * (L) × Spec( ) L . × 2 1
By universal property there exists a unique morphism of line bundles ∶ * 1 (L) → * (L), which is linear on the fibers since so is . Let ∈ ( ). To see that is an isomorphism we can use the correspondence [START_REF] Bruhat | Groupes réductifs sur un corps local II[END_REF]. In this case, if ∈ , ∈ ( ) and ( , ) ∶ L → L denotes the respective morphism between the stalks, then ( , ) is an isomorphism, being ( , -1 ) the inverse.

Let , ℎ ∈ ( ). Applying ( × ) * to we get the morphism ∶ L → * (L) and given that is a right action

( ℎ • = ℎ ), it fits into the following commutative diagram L * (L) * * ℎ (L) = * ℎ (L). ℎ * ( ℎ )
Moreover, since ∶ L → * (L) is an isomorphism for every ∈ ( ) (the fiber over ∈ coincides with ( , ) ) then we can consider the morphism Ψ ∶= -1 ∶ * (L) → L which coincides with the fibers of the morphism

Ψ ∶= -1 ∶ * (L) → * 1 (L).
By construction, these morphism satisfy the cocycle condition of the proposition. This means that for every , ℎ ∈ ( ), we have

Ψ ℎ = Ψ • * (Ψ ℎ ).

Associated Rees rings and arithmetic differential operators with congruence level

Throughout this sections will denote a smooth scheme over . As usual, we will denote by D ( ) the sheaf of level differential operators on . As we have remarked in the previous chapter, those sheaves come equipped with a filtration

O ⊆ D ( ) ,1 ⊆ ... ⊆ D ( ) , ⊆ ... ⊆ D ( ) , with D ( )
, the sheaf of level differential operators of order less or equal than . 8 Now, let A be a sheaf of -algebras endowed with a positive filtration ( A) ∈ℕ and such that ⊂ 0 A. The sheaf A gives rise to a subsheaf of graded rings (A) of the polynomial algebra A[ ] over A. This is defined by

(A) ∶= ⨁ ∈ℕ A ⋅ ,
its associated Rees ring. This subsheaf comes equipped with a filtration by the sheaves of subgroups

(A) ∶= ⨁ =0 A ⋅ ⊆ (A).
Specializing (A) in an element ∈ we get a sheaf of filtered subrings A of A. More exactly, A equals the image under the homomorphism of sheaves of rings ∶ (A) → A, sending  → , and it is equipped with the filtration induced by A. Moreover, if the sheaf of graded rings gr(A), associated to the filtration ( A) ∈ℕ , is flat over then9

A = ∑ =0 A. ( 11
) If ∶ A → B is a morphism of positive filtered -algebras (with ⊆ 0 A and ⊆ 0 B), then the commutative diagram (A) (B) A B  → ( )
gives us a filtered morphism of rings ∶ A → B . This in particular implies that for ∈ fixed, the preceding process is functorial.

Remark 3.2.1. The previous digression is well-known for rings. In this setting we have results completely analogues to the ones presented so far ([28, Chapter 12, section 6]

). We will use these results in the next sections. Now, let be a non-negative integer called a congruence level [START_REF] Huyghe | Arithmetic structures for differential operators on formal schemes[END_REF]Subsection 2.1]. By using the order filtration (D ( ) ̃ ) ∈ℕ of the sheaf D ( ) , we can define the sheaf of arithmetic differential operators of congruence level k, D ( , ) , as the subsheaf of D ( ) given by the specialization of (D ( ) ) in ∈ . This means

D ( , ) ∶= ∑ ∈ℕ D ( ) , .
By [START_REF] Berthelot | Notes on crystalline cohomology[END_REF] and [START_REF] Huyghe | D † -affinité de l'espace projectif[END_REF]Proposition 1.3.4.2] we can also conclude that, if (D ( , ) , ) ∈ℕ denotes the order filtration induced by D ( ) , then

D ( , ) , = ∑ =0 D ( ) , .
In local coordinates we can describe the sheaf D ( , ) in the following way. Let ⊆ be an open affine subset endowed with coordinates 1 , ..., . Let 1 , ..., be a basis of Ω ( ) and 1 , ..., the dual basis of T ( ). By using the notation in section 2.2, one has the following description [24, Subsection 2.1]

D ( , ) ( ) = <∞ ∑ | | < > | ∈ O ( ) .

Arithmetic differential operators acting on a line bundle

Throughout this subsection ∶= ∕ will always denote the flag scheme. For technical reasons (cf. Proposition 2.5.3) in this work we will always suppose that the group , and the scheme are endowed with the right regular -action. This means that for any -algebra and 0 , ∈ ( ) we have

0 • = -1 0 , 0 N( ) • = -1 0 N( ) and 0 ( ) • = -1 0 ( ).
Under this action, the canonical projection → is clearly -equivariant.

Finally, we recall for the reader that the sheaf D ( ) is endowed with a left and a right structure of O -module. These structures come from the canonical morphisms of rings 1 , 2 ∶ O → P ,( ) , which are induced by universal property and the projections. By construction, these actions also endow the sheaf D ( , ) with a left and a right structure of Omodule.

Dominant and regular characters.

Let us consider the positive system Λ + ⊂ Λ ⊂ ( ) ( ( ) = Hom( , ) the group of algebraic characters) associated to the Borel subgroup scheme ⊂ . The Weyl subgroup = ( )∕ acts naturally on the space * ∶= Hom ( , ), and via differentiation ∶ ( ) → * we may view ( ) as a subgroup of * in such a way that * ( )

⊗ = * . Let = 1 2 ∑
∈Λ + be the so-called Weyl vector. Let ̌ be a coroot of ∈ Λ viewed as an element of . An arbitrary weight ∈ * is called dominant if ( ̌ ) ≥ 0 for all ∈ Λ + . The weight is called regular if its stabilizer under the -action is trivial.

Line bundles on homogeneous spaces.

Let us suppose now that ∶= ∕ is again the smooth flag -scheme. We dispose of a canonical isomorphism ≃ ∕N which in particular implies that every algebraic character ∈ Hom( , ) induces a character of the Borel subgroup ∶ → . Let us consider the locally free action of on the trivial fiber bundle × over given by

.( , ) ∶= ( -1 , ( ) ); ( ∈ , ∈ , ∈ ).
We denote by L( ) ∶= ⟍( × ) the quotient space obtained by this action. Let ∶ → be the canonical projection. Since the map × → , ( , )  → ( ) is constant on -orbits, it induces a morphism ∶ L( ) → . Moreover, given that is locally trivial [26, Part II, 1.10 (2)] ∶ L( ) → defines a line bundle over [START_REF] Jantzen | Representation of algebraic groups[END_REF]Part I,5.16]. Furthermore, the right -action on × given by

( 0 , ) •  → ( -1 0 , ) ( ∈ , ( 0 , ) ∈ × )
induces a right action on L( ) for which L( ) turns out to be a -linearized line bundle on . By proposition 3.1.4, the sheaf of sections L( ) of the line bundle L( ) is a -equivariant invertible sheaf.

Definition 3.3.3. Let ∈ Hom( , ) be an algebraic character. For every congruence level ∈ ℕ, we define the sheaf of level arithmetic differential operators acting on the line bundle L( ) by

D ( , ) ( ) ∶= L( ) ⊗ O D ( , ) ⊗ O L( ) ∨ .
The multiplicative structure of the sheaf D ( , ) ( ) is defined as follows. If ∨ , ∨ ∈ L( ) ∨ , , ∈ D ( , ) and , ∈ L( ) then

⊗ ⊗ ∨ • ⊗ ⊗ ∨ = ⊗ ⟨ ∨ , ⟩ ⊗ ∨ . ( 12
)
Moreover, the action of D ( , ) ( ) on L( ) is given by

⊗ ⊗ ∨ • ∶= • < ∨ , > ( , ∈ L( ) and ∨ ∈ L( ) ∨ ).
Remark 3.3.4. Given that the locally free O -modules of rank one L( ) ∨ and L( ) are in particular flat, the sheaf D ( , ) ( ) is filtered by the order of twisted differential operators. This is, the subsheaf D ( , ) , of D ( , ) , of differential operators of order less that , induces a subsheaf of twisted differential operators of order less than defined by

D ( , ) , ( ) ∶= L( ) ⊗ O D ( , ) , ⊗ O L( ) ∨ ,
and given than the tensor product preserves inductive limits, we obtain

D ( , ) ( ) = lim ← ← ← ← ← ← ← ← ← ← → D ( , ) , ( ).
Moreover, the exact sequence

0 → D ( , ) , -1 → D ( , ) , → D ( , ) , ∕D ( , ) , -1 → 0 induces the exact sequence 0 → D ( , ) , -1 ( ) → D ( , ) , ( ) → L( ) ⊗ O D ( , ) , ∕D ( , ) , -1 ⊗ O L( ) ∨ → 0 which tells us that D ( , ) ( ) ≃ L( ) ⊗ O D ( , ) ⊗ O L( ) ∨ ≃ D ( , ) .
The second isomorphism is defined by ⊗ ⊗ ∨  → ∨ ( ) . This is well defined because D ( , ) is in particular a commutative ring.

Proposition 3.3.5. There exists a canonical isomorphism of graded sheaves of algebras

• D ( , ) ( ) ≃ ← ← ← ← ← ← ← → Sym ( ) ( T ).
Proof. By [START_REF] Berthelot | Notes on crystalline cohomology[END_REF], and the fact that D ( , ) and T are locally free sheaves (and therefore free -torsion) we have the following short exact sequence

0 → D ( , ) , -1 → D ( , ) , → Sym ( ) T → 0, which gives us the isomorphisms Sym ( ) T ≃ • D ( , ) ≃ • D ( , ) ( ) .
In the next proposition we will use the notation introduced in subsections 2.1 and 2.2.

< > = ∑ ′ + ′′ = ′ < ′ > ( ) < ′′ > ∈ D ( ,0) = D ( ) .
Now, let's take an affine covering S of such that for every ∈ S, is endowed with local coordinates and there exists a local section ∈ L( )( ) such that L( )| = O and L ( ) ∨ | = ∨ O , where ∨ denotes the dual element associated to . Let us show that

D ( , ) ( ) = ⨁ | | O ⊗ < > ⊗ ∨ . ( 13
)
It is enough to show that for every ∈ ℕ and , ∈ O the section ⊗ | | < > ⊗ ∨ belongs to the right side of (13). In fact, from the first part of the proof

⊗ | | < > ⊗ ∨ = ⊗ | | < > ⊗ ∨ = ∑ ′ + ′′ = | | ′ < ′ > ( ) ⊗ < ′′ > ⊗ ∨
and we get the relation [START_REF] Emerton | Locally analytic representation theory of -adic reductive groups: A summary of some recent developments[END_REF]. Let us define ∶ D ( , ) ( ) → D ( , ) by

| | ⊗ < > ⊗ ∨ = | | < > and
let us see that is a homomorphism of rings (the multiplication on the left is given by ( 12)). By [START_REF] Emerton | Locally analytic representation theory of -adic reductive groups: A summary of some recent developments[END_REF], the elements in D ( , ) ( ) are linear combinations of the terms

| | ⊗ < > ⊗ ∨
and therefore, it is enough to show that preserves the multiplicative structure over the elements of this form. So, let us take , ∈ ℕ and , ∈ O . On the one hand

( | | ⊗ < > ⊗ ∨ • | | ⊗ < > ⊗ ∨ ) = ( | | ⊗ < > | | < > ⊗ ∨ ) = ∑ ′ + ′′ = | | ′ < ′ > ( | | ) < ′′ > < > ,
and on the other hand

( | | ⊗ < > ⊗ ∨ ) • ( | | ⊗ < > ⊗ ∨ ) = | | < > • | | < > = ∑ ′ + ′′ = | | ′ < ′ > ( | | ) < ′′ > < > .
Both equations show that is a ring homomorphism. Finally, a reasoning completely analogous shows that the morphism -1 ∶ D ( , ) → D ( , ) ( ) defined by

-1 ( | | < > ) = | | ⊗ < > ⊗ ∨
it is also a homomorphism of rings and • -1 = -1 • = . This ends the proof of the lemma.

Congruence subgroups and wide open congruence subgroups

Let us denote by = ∕( ) the residue field of , and let us consider ∶= × Spec( ) Spec( ) the generic fiber of and ∶= × Spec( ) Spec( ) the special fiber. For every ∈ ℕ, there exists a smooth model ( ) of such that Lie( ( )) =

. In fact, we take (0) ∶= and we construct (1) as the dilatation of the trivial subgroup of in [7, Chapter 3, Section 3.2]. This is a flat -scheme which is an integral model of [START_REF] Waterhouse | One-dimensional affine group schemes[END_REF]Proposition 1.1]. In general, we let ( + 1) be the dilatation of the trivial subgroup of ( ) in ( ), in such a way that for every ∈ ℕ we dispose of a canonical morphism ( + 1) → ( ).

We end this briefly discussion about the congruence subgroups with the following description of the distribution algebra ( ) ( ( )). 10 . Let us take a triangular decomposition = ⊕ ⊕ and let us consider basis ( ), (ℎ ) and ( ) of the -Lie algebras , and , respectively. Then ( ) ( ) equals the -subalgebra of U ( ) ⊗ generated as an -module by the elements

! | | ! ′ ! | ′ | ℎ ′ ′′ ! | ′′ | ′′ ′′ ! . ( 14
)
An element of the preceding form has order

= | | + | ′ | + | ′′
| and the -spam of elements of order less or equal than forms an -submodule ( ) ( ( )) ⊂ ( ) ( ( )). In this way ( ) ( ( )) becomes a filtered -algebra, such that by [START_REF] Emerton | Locally Analytic Vectors in Representations of Locally -adic Analytic Groups[END_REF] and the well known Poincaré-Birkhoff-Witt theorem we have ( ) ( ( )) ⊗ = U ( ) ⊗ .

The preceding discussion also tells us that

( ) ( (0)) = ( ) ( ( )).
Finally, let us introduce a family of certain rigid-analytic "wide-open" groups ( ) • , which will be important in our work.

To do this, let us first consider the formal completion ( ) of the group scheme ( ) along its special fiber, which is a formal scheme of topological finite type over Spf( ). Now, we consider ̂ ( ) • be the completion of ( ) • along its unit section Spf( ) → ( ), and we denote by ( ) • its associated rigid-analytic space [4, (0.2.6)], which is a rigid-analytic group.

We recall for the reader that in subsection 2.5 we have introduced the sheaves A ( , ) ∶= O ⊗ ( ) ( ( )), which carries a structure of filtered O -ring, such that (A ( , ) ) = O ⊗ Sym ( ) ( ).

Proposition 3.3.8.

There exists a canonical surjective homomorphism of sheaves of filtered -algebras

Φ ( , ) ∶ A ( , ) → D ( , ) ( ).
Proof. Let us star by showing the existence of such a morphism. By [22, Corollary 4.5.2], there exists a morphism of sheaves of filtered -algebras

A ( ,0) → D ( ,0) ( ). (15)
Let's first show that after specialising in the Rees ring associated to the twisted order filtration of D ( ,0) 0 , we get D ( , ) ( ). To do that, we consider D ( ,0) filtered by the order of differential operators and we define the following homomorphisms of O -modules

L( ) ⊗ O D ( ,0) ⊗ O L( ) ∨ D ( ,0) ( ) , -1 (16)
by

⊗ ∑ ⊗ ∨ = ∑ ( ⊗ ⊗ ∨ ) and -1 ∑ ( ⊗ ⊗ ∨ ) = ∑ ⊗ ⊗ ∨
with ord( ) = for every in the sum (resp. ord( ) = for every ). It's clear that • -1 = -1 • = and therefore ( 16) is an isomorphism of O -modules. We remark that an easy calculation shows that ( 16) is in fact an isomorphism of rings.

Let's denote by 1 ∶ D ( ,0) ( ) → D ( , ) ( );  → and by 2 ∶ D ( ,0) → D ( , ) ;  → , and let's consider the following diagrams

L( ) ⊗ O D ( ,0) ⊗ O L( ) ∨ D ( ,0) ( ) D ( , ) ( ) L( ) ⊗ 2 ⊗ L( ) ∨ -1 1
It is straightforward to check that both diagrams are commutative and we can conclude that

D ( ,0) 0 ( ) = Im( 1 ) = Im( L( ) ⊗ 2 ⊗ L( ) ∨ ) = L( ) ⊗ O Im( 2 ) ⊗ O L( ) ∨ = D ( , ) ( ).
On the other hand, taking the natural filtration of A ( ,0) we have

A ( ,0) = O ⊗ ( ) ( (0))
and therefore (A ( ,0) ) = A ( , ) . The above two calculations tell us that passing to the Rees rings and specialising in the map [START_REF] Grothendieck | Éléments de géométrie algébrique: I. Le langage des schémas[END_REF], we get the desired homomorphism of filtered sheaves of -algebras

Φ ( , ) ∶ A ( , ) → D ( , ) ( ). (17)
Let us finally show that this morphism is surjective. To do that, let us recall that the right -action on induces a canonical application

O ⊗ → T (18)
which is surjective by [

Finiteness properties

Throeught this section ∈ ( ) will denote an algebraic character. By abuse of notation, we will denote again by the character ∈ Hom -mod ( , ) induced via differentiation. In this subsection we will show one important property about the -torsion of the cohomology groups of coherent D ( , ) , -modules, when the character + ∈ Hom -mod ( , ) is dominant and regular. We will follow the arguments of [START_REF] Huyghe | Un théorème de Beilinson-Bernstein pour les D-modules arithmétiques[END_REF]. (i) 0 ( , A ( , ) ) = ( ) ( ( )) is a noetherian -algebra.

Let be a projective

(ii) There exists a surjection on A ( , ) -modules A ( , ) (-)

⊕ → E → 0 for suitable ∈ ℤ and ∈ ℕ.

(iii) For any ≥ 0 the group ( , E ) is a finitely generated ( ) ( ( ))-module.

Inspired in proposition 3.3.8, in a first time we will be concentrated in coherent A ( , ) -modules. The next two results will play an important role when we consider formal completions.

Lemma 3.4.2. For every coherent A ( , ) -module E , there exists = (E ) ∈ ℤ such that ( , E ( )) = 0 for every ≥ .

Proof. Let us fix 0 ∈ ℤ such that ( , O( )) = 0 for every > 0 and ≥ 0 . We have

( , A ( , ) ( )) = ( , O( )) ⊗ ( ) ( ( )) = 0.
The rest of the proof follows the same inductive argument given in [21, Proposition 2.2.1].

Let us suppose now that ∶= ∕ is the smooth flag -scheme of . From proposition 3.3.8 and lemma 3.4.2 we have the following result.

Lemma 3.4.3. For every coherent D ( , ) , -module E , there exist = (E ) ∈ ℤ, a natural number ∈ ℕ and an epimorphism of D ( , ) , -modules (ii) For every coherent D ( , ) , -module E , the cohomology group ( , E ) has bounded -torsion for all > 0.

D ( , ) , (-) ⊕ → E → 0.
Proof. To show ( ) we remark for the reader that by construction D ( , ) , ,ℚ = D is the usual sheaf of twisted differential operators on the flag variety [26, Part I, 5.17]. As D ( , ) , ,ℚ ( ) is a coherent D -module, the classical Beilinson-Bernstein theorem [START_REF] Beilinson | Localisation de -modules[END_REF] allows us to conclude that ( , D ( , ) , ( )) ⊗ = 0 for every positive integer ∈ ℤ >0 . This in particular implies that the sheaf D ( , ) , ( ) has -torsion cohomology groups ( , D ( , ) , ( )), for every > 0 and ∈ ℤ. Now, by proposition 3.3.8, we know that D ( , ) , ( ) is in particular a coherent A ( , ) -module and hence, by the third part of proposition 3.4.1 we get that for every ≥ 0 the cohomology groups ( , D ( ) , ( )) are finitely generated ( ) ( ( ))modules. Consequently, of finite -torsion for every integer 0 < ≤ dim( ) and ∈ ℤ.

Passing to formal completions

We recall for the reader that throughout this work

∶= lim ← ← ← ← ← ← ← ← ← ← → ∈ℕ , ∶= × Spec( ) Spec( ∕ +1 )
denote the formal completion of along its special fiber.

Definition 3.5.1. We will denote by

D( , ) , ∶= lim ← ← ← ← ← ← ← ← ← ← ← ∈ℕ D ( , ) , ∕ +1 D ( , )
, the -adic completion of D ( , ) , and we consider it as a sheaf on . Following the notation given at the beginning of this work, the sheaf D( , )

, ,ℚ will denote our sheaf of level twisted differential operators with congruence level on the formal flag scheme .

Proposition 3.5.2.

(i) There exists a basis B of the topology of , consisting of open affine subset, such that for every ∈ B the ring D( , )

, ( ) is two-sided noetherian.

(ii) The sheaf of rings D( , ) , ,ℚ is coherent. Proof. To show (i) we can take an open affine subset ∈ S and to consider its formal completion along the special fiber. We have

0 ( , D( , ) , ) ≃ ̂ 0 ( , D ( , ) , ) ≃ ̂ 0 ( , D ( , ) ) ≃ 0 ( , D( , ) )
The first and third isomorphisms are given by [15, (0 , 3.2.6)] and the second one arises from proposition 3. The objective of this subsection is to prove an analogue of proposition 3.4.4 for coherent D( , ) , ,ℚ -modules and to conclude that 0 ( , •) is an exact functor over the category of coherent D( , )

, ,ℚ -modules.

Proposition 3.5.3. Let E be a coherent D ( , ) , -module and Ê its -adic completion, which we consider as a sheaf on .

(i) For all ≤ 0 one has

( , Ê ) = lim ← ← ← ← ← ← ← ← ← ← ← ( , E ∕ E ).
(ii) For all > 0 one has ( , Ê ) = ( , E ).

(iii) The global section functor 0 ( ,

•) satisfies 0 ( , Ê ) = lim ← ← ← ← ← ← ← ← ← ← ← 0 ( , E )∕ 0 ( , E ).
Proof. Let E denote the torsion subpresheaf of E . As is a noetherian space and D ( , ) , has noetherian rings sections over open affine subsets of (proposition 3.3.9), we can conclude that E is in fact a coherent D ( , ) , -module. This is generated by a coherent O -module which is annihilated by a power of , and so is E . The quotient G ∶= E ∕E is again a coherent D ( , ) , -module and therefore we can assume, after possibly replacing by a larger number, that E = 0 and

( , E ) = ( , G) = 0 , for all > 0. From here on the proof of the proposition follows the same lines of reasoning that in [20, and its -adic completion ̂ ( ) ( ( )) . It is clear that ( ) ( ( )) is an integral model of U ( ) . We will denote by † ( ( )) the limit of the inductive system ̂ ( ) ( ( )) ⊗ → ̂ ( +1) ( ( )) ⊗ .

Theorem 3.6.1. The homomorphism of -algebras Φ ( , ) ∶ ( ) ( ( )) → 0 ( , D ( , ) , ), defined by taking global sections in (3.3.8), induces an isomorphism of -algebras

̂ ( ) ( ( )) ⊗ ≃ ← ← ← ← ← ← ← → 0 , D( , ) , ,ℚ .
Proof. The key in the proof of the theorem is the following commutative diagram

( ) ( ( )) 0 ( , D ( , ) , ) U ( ) 0 ( , D ). Φ ( , ) Φ
Here Φ is the morphism in [19, (11.2.2)] 13 . By the classical Beilinson-Bernsein theorem [START_REF] Beilinson | Localisation de -modules[END_REF] and the preceding commutative diagram, we have that Φ ( , ) factors through the morphism Φ ( , ) ∶ ( ) ( ( )) → 0 ( , D ( , ) , ) which becomes an isomorphism after tensoring with . By [START_REF] Huyghe | D-modules arithmétiques sur la variété de drapeaux[END_REF]Lemma 3.3] we have that Φ ( ) gives rise to an isomorphism

̂ ( ) ( ( )) ⊗ ≃ ← ← ← ← ← ← ← → ̂ 0 ( , D ( , )
, ) ⊗ , and proposition 3.5.3 together with the fact that is in particular a noetherian topological space end the proof of the theorem.

The localization functor

In this section we will introduce the localization functor. Let be a finitely generated ̂ ( ) ( ( )) ⊗ -module. We define L ( , ) , ( ) as the associated sheaf to the presheaf on defined by

 → D( , ) , ,ℚ ( ) ⊗ ̂ ( ) ( ( )) ⊗ . It is clear that L ( , )
, is a functor from the category of finitely generated ̂ ( ) ( ( )) ⊗ -modules to the category of coherent D( , ) , ,ℚ -modules.

The arithmetic Beilinson-Bernstein theorem

We are finally ready to prove one of the principal results of this work. To start with, we will enunciate the following proposition whose proof can be founded in [

The sheaves D † , ( )

In this section we will study the problem of passing to the inductive limit when varies, this means

D † , ( ) ∶= lim ← ← ← ← ← ← ← ← ← ← → ∈ℕ D( , ) , ⊗ and † ( ( )) ∶= lim ← ← ← ← ← ← ← ← ← ← → ∈ℕ ̂ ( ) ( ( )) ⊗ . ( 19
)
As in 3.6.2 let us consider the following localization functor L † , ( ) from the category of finitely presented † ( ( ))modules to the category of coherent D † , ( ). Let be a finitely presented † ( ( )) -module, then L † , ( )( ) denotes the associated sheaf to the presheaf on defined by

⊆  → D † , ( ) ⊗ † ( ( )) .
As before, it is clear that L † , ( ) is a functor from the category of finitely presented † ( ( )) -modules to the category of coherent D † , ( )-modules.

Analytic distribution algebra

The wide-open rigid analytic groups, defined in 3.3.7, play an important role in the work developed by Emerton in [START_REF] Emerton | Locally analytic representation theory of -adic reductive groups: A summary of some recent developments[END_REF], to treat locally analytic representations of -adic groups. The analytic distribution of ( ) • is defined to be the continuous dual space of the space of rigid-analytic functions on ( ) • . This is,

D an ( ( ) • ) ∶= O ( ) • ( ( ) • ) ′ = Hom cont O ( ) • ( ( ) • ), ,
this is a topological -algebra of compact type. In [22, Proposition 5.2.1] Huyghe-Schmidt have shown that † ( ( )) ≃ D an ( ( ) • ).

As is a noetherian space, theorem 3.6.1 and the preceding relation tell us that

0 ( , D † , ( )) = † ( ( )) = D an ( ( ) • ) ∶= D an ( ( ) • ) D an ( ( ) • )(Ker( + )). (20)
We will concentrate our efforts to prove the following Beilinson-Bernstein theorem for the sheaves D † , ( ). Proof. The previous proposition tells us that we only need to show that the transition morphisms D( , ) , ,ℚ → D( +1, ) , ,ℚ are flat. As this is a local property we can take ∈ S (covering in proposition 3.3.6) and to verify this property over the formal completion . In this case, the argument used in the proof of the first part of proposition 3.5.2 give us the following commutative diagram

D( , )

, ,ℚ ( ) D( +1, ) , ,ℚ ( )

D( , ) ,ℚ ( ) D( +1, ) ,ℚ ( ) ≃ ≃
The flatness theorem [24, Proposition 2.2.11 (iii)] states that the lower morphism is flat and so is the morphism on the top.

Lemma 3.7.5. For every coherent D † , ( )-module E there exists ≥ 0, a coherent D( , ) , ,ℚ -module E and an isomorphism of D † , ( )-modules

∶ D † , ( ) ⊗ D( , ) , ,ℚ E ≃ ← ← ← ← ← ← ← → E . Moreover, if ( ′ , E ′ , ′
) is another such triple, then there exists ≥ max{ , ′ } and an isomorphism of D( , ) , ,ℚ -modules

∶ D( , ) , ,ℚ ⊗ D( , ) , ,ℚ E ≃ ← ← ← ← ← ← ← → D( , ) , ,ℚ ⊗ D( ′ , ) , ,ℚ E ′ such that ′ • D † , ( ) ⊗ = .
Proof. This is [3, proposition 3.6.2 (ii)]. We remark that is quasi-compact and separated, and the sheaf D( , ) , ,ℚ satisfies the conditions in [3, 3.4.1]. Proposition 3.7.6. Let E be a coherent D † , ( )-module.

(i) There exists an integer (E ) such that, for all ≥ (E ) there is ∈ ℕ and an epimorphism of D † , ( )-modules

D † , ( )(-) ⊕ → E → 0.
(ii) For all > 0 one has ( , E ) = 0.

Proof. 14 Let E be a coherent D † , ( )-coherent module. The preceding proposition tells us that there exists ∈ ℕ, a coherent D( , )

, ,ℚ -module E and an isomorphism of D † , ( )-modules

∶ D † , ( ) ⊗ D( , )
, ,

E ≃ ← ← ← ← ← ← ← → E .
Now we use proposition 3.5.7 for E and we get the desired surjection in (i) after tensoring with D † , ( ). To show (ii) we may use the fact that, as is a noetherian topological space, cohomology commutes with direct limites. Therefore, given that D( , )

, ,ℚ ⊗ D( , )

, ,ℚ E is a coherent D ( , ) , ,ℚ -module for every ≥ , we have for every > 0

( , E ) = lim ← ← ← ← ← ← ← ← ← ← → ≥ , D( , ) , ,ℚ ⊗ D( , ) , ,ℚ E = 0.
Proposition 3.7.7. Let E be a coherent D † , ( )-module. Then E is generated by its global sections as D † , ( )-module. Moreover, E has a resolution by finite free D † , ( )-modules and 0 ( , E ) is a † ( ( )) ⊗ -module of finite presentation.

Proof. 15 Theorem 3.7.5 gives us a coherent D( , )

, ,ℚ -module

E such that E ≃ D † , ( ) ⊗ D( , )
, ,ℚ E . Moreover, E has a resolution by finite free D( , ) , ,ℚ -modules ( proposition 3.6.2). Both results clearly imply the first and the second part of the lemma. The final part of the lemma is therefore a consequence of the first part and the acyclicity of the the functor 0 ( , •).

Proof of theorem 3.7.2. All in all, we can follow the same arguments of [21, corollary 2.3.7]. We start by taking

( † ( ( )) ⊗ ) ⊕ → ( † ( ( )) ⊗ ) ⊕ → → 0
a finitely presented † ( ( )) ⊗ -module. By localizing and applying the global sections functor, we obtain a commutative diagram

( † ( ( )) ⊗ ) ⊕ ( † ( ( )) ⊗ ) ⊕ 0 ( † ( ( )) ⊗ ) ⊕ ( † ( ( )) ⊗ ) ⊕ 0 ( , ℒ † , ( )( )) 0.
which tells us that → 0 ( , L † , ( )( )) is an isomorphism. To show that if E is coherent D † , ( )-module then the canonical morphism D † , ( ) ⊗ † ( ( )) ⊗ 0 ( , E ) → E is an isomorphism the reader can follow the same argument as before. As we have remarked, the second assertion follows because any equivalence between abelian categories is exact.

Twisted differential operators on formal models of flag varieties

Through out this section = ∕ will denote the smooth flag -scheme and ∈ ( ) = Hom( , ) will always denote an algebraic character. As before, we will denote be L( ) the (algebraic) line bundle on induced by (subsection 3.1).

In this section we will generalize the construction given in [START_REF] Huyghe | D † -affinity of formal models of flag varieties[END_REF] by introducing sheaves of twisted differential operators on an admissible blow-up of the smooth formal flag -scheme . The reader will figure out that some reasoning are inspired in the results of Huyghe-Patel-Strauch-Schmidt in [START_REF] Huyghe | D † -affinity of formal models of flag varieties[END_REF].

Differential operators on admissible blow-ups

We start with the following definition. |module. The idea is to find those congruence levels ∈ ℕ such that the preceding structure extends to a module structure on O over pr -1 D ( , ) . Let us denote by

D ( , ) ∶= pr * D ( , ) = O ⊗ pr -1 O pr -1 D ( , ) . ( 21
)
The problem to find those congruence levels was studied in [START_REF] Huyghe | D † -affinity of formal models of flag varieties[END_REF] and [START_REF] Huyghe | Arithmetic structures for differential operators on formal schemes[END_REF]. In fact, we have the following condition [

( 1 ⊗ 1 ) • ( 2 ⊗ 2 ) = 1 1 ( 2 ) ⊗ 2 + 1 2 ⊗ 1 2 .
We have all the ingredients that allow us to construct the desired sheaves over , this is, to extend the sheaves of rings defined in the preceding chapter to an admissible blow-up of . Let ≥ fix. Let us first recall that taking arbitrary sections , ∈ D ( , ) , , ∈ L( ) and ∨ , ∨ ∈ L( ) ∨ (the last two not necessarily the duals of and ) over an arbitrary open subset ⊂ , the multiplicative structure of the sheaf D ( , ) , is defined by (cf. [START_REF] Dolgachev | Lectures on invariant theory[END_REF])

⊗ ⊗ ∨ • ⊗ ⊗ ∨ = ⊗ ⟨ ∨ , ⟩ ⊗ ∨ .
Now, if pr∶ → denotes the projection, we put

D ( , ) ( ) ∶= pr * D ( , ) , = pr * L( ) ⊗ O pr * D ( , ) ⊗ O pr * L( ) ∨ .
Proposition 4.1.2 allows us to endow the sheaf of O -modules D ( , ) ( ) with a multiplicative structure for every ≥ .

On local sections we have

⊗ ⊗ ∨ • ⊗ ⊗ ∨ = ⊗ ⟨ ∨ , ⟩ ⊗ ∨ ,
where , ∈ pr * L( ), ∨ , ∨ ∈ pr * L( ) ∨ and , ∈ D ( , ) , are local sections.

Let be the completion of along its special fiber = × Spec( ) Spec( ∕ ).

In this work we will only consider formal blow-ups arising from the formal completion along the special fiber of an admissible blow-up → [25, Proposition 2.2.9]

. Under this assumption we will identify = .

Definition 4.1.4. Let pr ∶ → be an admissible blow-up of the flag variety and let ≥ . The sheaves

D( , ) ,ℚ ( ) ∶= lim ← ← ← ← ← ← ← ← ← ← ← ∈ℕ D ( , ) ( )∕ +1 D ( , ) ( ) ⊗ and D † , ( ) ∶= lim ← ← ← ← ← ← ← ← ← ← → ∈ℕ D( , ) ,ℚ ( ).
are called sheaves of -twisted arithmetic differential operators on .

Proposition 4.1.5.

(i) The sheaves D ( , ) ( ) are filtered by the order of twisted differential operators and there is a canonical isomorphism of graded sheaves of algebras gr D ( , ) ( ) ≃ Sym ( ) pr * T , where ≥ .

(ii) There is a basis for the topology of , consisting of affine open subsets, such that for any open subset ∈ in this basis, the ring D ( , ) ( ) ( ) is noetherian. In particular, the sheaf of rings D ( , ) ( ) is coherent.

(iii) The sheaf D( , ) ,ℚ ( ) is coherent.

Proof. By 5, we have an exact sequence of O -modules

0 → D ( , ) , -1 → D ( , ) , → Sym ( ) T → 0.
Taking the tensor product with L( ) and L( ) ∨ on the left and on the right, respectively, and applying pr * we obtain the exact sequence (since Sym ( ) ( T ) is a locally free O -module of finite rank)

0 → D ( , ) , -1 ( ) → D ( , ) , ( ) → pr * L( ) ⊗ O Sym ( ) pr * T ⊗ O pr * L( ) ∨ → 0, which implies (i) because pr * L( ) ⊗ O Sym ( ) pr * T ⊗ O pr * L( ) ∨ ≃ Sym ( ) pr * T
by commutativity of the symmetric algebra.

Let ⊆ be an affine open subset endowed with local coordinates 1 , ..., and such that L( )| = O for some ∈ L( )( ). Then, by lemma 3.3.6 we have the following local description for D ( , ) ( ) on = pr -1 ( )

D ( , ) ( )( ) = <∞ ∑ | | ⟨ ⟩ | = ( 1 , ..., ) ∈ ℕ ∈ O ( ) .
By (i), the graded algebra • D ( , ) ( )( ) is isomorphic to Sym ( ) pr * T ( ) which is known to be noetherian [START_REF] Huyghe | D † -affinité de l'espace projectif[END_REF]Proposition 1.3.6]. Therefore, taking as a basis the set of affine open subsets of that are contained in some pr -1 ( ) we get (ii). We also remark that, as Let us briefly study the problem of passing to the inductive limit when varies.

D ( , ) ( ) is O -quasi-coherent,
Let ⊂ such that D ( , ) ( )| ≃ D ( , )
| and let us take ⊆ an affine open subset such that ⊆ pr -1 ( ). We have the commutative diagram

, which implies that D ( , ) ( )| ≃ D ( , )
| , as sheaves of rings. In particular, if denotes the formal -adic completion of along the special fiber we have the commutative diagram (cf. proposition 3.7.4) As we will explain later, there exists a canonical epimorphism of sheaves of filtered -algebras16

D( , ) ,ℚ ( )( ) D( +1, ) ,ℚ ( )( ) D( , ) ,ℚ ( ) D( +1, ) ,ℚ ( ).
A ( , ) ∶= O ⊗ ( ) ( ( )) → D ( , ) ( )
which allows to conclude the following proposition exactly as we have done in the proof of proposition 3.6.2 (cf. [25, Proposition

Proposition 4.1.7. Let ∈ Hom( , ) be an algebraic character such that + ∈ * is a dominant and regular character of .

(i) Let E be a coherent D( , ) ,ℚ ( )-module. Then E is generated by its global sections as D( , ) ,ℚ ( )-module. Furthermore, E has a resolution by finite free D( , )

,ℚ ( )-modules.

(ii) Let E be a coherent D † , ( )-module. Then E is generated by its global sections as D † , ( )-module. Furthermore, E has a resolution by finite free D † , ( )-modules.

An Invariance theorem for admissible blow-ups

Let ∶ → be an admissible blow-up along a closed subset V(I ) defined by an open ideal sheaf I ⊆ O . Using (4.1.3), we can suppose that is obtained as the formal completion of an admissible blow-up → (we will abuse of the notation and we will denote again by pr ∶ → the canonical morphism of this (algebraic) blow-up) along a closed subset V(I) defined by an open ideal sheaf I ⊆ O , such that I is the restriction of the formal -adic completion of I. Let us denote by ∶= × Spec( ) Spec( ∕ +1 ) the redaction module +1 and by ∶ → the canonical closed embedding. In [START_REF] Huyghe | Arithmetic structures for differential operators on formal schemes[END_REF] the authors have studied the cohomological properties of the sheaves

D( , ) ,ℚ ∶= lim ← ← ← ← ← ← ← ← ← ← ← ∈ℕ * D ( , ) ⊗ and D † , ∶= lim ← ← ← ← ← ← ← ← ← ← → ∈ℕ D( , ) ,ℚ .
Let us consider the commutative diagram .

Here pr ∶ → denotes the redaction of the morphism module +1 . We put L ( ) ∨ ∶= lim

← ← ← ← ← ← ← ← ← ← ← * pr * L( ) ∨ and L ( ) ∶= lim ← ← ← ← ← ← ← ← ← ← ← * pr * L( ). By using the preceding commutative diagram we have * D ( , ) ( ) = * pr * L( ) ⊗ O D ( , ) ⊗ O pr * L( ) ∨ = * pr * L( ) ⊗ O * D ( , ) ⊗ O * pr * L( ) ∨ .
Taking the projective limit we get the following description of the sheaves D( , ) ,ℚ ( )

D( , ) ,ℚ ( ) = L ( ) ℚ ⊗ O ,ℚ D( , ) ,ℚ ⊗ O ,ℚ L ( ) ∨ ℚ ,
and by taking the inductive limit we get the characterization

D † , ( ) = L ( ) ℚ ⊗ O ,ℚ D † , ⊗ O ,ℚ L ( ) ∨ ℚ . ( 23
)
As in the preceding section, the sheaf L ( )

ℚ
is endowed with the following (left) D † , ( )-action

⊗ ⊗ ∨ • ∶= • < ∨ , > ( , ∈ L ( ) and ∨ ∈ L ( ) ∨ ).
We end this first discussion by remarking that the relation pr * • * = * • pr * , which comes from the preceding commutative diagram, implies that

D † , ( ) = pr * D † , ( ). (24)
Let us suppose that ∶ ′ → is a morphism of admissible blow-ups (abusing of the notation, we will also denote by ∶ ′ → the respective morphism of formal admissible blow-ups in the sense of [6, Part II, chapter 8, section 8.2, definition 3]). This means that we have a commutative diagram

′ . ′ resp. ′ . ′ Let ≥ { ′ , }. Let us denote by D ( , )
, ( ) ∶= D ( , ) ( )∕ +1 D ( , ) ( ) (we will use the same notations over ′ and ) and by ∶ ′ → the redaction module +1 . The preceding commutative diagram implies that

D ( , ) ′ , ( ) = ( ′ ) * D ( , ) ( ) = * D ( , ) ( ). (25)
In this way, she sheaf D ( , ) ′ , ( ) can be endowed with a structure of right -1 D ( , ) ( )-module. Passing to the projective limit, the sheaf D( , ) ′ ( ) is a sheaf of right -1 D( , ) ( )-modules. So, passing to the inductive limit over we can

conclude that D † ′ , ( ) is a right -1 D † , ( )-module. For a D † , ( )-module E , we define ! E ∶= D † ′ , ( ) ⊗ -1 D † , ( ) -1 E ,
with analogous definitions for D( , ) ,ℚ ( ).

Theorem 4.2.1. Let ∶ ′ → be a morphism over of admissible blow-ups. Let ≥ max{ ′ , }.

(i) If E is a coherent D † ′ , ( ), then R * E = 0 for every > 0. Moreover, * D † ′ , ( ) = D † , (
), so * induces an exact functor between coherent modules over D † ′ , ( ) and D † , ( ), respectively.

(ii) The formation ! is an exact functor from the category of coherent D † , ( )-modules to the category of coherent D † ′ , ( )-modules.

(iii) The functors * and ! are quasi-inverse equivalences between the categories of coherent D † ′ , ( )-modules and coherent D † , ( )-modules.

We remark for the reader that this theorem has an equivalent version for the sheaves D( , ) ,ℚ ( ) and D( , ) ′ ,ℚ ( ).

Proof. Let us first assume that E = D † ′ , ( ). Let us consider the covering B of , defined in proposition 3.5.2 and let us take U ∈ B. We put V ′ ∶= pr ′-1 (U ) and V ∶= pr -1 (U ). By assumption V ′ = -1 (V) in such a way that

| V ′ .
No we can use [START_REF] Huyghe | Arithmetic structures for differential operators on formal schemes[END_REF]Theorem 2.3.8 (i)] to conclude that * D † ′ , ( ) = 0 for every > 0. Furthermore, by ( 25) there exists a canonical map

D † , ( ) → * D † ′ , ( )
which is in fact an isomorphism by the preceding reasoning and [24, Theorem 2.3.8 (i)].

To handle with the second part let us consider the following assertion for every ≥ 1. Let : for any coherent D † ′ , ( )module E and for all ≥ , * E = 0. The assertion is true for = dim( ) + 1. Let us suppose that +1 is true and let us take a coherent D † ′ , ( )-module E . By proposition 4.1.7 there exists ∈ ℕ and a short exact sequence of coherent

D † ′ , ( )-modules 0 → F → D † ′ , ( ) ⊕ → E → 0.
Since * D † ′ , ( ) = 0, the long exact sequence for * gives us * E ≃ +1 * F , which is 0 by induction hypothesis. This ends the proof of (i).

Let us show (ii) for the sheaves D † , ( ). The case for the sheaves D( , ) ,ℚ ( ) being equal. Given that ! D † , ( ) = D † ′ , ( ), and since the tensor product is right exact, we can conclude that ! preserves coherence.

Now, we have a morphism

-1 E → ! E sending  → 1 ⊗ . This maps induces the morphism E → * ! E . To
show that this is an isomorphism is a local question on . If V ⊆ is the formal completion of an affine open subset ⊆ pr -1 ( ), and ⊆ is an affine open subset such that D ( , ) ( )| ≃ D ( , ) | (lemma 3.3.6), then by ( 22) and [24, Corollary 2.2.15] we can conclude that the previous map is in fact an isomorphism over V17 . Finally, if F is a coherent D † ′ , ( )-module, then we have the map ! * F → F , sending ⊗  → . To see that this is an isomorphism we can use the preceding reasoning.

Let us recall that if ∈ Hom( , ) is an algebraic character such that + ∈ * is a dominant and regular character of , then by [START_REF] Huyghe | D † -affinité de l'espace projectif[END_REF] we have

0 , D † , ( ) = † ( ( )) = D an ( ( ) • ) ∶= D an ( ( ) • ) D an ( ( ) • )(Ker).
The previous theorem implies Corollary 4.2.2. Let ∈ Hom( , ) be an algebraic character such that + ∈ * is a dominant and regular character of . In the situation of the preceding theorem we have (ii) The functor 0 ( , •) is an equivalence between the category of coherent D † , ( )-modules and the category of finitely presented † ( ( )) -modules.

0 , D † , ( ) = 0 , D † , ( ) = † ( ( )) = 0 ′ , D † ′ , ( ) .
The same statement holds for coherent modules over D( , )

,ℚ ( ).

Proof. The first part of the theorem follows from the fact that 0 ( , •) = 0 ( , •) • * . Now we only have to apply the preceding theorem and theorem 3.7.2.

Let us denote by L † , ( ) the exact functor defined by the composition Finitely presented † ( ( )) -modules

L † , ( ) ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← → Coherent D † , ( ) -modules ! ← ← ← ← ← ← ← ← ← → CoherentD † , ( ) -modules.
Let us compute this functor. To do that, we may fix a finitely presented † ( ( )) -module . Then

! L † , ( )( ) = D † , ( ) ⊗ -1 D † , ( ) -1 D † , ( ) ⊗ † ( ( )) = L † ( )( ).
Now, to show that

0 , ! L † , ( )( ) = 0 , D † , ( ) ⊗ † ( ( )) = ,
we can take a resolution † ( ( ))

⊕ → † ( ( )) ⊕ → → 0, to get the following diagram † ( ( )) ⊕ † ( ( )) ⊕ 0 † ( ( )) ⊕ † ( ( )) ⊕ 0 , D † , ( ) ⊗ † ( ( )) 0.
where the sequence on the top is clearly exact. By definition L † , ( )(•) is an exact functor and by ( ) the global section functor 0 ( , •) is also exact. This shows that the sequence at the bottom is also exact and we end the proof of the theorem.

In the sequel we will denote by 0 the compact locally -analytic group 0 ∶= ( ).

Group action on blow-ups

Let be the formal completion of the group -scheme , along its special fiber ∶= × Spec( ) Spec( ∕ ). Let us denote by ∶ × Spf( ) → the induced right -action on the formal flag -scheme (cf. subsection 3.3). For every ∈ ( ) = 0 we have an automorphism of given by

∶ = × Spf( ) Spf( ) × ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← → × Spf( ) ← ← ← ← ← ← → .
As acts on the right, we have the following relation

* ♮ ℎ • ♮ = ♮ ℎ ( , ℎ ∈ 0 ). (26)
Here We will need the following result in the next sections. The reader can find its proof in [START_REF] Huyghe | D † -affinity of formal models of flag varieties[END_REF]Lemma 5.2.3]. Lemma 4.3.2. Let ∶ → be an admissible blow-up, and let us assume that ≥ = . Then is = ( )( )equivariant and the induced action of every ∈ +1 on the special fiber of is the identity. Therefore, +1 acts trivially on the underlying topological space of . By proposition 3.1.4 (cf. [22, 3.3.2]), for every ∈ ( ) = ( ) = 0 there exists an isomorphism

∶ × ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← → × Spec( ) ← ← ← ← ← ← → ,
which induces an O -linear isomorphism Φ ∶ L ( ) → ( ) * (L ( )) verifying the cocycle condition

Φ ℎ = ( ) * (Φ ℎ ) • Φ and ( , ℎ ∈ ( )). (28)
In particular, we have an induced 0 -action on the sheaf D † , ( )

∶ D † , ( ) → ( ) * D † , ( ),  → Φ • • (Φ ) -1 . ( 29
)
Locally, if U ⊆ and ∈ D † , ( )(U ) then the cocycle condition [START_REF] Mcconnell | Noncommutative notherian rings[END_REF] tells that the diagram

L ( )(U .(ℎ ) -1 ) = L ( )(U . -1 ℎ -1 ) L ( )(U . -1 ℎ -1 ) L ( )(U . -1 ) L (U . -1 ) L ( )(U ) L ( )(U ) ℎ, (U ( ) Φ -1 ℎ, U . -1 = ( ) * Φ -1 ℎ, U Φ -1 , U Φ ℎ, U . -1 = ( ) * Φ ℎ, U Φ , U (30)
is commutative and we get the relation

ℎ = * ℎ • ( , ℎ ∈ 0 ). (31)
Let us suppose that ⊆ 0 is an open subgroup and that pr ∶ → is an -equivariant admissible blow-up. Pulling back the isomorphism ( ) * L ( ) → L ( ), via (pr) * , and using the commutative diagram [START_REF] Liu | Algebraic geometry and arithmetic curves[END_REF] we get pr * ( ) * L ( ) = ( ) * pr * L ( ) = ( ) * L ( ) (notation given at the beginning of the preceding subsection). By adjontion we get the map

R ∶ L ( ) ≃ ← ← ← ← ← ← ← → * L ( )
which satisfies, by functoriality, the cocycle condition

ℎ = * ℎ • ( , ℎ ∈ ). (32)
As in [START_REF] Miličić | Localization and representation theory of reductive Lie groups[END_REF] we can define (from now on we will work on admissible blow-ups of so we will use the same notation)

∶ D † , ( ) → * D † , ( ); → • • -1 (33)
and exactly as we have done in [START_REF] Patel | Locally analytic representations and sheaves on the Bruhat-Tits building[END_REF] we can conclude that

ℎ = * ℎ • ,
for every , ℎ ∈ .

Localization of locally analytic representations

We recall for the reader that 0 denotes the compact locally -analytic group 0 = ( ). In this section we will show how to localize admissible locally analytic representations of 0 . We will denote by C an ( 0 , ) the space of -valued locally -analytic functions on 0 and by ( 0 , ) its strong dual (the space of locally analytic distributions in the sense of [START_REF] Schneider | Continuous and locally analytic representation theory[END_REF]Section 11]). This space contains a set of delta distributions { } ∈ 0 defined by ( ) = ( ), if ∈ C an ( 0 , ), in such a way that the map  → is an injective group homomorphism from 0 into ( 0 , ) × . We also recall that given that 0 is compact, this space carries a structure of nuclear Fréchet-Stein algebra [START_REF] Schneider | Continuous and locally analytic representation theory[END_REF]Theorem 24.1]. To our work it will be enough to define a weak Fréchet-Stein structure (in the sense of [START_REF] Emerton | Locally Analytic Vectors in Representations of Locally -adic Analytic Groups[END_REF]Definition 1.2.8]) on the algebra ( 0 , ).

We finally recall that in (3.7.1) we have introduced Emerton's distribution algebra as the continuous dual space of the space of rigid-analytic functions on ( )

• D an ( ( ) • ) ∶= Hom Cont O ( ) • ( ( ) • ), .

Coadmissible modules

let us start by recalling that 0 acts on the space C cts ( 0 , ), of continuous -valued functions, by the formula ( • )( ) ∶= ( -1 ) ( , ∈ 0 , ∈ C cts ( 0 , )).

Moreover, given an admissible locally analytic representation of 0 [34, First definition of lecture VI] then, by definition, its strong dual ∶= ( ) ′ is a coadmissible module over ( 0 , ) 18 .

Given a continuous representation of 0 , we can consider the subspace ( ) • ⊆ of ( ) • -analytic vectors [14, 3.4.1]. In particular, the 0 -action on C cts ( 0 , ), defined at the beginning of this subsection, gives us

lim ← ← ← ← ← ← ← ← ← ← → C cts ( 0 , ) ( ) • -an ≃ ← ← ← ← ← ← ← → C an ( 0 , ). (34)
As in [14, Proposition 5.3.1], for each ∈ ℤ >0 we put

( ( ) • , 0 ) ∶= C cts ( 0 , ) ( ) • -an ′
the strong dual of the space of ( ) • -analytic vectors of C cts ( 0 , ) [14, 3.4.1]. The ring structure on D an ( ( ) • ) extends naturally to a ring structure on ( ( ) • , 0 ), such that

( ( ) • , 0 ) = ⨁ ∈ 0 ∕ D an ( ( ) • ) . ( 35
)
Dualizing the isomorphism (34) yields an isomorphism of topological -algebras

( 0 , ) ≃ ← ← ← ← ← ← ← → lim ← ← ← ← ← ← ← ← ← ← ← ∈ℤ >0 ( ( ) • , 0 ). (36)
This is the weak Fréchet-Stein structure on the locally analytic distribution algebra ( 0 , ) ([14, Proposition 5.3.1]).

Let be an admissible locally analytic representation and ∶= ′ . By [14, Lemma 6.1.6] the subspace ( ) • -an ⊆ is a nuclear Fréchet space and therefore its strong dual ∶= ( ) • -an ′ is a space of compact type and a finitely generated topological ( ( ) • , 0 )-module by [START_REF] Emerton | Locally Analytic Vectors in Representations of Locally -adic Analytic Groups[END_REF]Lemma 6.1.13]. By [START_REF] Emerton | Locally Analytic Vectors in Representations of Locally -adic Analytic Groups[END_REF]Theorem 6.1.20] the module is a coadmissible ( 0 , )-module relative to the weak Fréchet -Stein structure of ( 0 , ) defined in the previous paragraph. (i) The ( ( ) • , 0 )-module is finitely generated.

(ii) There are natural isomorphisms

( ( -1) • , 0 ) ⊗ ( ( ) • , 0 ) ≃ ← ← ← ← ← ← ← → -1 .
(iii) The natural map ( ( -1) • , 0 ) ⊗ ( 0 , ) → is bijective. Now, let ∈ Hom( , ) be an algebraic character such that + + * is a dominant and regular character of . Let us recall that we have identifications

D an ( ( ) • ) = † ( ( )) = lim ← ← ← ← ← ← ← ← ← ← → ∈ℕ ̂ ( ) ( ( )) ⊗ . morphism of sheaves Ψ ∶ * D † ′ , ′ ( ) = D † , ′ ( ) → D † , ( ). (38)
Moreover, if ∈ 0 we have (

• Ψ = * (
) 40
As M ′ is a coherent D † ′ , ′ ( ), by proposition 3.7.7 and theorem 4.2.1 we can find a finite presentation of

M ′ D † ′ , ′ ( ) ⊕ → (D † ′ , ′ ( )) ⊕ → M ′ → 0
which induces, by exactness of ( ) * and * (theorem 4.2.1.), the exact sequence (of course, here we have used theorem 4.2.1 to identify * D † ′ , ′ ( ) = D † , ′ ( )). This shows (40) and therefore we dispose of a diagonal action

∶ D † , ( ) ⊗ * D † ′ , ′ ( ) * M ′ → * D † , ( ) ⊗ * D † ′ , ′ ( ) * M ′
defined on simple tensor products by

• ( ⊗ ) ∶= • ⊗ • , ( 41
)
for ∈ 0 , and and local sections of D † , ( ) and * M ′ , respectively (in order to soft the notation we use the accord introduced in the commentary 1 after the definition 5.1.3). Now to see that (39) is a strongly 0 -equivariant D † , ( )module, we only need to check that (K ) ⊂ K . This is, the diagonal action fix the submodule K . We have

• ( ℎ ⊗ -⊗ ℎ • ) = • ( ℎ ) ⊗ • -• ⊗ • (ℎ • ) = ( • )( • ℎ ) ⊗ • -• ⊗ ( ℎ -1 ) • ( • ) = ( • ) ℎ -1 ⊗ • -• ⊗ ( ℎ -1 ) • ( • ),
as +1 is a normal subgroup we can conclude that ℎ -1 ∈ +1 and 0 fix K . Moreover, since the target of the preceding morphism is strongly 0 -equivariant, this factors through the quotient and we thus obtain a morphism of D † , ( )modules

∶ D † , ( ) ⊗ * D † ′ , ′ ( ), +1 * M ′ → M . (42)
By construction ∈ Coh D † , ( ), 0 .

Admissible blow-ups and formal models

The following discussion is given in [25, 3.1.1 and 5.2.13]. Let us start by considering the generic fiber ∶= × Spec( ) Spec( ) of the flag scheme (the flag variety). For the rest of this work rig will denote the rigid-analytic space associated via the GAGA functor to [6, Part I, chapter 5, section 5.4, Definition and proposition 3]. Any admissible formalscheme (in the sense of [6, Part II, chapter 7, section 7.4, Definitions 1 and 4]) whose associated rigid-analytic space is isomorphic to rig will be called a formal model of rig . For any two formal models 1 and 2 there exists a formal model ′ and admissible formal blow-up morphisms ′ → 1 and ′ → 2 [6, Part II, chapter 8, section 8.2, remark 10]. Now, let us denote by F the set of admissible formal blow-ups → . This set is ordered by ′ ⪰ if the blow-up morphism ′ → factors as the composition of a morphism ′ → and the blow-up morphism → . In this case, the morphism ′ → is unique [6, Part II, chapter 8, section 8.2, proposition 9], and it is itself a blow-up morphism [27, Chapter 8, section 8.1.3, proposition 1.12 (d) and theorem 1.24]. By [6, Part II, chapter 8, section 8.2, remark 10] the set F is directed and it is cofinal in the set of all formal models. Furthermore, any formal model of rig is dominated by one which is a 0 -equivariant admissible blow-up of [START_REF] Huyghe | D † -affinity of formal models of flag varieties[END_REF]Proposition 5.2.14]. In particular, if ∞ denotes the projective limit of all formal models of rig , then

∞ = lim ← ← ← ← ← ← ← ← ← ← ← F .
We will be interested in the following directed subset of F . Definition 6.0.1. We denote by F the set of pairs ( , ), where ∈ F and ∈ ℕ satisfies ≥ . This set is ordered by ( ′ , ′ ) ⪰ ( , ) if and only if ⪰ and ′ ≥ .

We will need the following auxiliary result. Lemma 6.0.2. Let ′ , ∈ F be 0 -equivariant admissible blow-ups (definition 4.3.1). Suppose ( ′ , ′ ) ⪰ ( , ) with canonical morphism ∶ ′ → over and let be a coherent ( ( ′ ) • , 0 ) -module with localization M = L † ′ , ′ ( )( ) ∈ Coh D † ′ , ′ ( ), 0 . Then there exists a canonical isomorphism in Coh D † , ( ), 0 given by

D † , ( ) ⊗ * D † ′ , ′ ( ), +1 * M ≃ ← ← ← ← ← ← ← → L † , ( ) ( ( ) • , 0 ) ⊗ ( ( ′ ) • , 0 ) .
Proof. The proof follows word for word the reasoning given in [25, Lemma 5.2.12] when ∈ Hom( , ) is equal to the trivial character. We will only indicate to the reader how the isomorphism is obtained. Let Σ be a system of representatives in +1 for the cosets in +1 ∕ ′ +1 . By [START_REF] Schneider | Locally analytic distributions and -adic representation theory, with applications to GL 2[END_REF] we have a canonical map

D an ( ( ) • ) → ( ( ) • , 0 ) (43)
which is compatible with variation in . Now, let us take a ( ( ′ ) • , 0 ) -module and let us consider the free D an ( ( )

• ) -module D an ( ( ) • ) ×Σ ∶= ⨁ ( ,ℎ)∈ ×Σ D an ( ( ) • ) ,ℎ ,
whose formation is functorial in and it comes with a linear map

∶ D an ( ( ) • ) ×Σ → D an ( ( ) • ) ⊗ D an ( ( ′ ) • ) ,ℎ ,ℎ  → ( ,ℎ ℎ ) ⊗ -,ℎ ⊗ ( ℎ . ).
which fits into an exact sequence This means that the above definition extends to a map

D an ( ( ) • ) ×Σ ← ← ← ← ← ← ← ← ← ← ← ← ← ← → D an ( ( ) • ) ⊗ D an ( ( ′ ) • ) can ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← → ( ( ) • , 0 ) ⊗ ( ( ′ ) • , 0 ) → 0,
∶ D † . , ( ) ⊗ D an ( ( ) • ) → * L † , ( )( ) .
The family { } ∈ 0 satisfies ( ), ( ) and ( ) in ( ). Let us verify condition (b). We suppose that ′ , are 0 -equivariant and that ( ′ , ) ⪰ ( , ) with canonical morphism ∶ ′ → over . As * is exact we have an isomorphism

* L † ′ , ′ ( )( ′ ) ≃ ← ← ← ← ← ← ← → * D † ′ , ′ ( ) ⊗ D an ( ( ) • ) ′ .
(This is an argument already given in the text for the functor ( ) * ). On the other hand, we have that ( ′ ) • ⊆ ( ) • and we have a map ̃ ′ , ∶ ′ → obtained as the dual map of the natural inclusion ( ) • -an → ( ′ ) • -an . Let U ⊆ be an open subset and ∈ * D † ′ , ′ ( )(U ), ∈ ′ . We define ′ , ( ⊗ ) ∶= Ψ ′ , ( ) ⊗ ̃ ′ , ( ),

where Ψ is the canonical injection * D † ′ , ′ ( ) → D † , ( ). By using the preceding isomorphism we can conclude that this morphisms extends naturally to a map ′ , is in fact an isomorphism. The morphism ′ , corresponds under the isomorphism of lemma 6.0.2 to the linear extension

( ( ) • , 0 ) ⊗ ( ( ′ ) • , 0 ) ′ →
of ̃ ′ , via functoriality of L † , ( ). By lemma 5.1.1 this linear extension is an isomorphism and hence, so is ′ , . We conclude that L 0 ( ) ∈ C 0 , . Given a morphism → in C 0 , , we get, by definition, morphisms → By definition, the open subsets of the form ∶= sp -1 (U ) form a basis for the topology of ∞ and D( ) is a presheaf on this basis. The associated sheaf on ∞ to this presheaf will also be denoted by D( ). This action is on the left, in the sense that if , ℎ ∈ 0 then ( ) * ℎ • = ℎ . Let us suppose now that M = (M , ) ∈ C 0 , . We have the transition maps ′′ , ′ ∶ ′′ * M ′′ , ′′ → M ′ , ′ which are linear relative to the morphism (38). As before, we have the map

M ′′ , ′′ sp ′′ ( ) = ′′ * M ′′ , ′′ sp ′ ( ) sp ′ ( ) ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← → M ′ , ′ sp ′ ( )
which allows us to define M ∞ as the sheaf on ∞ associated to the presheaf

M ∞ ( ) ∶= lim ← ← ← ← ← ← ← ← ← ← ← ′ → M ′ , ′ sp ′ ( ) .
where runs over all special vertices of B. We recall for the reader that ∞ is equal to the projective limit of all formal models of rig . (53)

As usual, for every special vertex ∈ B, we will denote by L ( ) the -adic completion of the sheaf L ( ), which is considered as an invertible sheaf on . Let ( , ) ∈ F with blow-up morphism pr ∶ → . At the level of differential operators, we will denote by D † , ( ) the sheaf of arithmetic differential operators on acting on the line bundle L ( ) 23 . We have the following important properties. Let ∈ . As in [START_REF] Schneider | Algebras of -adic distributions and admissible representations[END_REF] the isomorphism (53) induces a left action

∶ D † , ( ) ≃ ← ← ← ← ← ← ← → * D † , ( )  → ( ) -1 .

≃

  Here the left-hand vertical arrow is the restriction functor coming from the homomorphism( 0 , ) → ( , )and the right-hand vertical arrow is the forgetful functor.Acknowledgements:The present article contains a part of the author PhD thesis written at the Universities of Strasbourg and Rennes 1 under the supervision of C. Huyghe and T. Schmidt. Both have always been very patient and attentive supervisors. For this, I express my deep gratitude to them.

Example 2 . 1 . 1 .

 211 Let be a discrete valuation ring of unequal characteristic (0, ) and uniformizing parameter . Let us write = , with a unit of and a positive integer (called the absolute ramification index of ). Let ∈ ℕ. Then ( ) ∶= ∕ ! defines a PD-structure on ( ) if and only if ≤ ( -1) [5, Section 3, examples 3.2 (3)]. In particular, we dispose of a PD-structure on ( ) ⊆ ℤ ( ) . We let [ ] ∶= ( ) and we denote by (( ), []) this PD-ideal. Moreover, if ≤ -1 and ≥ ( ), then ( ) endowed with the preceding PD-structure defines an m-PD-structure on ( ) [3, Section 1.3, examples (i)].

Finally, let us

  give a local description of D ( ) , . Let be a smooth open affine subset of endowed with a family of local coordinates 1 , . . . , . Let 1 , . . . , be a basis of Ω ( ) and 1 , . . . , the dual basis of T ( ) (as usual, T and Ω denote the tangent and cotangent sheaf on , respectively). Let ∈ ℕ . Let us denote by | | = ∑ =1 and [ ] = ∕ ! for every 1 ≤ ≤ . Then, using multi-index notation, we have [ ] = ∏ =1 [ ] and < > = ! [ ] . In this case, the sheaf D ( ) , has the following description on

  are therefore -modules generated by the elements { } with | | ≤ [3, Proposition 1.5.3 (ii)]. Moreover, there exists an isomorphism of

Proposition 3 . 1 . 4 .

 314 have the relations ( × ) * * (L) = * ( ) and ( × ) * * 1 (L) = L. So every morphism of line bundles Ψ ∶ * (L) → * 1 (L) induces morphisms Ψ ∶ * ( ) → for all ∈ ( ). The following reasoning can be found in [12, Page 104] or [8, Lemma 3.2.4]. Let ∶ L → be a line bundle over endowed with a -linearization ∶ L × Spec( ) → L. Then there exists an isomorphism

21 , 11 Proposition 3 . 3 . 9 .

 2111339 Subsection 1.6]. By using proposition 3.3.5 and (A ( , ) ) = O ⊗ Sym ( ) ( ), we can conclude that Φ ( , ) is surjective. Proposition 3.3.6 and the same reasoning given in [24, Proposition 2.2.2 (iii)] imply the following meaningful result. The sheaf D ( , ) ( ) is a sheaf of O -rings with noetherian sections over all open affine subsets of .

  scheme. There exists a very ample sheaf O(1) on [17, Chapter II, remark 5.16.1]. Therefore, for any arbitrary O -module E we can consider the twist E ( ) ∶= E ⊗ O O( ) where ∈ ℤ means the -th tensor product of O(1) with itself. We recall to the reader that there exists 0 ∈ ℤ, depending of O(1), such that for every ∈ ℤ >0 and for every ≤ 0 , ( , O( )) = 0 [17, Chapter II, theorem 5.2 (b)].Let us start the results of this subsection with the following proposition which states three important properties of coherent A ( , ) -modules[START_REF] Huyghe | D-modules arithmétiques, distributions et localisation[END_REF] Proposition A.2.6.1]. This is a key result in our work. Let E be a coherent A ( , ) -module.

Proposition 3 . 4 . 4 .

 344 Let us suppose that + ∈ * is a dominant and regular character (cf.3.3.1).(i) Let us fix ∈ ℤ. For every positive integer ∈ ℤ >0 , the cohomology group ( , D ( , ) , ( )) has bounded -torsion.

By lemma 3

 3 .4.3 we can use the same inductive reasoning that in [21, Corollary 2.2.4] to show (ii).

  3.6. By [24, Proposition 2.2.2 (v)] 0 ( , D( , ) ) is twosided noetherian. Therefore, we can take B as the set of affine open subsets of contained in the -adic completion of an affine open subset ∈ S. This proves (i). By [3, proposition 3.3.4] we can conclude that (ii) is an immediately consequence of (i) because 0 ( , D( , ) , ,ℚ ) = 0 ( , D( , ) , ) ⊗ [3, (3.4.0.1)].

12 Proposition 3 . 5 . 4 .

 12354 proposition 3.2]. Next proposition is a natural result from lemmas 3.4.2 and 3.4.3. The proof is exactly the same that in [25, Proposition 4.2.2]. Let E be a coherent D( , ) , -module. (i) There exists 2 = 2 (E ) ∈ ℤ such that for all ≥ 2 there is ∈ ℤ and an epimorphism of D( , ) , -modules D( , ) , (-) ⊕ → E → 0 which induces the central reduction U ( ) ∶= U ( ) ⊗ ( ), + . If Ker( + ) ∶= ( ) ( ( )) ∩ Ker( + ), we can consider the central redaction ( ) ( ( )) ∶= ( ) ( ( ))∕ ( ) ( ( ))Ker( + )

Theorem 3 . 7 . 2 .Proposition 3 . 7 . 3 .Proposition 3 . 7 . 4 .

 372373374 Let ∈ ( ) be an algebraic character, such that + ∈ * is dominant and regular. The functors L † , ( ) and 0 ( , •) are quasi-inverse equivalence of categories between the abelian categories of finitely presented (left) † ( ( )) -modules and coherent D † , ( )-modules. Let us start by recalling the following proposition [3, Proposition 3.6.1]. Let be a topological space, and {D } ∈ be a filtered inductive system of coherent sheaves of rings on , such that for any ≤ the morphisms D → D are flat. Then the sheaf D † ∶= lim ← ← ← ← ← ← ← ← ← ← → ∈ D is a coherent sheaf of rings. The sheaf of rings D † , ( ) is coherent.

Definition 4 . 1 . 1 .

 411 Let I ⊆ O be a coherent ideal sheaf. We say that a blow-up pr ∶ → along the closed subset(I) is admissible if there is ∈ ℕ such that O ⊆ I.Let us fix I ⊆ O an open ideal and pr ∶ → an admissible blow-up along (I). We point out to the reader that I is not uniquely determined by the space . In the sequel we will denote by∶= min I min{ ∈ ℕ | ∈ I},where the first minimum runs over all open ideal sheaves I such that the blow-up along (I) is isomorphic to . Now, as I is an open ideal sheaf, the blow-up induces a canonical isomorphism ≃ between the generic fibers. Moreover, as is invertible on , we have D ( , ) | = D | = D , the usual sheaf of (algebraic) differential operators on . Therefore pr -1 D ( , ) | = D . In particular, O has a natural structure of (left) pr -1 D ( , )

  and by (ii) in the actual proposition, it has noetherian sections over the affine open subsets of (cf. [24, Proposition 2.2.2 (iii)]), it is certainly a sheaf of coherent rings [3, proposition 3.1.3]. Finally, by definition, we see that D( , ) ( ) satisfies the conditions (a) and (b) of 3.3.3 in [3] and hence [3, Proposition 3.3.4] gives us (iii).

≃ ≃ ( 22 )Proposition 4 . 1 . 6 .

 22416 Given that the morphism of sheaves D( , ) ,ℚ → D( +1, ) ,ℚ is left and right flat [24, Proposition 2.2.11 (iii)], the preceding diagram allows us to conclude that the morphism D( , ) ,ℚ ( ) → D( +1, ) ,ℚ ( ) is also left and right flat. By proposition 3.7.3 we have the following result. The sheaf of rings D † , ( ) is coherent.

Theorem 4 . 2 . 3 .

 423 Let pr∶ → be an admissible blow-up. Let us suppose that ∈ Hom( , ) is an algebraic character such that + ∈ * is a dominant and regular character of .(i) For any coherent D † , ( )-module E and for all > 0 one has ( , E ) = 0.

  ♮ ∶ O → ( ) * O denotes the comorphism of . Let ⊆ 0 be an open subgroup. We say that an open ideal sheaf I ⊆ O is -stable if for all ∈ the comorphism ♮ maps I ⊆ O into ( ) * I ⊆ ( ) * O . In this case ♮ induces a morphism of sheaves of graded rings ⨁ ∈ℕ I → * ⨁ ∈ℕ I on . This morphism of sheaves induces an automorphism of the blow-up = Proj ⊕ ∈ℕ I , let us say by abuse of notation, and the action of on lifts to a right action of on , in the sense that for every , ℎ ∈ 0 the relation (26) is verified and we have a commutative diagram . Let ⊆ 0 be an open subgroup and pr ∶ → and admissible blow-up defined by an open ideal subsheaf I ⊂ O . We say that is -equivariant if I is -stable.

  We have the following result from [25, Lemma 5.1.7]. Lemma 5.1.1.

  Ψ) • * ′ and therefore Ψ is 0 -equivariant. Now, let us consider M ′ ∈ Coh D † ′ , ′ ( ), 0 and M ∈ Coh D † , ( ), 0 together with a morphism ∶ * M ′ → M linear relative to Ψ ∶ * D † ′ , ′ ( ) → D † , ( ) and which is 0 -equivariant, i.e. satisfying M • = * • * M ′ for all ∈ 0 . By using Ψ we obtain a morphism of D † , ( )-modules D † , ( ) ⊗ * D † ′ , ′ ( ) * M ′ → M . Let us denote by K the submodule of D † , ( ) ⊗ * D † ′ , ′ ( ) * M ′ locally generated by all the elements of the form ℎ ⊗ -⊗ (ℎ • ), where ℎ ∈ +1 , is a local section of * M ′ and is a local section of D † , ( ). As in [25, Page 35] we will denote the quotient D † , ( ) ⊗ * D † ′ , ′ ( ) * M ′ K by D † , ( ) ⊗ * D † ′ , ′ ( ), +1 * M ′ . (39) Let us see that this module lies in Coh D † , ( ), 0 . To do that let us first show that * D † , ( ) ⊗ ( ) * * D † ′ , ′ ( ) * * M ′ = * D † , ( ) ⊗ * D † ′ , ′ ( ) * M ′ .

  if is a finitely presented ( ( ′ ) • , 0 ) -module[25, Claim 1 in the proof of lemma 5.2.12]. Now, let be a finitely presented D an ( ( ′ ) • ) -module and M ∶= L † ′ , ′ ( )( ). Then the natural morphismL † , ( ) D an ( ( ) • ) ⊗ D an ( ( ′ ) • ) → D † , ( ) ⊗ * D † ′ , ′ ( ) * M(44)is bijective. In fact, by theorem 4.2.1 we know that the functor * is exact on coherent D † ′ , ′ ( )-modules, so taking a finite presentation of we reduce to the case = D an ( ( ′ ) • ) which is clear.Finally, let us take a finitely presented ( ( ′ ) • , 0 ) -module. Let 1 , ..., be generators for as a D an ( ( ′ ) • )module. We have a sequence of D an ( ( )• ) -modules ⨁ ( ,ℎ) D an ( ( ) • ) ,ℎ ← ← ← ← ← ← ← ← ← ← → D an ( ( ) • ) ⊗ D an ( ( ′ ) • ) can ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← → ( ( ) • , 0 ) ⊗ ( ( ′ ) • , 0 ) → 0where denotes the restriction of the map to the free submodule of D an ( ( ) • ) ×Σ generated by the finitely many vectors ,ℎ , with 1 ≤ ≤ and ℎ ∈ Σ. Since im( ) = im( ) the sequence is exact. Since it consists of finitely Given that ( ) * is exact we can choose a finite presentation of as a D an ( ( ) • ) -module to conclude that we have a canonical isomorphism * L † , , ( ) ⊗ D an ( ( ) • ) .

′

  , ∶ * L † ′ , ′ ( )( ′ ) → L † , ( )( ).The cocycle condition translates into the diagram * * L † ′ , ′ ( )( ′ ) = ( . ) * * therefore (48) is also a commutative diagram. The transitivity properties are clear. Let us see that the induced morphism

,

  then relation (51), and commutativity of the first diagram in (49) tells us thatD † ′′ , ′′ ( ) sp ′′ ( ) = ′′ * D † ′′ , ′′ ( ) sp ′ ( ) D † ′ , ′ ( ) sp ′ ( ) ′ , ′ ( ′ , ′ )∈F ∈ ∏ ( ′ , ′ )F D † ′ , ′ ( ) sp ′ ( ) | Ψ ′′ , ′ ( ′′ , ′′ ) = ′ , ′Using the commutativity of the preceding diagram we see thatΨ sp ′ ( -1 ( )) ′′ , sp ′′ ( ) ( ′′ , ′′ ) = ′ , sp ′ ( ) Ψ sp ′ ( ) ( ′′ , ′′ ) = ′ , sp ′ ( ) ( ′ , ′ )and therefore, for ∈ 0 , the actions assemble to an action ∶

Remark 7 . 0 . 1 . 2 .

 7012 The set F is partially ordered in the following way. We say that ′ ⪰ if the projection sp ′ ∶ ∞ → ′ factors through the projection sp ∶ ∞ → We will denote by F ∶= ⨆ F , where runs over all the special vertices of B. This set is partially ordered as follows. We say that ( ′ , ′ ) ⪰ ( ,) if ′ ⪰and Lie( ′ ( ′ )) ⊂ Lie( ( )) (or equivalent ′ Lie( ′ ) ⊂ Lie( )) as lattices in .For any special vertex ∈ B, any element ∈ induces an isomorphism∶ → . .The isomorphism induced by on the generic fibers ( ) ≃ ≃ ( . ) coincides with right translation by on∶ = × Spec( ) Spec( ) × ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← → × Spec( ) Spec( ) ← ← ← ← ← ← ← ← ← ← ← → ,where we have used ( ) = ( ). Moreover, induces a morphism → . , which we denote again by , and which coincides with the right translation on if ∈ ,0 (of course in this case= ). Let ( ) ♮ ∶ O → ( ) * O be the comorphism of . If ∶→ is an admissible blow-up of an ideal I ⊂ O , then blowing-up (( ) ♮ ) -1 (( ) * I) produced a formal scheme (cf. lemma 6.0.3), together with an isomorphism ∶ → . As in lemma 6.0.3 we have = . For any , ℎ ∈ and any admissible formal blow-up → , we haveℎ • = ℎ ∶ → ℎ .This gives a right -action on the family F and on the projective limit ∞ . Finally, if ′ ⪰ with morphism ∶ ′ → and ∈ , then ′ ⪰ , and we have the relation• = • ′ (here ∶ ′ →). Now, over every special vertex ∈ B the algebraic character induces an invertible sheaf L ( ) on , such that for every ∈ there exists an isomorphism ∶ L ( ) → ( ) * L ( ), satisfying the cocycle conditionℎ ℎ = ℎ * ℎ • ℎ (ℎ, ∈ ).

  32, Proposition 4.4.1]. Let us suppose that ∈ ( ) is an algebraic character such that + ∈ * is a dominant and regular character of . The functors L ( , ) The proof of [20, Proposition 5.2.1] carries over word by word.

	Theorem 3.6.3.
	Proposition 3.6.2. Let E be a coherent D( , ) , ,ℚ -module. Then E is generated by its global sections as D( ) , ,ℚ -module.
	Furthermore, every coherent D( , ) , ,ℚ -module admits a resolution by finite free D( , ) , ,ℚ -modules.

, and 0 ( , •) are quasi-inverse equivalence of categories between the abelian categories of finitely generated ̂ ( ) ( ( )) ⊗ -modules and coherent D( ) , ,ℚ -modules.

Proof.

This technical condition is clarified in proposition 4.1.2. It is also explained in (2) below.

Via the classical isomorphism of Harish-Chandra induces a character ∶ (Lie( ) ⊗ ) → which allows to consider the central redaction.

Here 0 is a (maximal) compact subgroup of . This compactness property allows to define the structure of weak Fréchet-Stein algebra.

This remark exemplifies the local situation when = Spec( ) with a ℤ ( ) -algebra [20, Subsection 1.3.1].

This digression can be found before the proof of[START_REF] Huyghe | D † -affinity of formal models of flag varieties[END_REF] Proposition 3.3.7].

This is [25, Claim 3.3.10.].

This is exactly as in[25, 3.3.2]

Of course, this is also an immediately consequence of proposition 3.3.8 and[20, proposition 1.3.6].

We skip the proof here, but the reader can take a look to [32, Proposition 4.1.2] when we have treated the case = 0. The proof for ∈ ℤ >0 is exactly the same.

We recall for the reader that L( ) is a -equivariant line bundle, which implies the existence of this morphism[START_REF] Hotta | D-modules, perverse sheaves and representation theory[END_REF] Section 11.1].

This is exactly as in[25, theorem 4.2.8].

This is exactly as in[20, theorem 5.1].

We construct this morphism in (55). The arguments given there are independent and we won't introduce a circular argument.

This is the same reasoning that we have used in (i).

We recall for the reader, that the category of coadmisisble ( 0 , )-modules is a full abelian subcategory of the category of ( 0 , )-modules and the "strong dual" functor induces an anti-equivalence of categories to the category of admissible locally analytic representations[START_REF] Schneider | Continuous and locally analytic representation theory[END_REF] Theorem 20.1].

This conditions makes sense because the elements ∈ +1 acts trivially on the underlying topological space of , cf. Lemma 4.3.2.

In this reference this space is denoted by ⟨ ⟩ , cf. [24, subsection 3.2].

This is exactly as in[25, 5.3.1].

Here we abuse of the notation and we denote again by L ( ) the invertible sheaf pr * L ( ) on .

(ii) There exists 3 = 3 (E ) ∈ ℤ such that for all ≥ 3 we have ( , E ) = 0, for all > 0.

The same inductive argument exhibited in proposition [START_REF] Huyghe | D † -affinité de l'espace projectif[END_REF]Proposition 3.4 (i)] shows Corollary 3.5.5. Let E be a coherent D( , )

, -module. There exists = (E ) ∈ ℕ such that for all > 0 the cohomology group ( , E ) is annihilated by . Now, we want to extend the part (i) of the preceding proposition to the sheaves D( , ) , ,ℚ . To do that, we need to show that the category of coherent D( , )

, ,ℚ -modules admits integral models (definition 2.5.1). Let Coh( D( , )

, ) be the category of coherent D( , ) , -modules and let Coh( D( , ) , ) ℚ be the category of coherent D( , ) ,modules up to isogeny. This means that Coh( D( , )

, ) ℚ has the same class of objects as Coh( D( , ) , ) and, for any two objects M and N in Coh ( D( , ) , ) ℚ one has Proof. By definition, the sheaf D( , ) , ,ℚ satisfies [3, conditions 3.4.1] and therefore [3, proposition 3.4.5] allows to conclude the proposition.

The proof of the next theorem follows exactly the same lines than in [25, theorem 4.2.8].

Theorem 3.5.7. Let E be a coherent D( , )

, ,ℚ -module.

(i) There is (E ) ∈ ℤ such that, for every ≥ (E ) there exists ∈ ℕ and an epimorphism of D( , ) , ,ℚ -modules D( , ) , ,ℚ (-)

⊕ → E → 0.

(ii) For all > 0 one has ( , E ) = 0.

Proof. By the preceding proposition, there exists a coherent D( , ) , -module F such that F ⊗ ≃ E . Therefore, applying proposition 3.5.4 to F gives ( ). Moreover, as is a noetherian space, corollary 3.5.5 allows us to conclude that

for every > 0 [3, (3.4.0.1)].

The arithmetic Beilinson-Bernstein theorem with congruence level

Calculation of global sections

We recall for the reader that throughout this section + ∈ * denotes a dominant and regular character, which is induced by an algebraic character ∈ ( ). Insppired in the arguments exhibited in [START_REF] Huyghe | D-modules arithmétiques sur la variété de drapeaux[END_REF], in this subsection we propose to calculate the global sections of the sheaf D( , ) , ,ℚ .

Let us identify the universal enveloping algebra U ( ) of the Cartan subalgebra with the symmetric algebra ( ), and let ( ) denote the center of the universal enveloping algebra U ( ) of . The classical Harish-Chandra isomorphism ( ) ≃ ( ) (the subalgebra of Weyl invariants) [11, theorem 7.4.5], allows us to define for every linear form ∈ * a central character [11, 7.4.6] ∶ ( ) →

The preceding relation and the fact that the ring structure of D an ( ( ) • ) extends naturally to a ring structure on ( ( ) • , 0 ) allow us to consider the ring ( ( ) • , 0 ) ∶= ( ( ) • , 0 )∕Ker( ) ( ( ) • , 0 ).

From now on, we will denote C 0 the full subcategory of Mod( ( 0 , )) consisting of coadmissible modules, with respect to the preceding weak Fréchet-Stein structure on ( 0 , ). Definition 5.1.2. We define the category C 0 , of coadmissible ( 0 , )-modules with central character ∈ Hom( , ) by C 0 , ∶= ( 0 , ) ( ) ( 0 , ) ∩ C 0 .

We point out that the preceding definition is completely legal because the center ( ) of the universal enveloping algebra U ( ) lies in the center of ( 0 , ) [START_REF] Schneider | Locally analytic distributions and -adic representation theory, with applications to GL 2[END_REF]Proposition 3.7]. We also recall that the group ∶= ( )( ) is contained in D an ( ( ) • ) as a set of Dirac distributions. For each ∈ we will write for the image of the Dirac distribution supported at in

Inspired in [START_REF] Huyghe | D † -affinity of formal models of flag varieties[END_REF]Definition 5.2.7] we have the following definition. (iii) 19 For all ∈ ∩ +1 the map

A morphism between two strongly -equivariant D † , ( )-modules (M , ( M ) ∈ ) and (N ,

We denote the category of strongly -equivariant coherent D † , ( )-modules by Coh D † , ( ), 0 .

In what follows we will use the notation ∶= , U ( ) ∈ M (U . -1 ), for U ⊆ an open subset, ∈ 0 and ∈ M (U ). This notation is inspired in property ( ) of the previous definition. In fact, if , ℎ ∈ 0 , then by ( ) we have ℎ( ) = (ℎ ) .

Theorem 5.1.4. Let ∈ Hom( , ) be an algebraic character such that + ∈ * is a dominant and regular character of . Let pr ∶ → be a 0 -equivariant admissible blow-up, and let ≥ . The functors L † , ( ) and 0 ( , •) induce quasi-inverse equivalences between the category of finitely presented ( ( ) • , 0 ) -modules and Coh D † , ( ), 0 .

Before starting the proof, we recall for the reader that the functor L † , ( ) has been defined in the proof of theorem 4.2.3. An explicitly expression is given in (37) below.

Proof. If M ∈ Coh D † , ( ), 0 , then in particular M is a coherent D † , ( )-module. Since by corollary 4.2.2 and theorem 4.2.3 we have that 0 ( , M ) is a finitely presented D an ( ( ) • ) -module, then by [START_REF] Schneider | Locally analytic distributions and -adic representation theory, with applications to GL 2[END_REF] we can conclude that 0 ( , M ) is a finitely presented ( ( ) • , 0 ) -module.

On the other hand, let us suppose that is a finitely presented ( ( ) • , 0 ) -module. By [START_REF] Schneider | Locally analytic distributions and -adic representation theory, with applications to GL 2[END_REF] we can consider

For every ∈ 0 we want to define an isomorphism of sheaves of -vector spaces

satisfying the conditions ( ), ( ) and ( ) in the preceding definition. As we have remarked, the Dirac distributions induce an injective morphism from 0 to the group of units of ( 0 , ), since by [START_REF] Waterhouse | One-dimensional affine group schemes[END_REF] is in particular a 0 -module, we have an isomorphism

and is the isomorphism defined in [START_REF] Schneider | Algebras of -adic distributions and admissible representations[END_REF].

One has an isomorphism * (M )

Indeed, ( ) * is exact and so choosing a finite presentation of as D an ( ( ) • ) -module reduces to the case = D an ( ( ) • ) which is trivially true. This implies that the preceding isomorphism extends to an isomorphism

which verifies the first condition. Now, by definition , U ( ) = , U ( ) , U ( ) and therefore , U ( ⊗ ) = , U ( ) , U ( ⊗ ), which gives (ii). Finally, given that the delta distributions for in the normal subgroup +1 of 0 are contained in D an ( ( ) • ) we have . ∶= ( ) = -1 , and therefore

and condition ( ) follows.

Remark 5.1.5. If ∈ Hom( , ) denotes the trivial character, then D † , ( ) = D † , is the sheaf of arithmetic differential operators introduced in [START_REF] Huyghe | D † -affinity of formal models of flag varieties[END_REF]. Moreover, by construction, if pr ∶ → denotes an -equivariant admissible blow-up, then D † , ( ) = D † , and for every ∈ the isomorphism equals the isomorphism ( ) defined in [25, (5.2.6)].

Now, let us take ∶ ′ → a morphism of 0 -equivariant admissible blow-ups of (whose lifted actions we denote by ′ and ), and let us suppose that ≥ and ′ ≥ max{ ′ , }. By (25) and theorem 4.2.1 we have an injective presented D an ( ( ) • ) -modules, we can apply the localisation functor L † , ( ) to it. As L † , ( )

then (44) gives us the exact sequence

where M ∶= L † ′ , ′ ( )( ). The cokernel of the first map in this sequence equals by definition Let pr ∶ → be an admissible blow-up and let us denote by L ( ) the invertible sheaf on induced by pulling back the invertible sheaf on induced by the character . This is L ( ) ∶= pr * L ( ). Furthermore, for ∈ 0 if ∶ → . is the morphism given by the previous lemma and pr ∶ . → is the blow-up morphism, then we will denote L ( ) ∶= * L ( ).

The notation being fixed, we prevent the reader that in order to simplify the notation, in the rest of this work we will avoid to underline these sheaves if the context is clear and there is not risk to any confusion.

Let us recall that in subsection 4. By construction satisfies the cocycle condition [START_REF] Alzate | A Beilinson-Bernstein theorem for twisted arithmetic differential operators on the formal flag variety[END_REF]. This means that for every , ℎ ∈ 0 we have

In particular is an isomorphism for every ∈ 0 .

Exactly as we have done in [START_REF] Schneider | Algebras of -adic distributions and admissible representations[END_REF], and given that by construction D † , ( ) acts on L ( ) (resp. D † . , ( ) acts on L ( )), we can build an isomorphism

Satisfying the following cocycle condition

From the previous lemma we get [ , ( )-modules M , for all ( , ) ∈ F , with the following properties: (a) For any ∈ 0 with morphism ∶ → . , there exists an isomorphism

of sheaves of -vector spaces, satisfying the following properties: 20 For all ∈ +1 the map

(b) Suppose , ′ ∈ F are both 0 -equivariant, and assume further that ( ′ , ′ ) ⪰ ( , ), and that ∶ ′ → is the unique morphism over . We require the existence of a transition morphism

By using the commutative diagram in the preceding corollary, we required The morphism induced by ′ ,

is required to be an isomorphism of D † , ( )-modules. Additionally, the morphisms ′ , are required to satisfy the transitivity condition

A morphism M → N between such modules consists of morphisms M , → N , of D † , ( )-modules, which is compatible with the extra structures imposed in ( ) and ( ). We denote the resulting category by C 0 , .

Let us build now the bridge to the category C 0 , of coadmissible ( 0 , ) -modules. Given such a module we have its associated admissible locally analytic 0 -representation ∶= ′ together with its subspace of ( ) • -analytic vectors ( ) • -an ⊆ . As we have remarked, this is stable under the 0 -action and its dual ∶=

In this situation we produce a coherent D † , ( )-module

for any element ( , ) ∈ F . We will denote the resulting family by

.

On the other hand, let M be an arbitrary coadmissible 0 -equivariant arithmetic D( )-module on F . The transition morphisms ′ , ∶ * M ′ , ′ → M , induce maps 0 ′ , M ′ , ′ → 0 , M , on global sections. We let

The projective limit is taken in the sense of abelian groups. We have the following theorem. Except for some technical details the proof follows word for word the reasoning given in [25, Theorem 5.2.23].

Theorem 6.0.6. Let us suppose that ∈ Hom( , ) is an algebraic character such that + ∈ * is a dominant and regular character of . The functors L 0 and Γ(•) induce quasi-inverse equivalences between the categories C 0 , (of coadmissible ( 0 , ) -modules) and C 0 , .

Proof. Let us take ∈ C 0 , and M ∈ C 0 , . As in [25, Proof of theorem 5.2.23] we will organise the proof in four steps.

Step 1. We have L 0 ( ) ∈ C 0 , and L 0 ( ) is functorial in .

Proof. Let us start by defining

satisfying ( ), ( ) and ( ) in the preceding definition. Let ̃ ∶ → denote the map dual to the map ( )

for any ∈ ℤ >0 compatible with ̃ and ̃ ′ , . By functoriality of L † , ( ), they give rise to linear maps

which are compatible with the maps and ′ , .

Step 2. Γ(M ) is an object in C 0 , . Proof. For ∈ ℕ we choose ( , ) ∈ F and we put ∶= 0 ( , M ( , ) ). By (42), lemma 6.0.2 and the fact that M ∈ C 0 , we get linear isomorphisms

for ′ ≥ . This implies that the modules form a ( ( ) • , 0 ) ∈ℕ -sequence and the projective limit is a coadmissible module.

Step 3. Γ • L 0 ( ) ≃ .

Proof. If ∶= ′ , then we have by definition compatible isomorphisms

which imply that the coadmissible modules Γ • L 0 ( ) and have isomorphic ( ( ) • , 0 ) ∈ℕ -sequences.

Step 4.

Proof. Let ∶= Γ(M ) and ∶= ′ the corresponding admissible representation. Let N ∶= L 0 ( ). According to the lemma 5.1.1

produces a ( ( ) • , 0 ) ∈ℕ -sequence for the coadmissible module which is isomorphic to its constituting sequence 0 ( , M , ) ( , )∈F from step 2. Now let ( , ) ∈ F . We have the following isomorphisms

By -linearity the action maps M , and N , , constructed in step 1, are the same. Similarly if ( ′ , ′ ) ⪰ ( , ) are 0 -equivariant then the transition maps M ′ , and N ′ , coincide, by Ψ ′ , -linearity. Hence N ≃ M in C 0 , .

Coadmissible 0 -equivariant D( )-modules on the Zariski-Riemann space

Let us recall that ∞ denotes the projective limit of all formal models of rig (the rigid-analytic space associated by the GAGA functor to the flag variety ). The set F of admissible formal blow-ups → is ordered by setting ′ ⪰ if the blow-up morphism ′ → factors as ′ ← ← ← ← ← ← ← → → , with a blow-up morphism. The set F is directed in the sense that any two elements have a common upper bound, and it is cofinal in the set of all formal models. In particular,

The space ∞ is also known as the Zariski-Riemann space [6, Part II, chapter 9, section 9.3] 21 . In this subsection we indicate how to realize coadmissible 0 -equivariant D( )-modules on F as sheaves on the Zariski-Riemann space ∞ . We start with the following proposition whose proof can be found in [25, Proposition 5.2.14].

Proposition 6.1.1. Any formal model of rig s dominated by one which is a 0 -equivariant admissible blow-up of .

Remark 6.1.2. As F is cofinal in the set of all formal models, the preceding proposition tells us that the set of all 0equivariant admissible blow-ups of is also cofinal in the set of all formal models of . From now on, we will assume that if ∈ F , then also 0 -equivariant, and we will denoted by ∶ → the morphism induced by every ∈ 0 .

For every ∈ F we denote by sp ∶ ∞ → the canonical projection map. Let ′ ⪰ with blow-up morphism

which defines a 0 -action on the space ∞ .

Let U ⊂ be an open subset and let us take

which implies that sp ′ ( ) is an open subset of ′ . Now, let us suppose that

In this situation, the morphism

and we can form the projective limit as in [25, (5.2.25)]

By definition, we have the following commutative diagram

Identifying

we see as before that if

then the preceding commutative diagram implies that

and therefore we get a family ( ) ∈ 0 of isomorphisms

of sheaves of -vector spaces. By definition 6.0.5 we have that if , ℎ ∈ 0 then ℎ = ( ) * ℎ • . Furthermore, under the preceding identifications, if = ( ′ , ′ ) ∈ D( )( ) and = ( ′ , ′ ) ∈ M ∞ ( ), then . = ( ′ , ′ . ′ , ′ ) ( ′ , ′ )∈F and therefore

In particular, M ∞ is a 0 -equivariant D( )-module on the topologial 0 -space ∞ . Let us see that the formation of M ∞ is functorial. Let ∶ M → N be a morphism in C 0 , . Then, in particular we have the following commutative diagram

Commutativity in the preceding diagram implies that Proof. We start the proof by remarking that sp ( ∞ ) = for every ∈ F . By remark 6.1.2, the global sections of

If (•) ∞ denotes the previous functor, then we will denote by L 0 ∞ ( ) the composition of the functor L 0 with (•) ∞ , i.e., {Coadmissible ( 0 , ) -modules}

Since L 0 is an equivalence of categories, the preceding proposition implies that L 0 ∞ ( ) is a faithful functor.

-equivariant modules

Thorough this section we will denote by = ( ) and by B the semi-simple Bruhat-Tits building of the -adic group ( [START_REF] Bruhat | Groupes réductifs sur un corps local I[END_REF] et [START_REF] Bruhat | Groupes réductifs sur un corps local II[END_REF]). This is a simplicial complex endowed with a natural right -action.

The purpose of this section is to extend the above results from 0 -equivariant objects to objects equivariants for the whole group .

We start by fixing some notation. 22 To each special vertex ∈ B the Bruhat-Tits theory associates a connected reductive group -scheme , whose generic fiber ( ) ∶= × Spec( ) Spec( ) is canonically isomorphic to . We denote by the smooth flag scheme of whose generic fiber ( ) is canonically isomorphic to the flag variety . We will distinguish the next constructions by adding the corresponding vertex to them. For instance, we will write for an (algebraic) admissible blow-up of the smooth model , ,0 for the group of points ( ) and , for the group of points ( )( ). We will use the same conventions if we deal with formal completions. For instance, will always denote an admissible formal blow-up of . We point out to the reader that the blow-up morphism → will make part of the datum of , and that even if for another special vertex ′ ≠ the formal -scheme is also a blow-up of the smooth formal model ′ , we will only consider it as a blow-up of . We will denote by F ∶= F , the set of all admissible formal blow-ups → of and by F ∶= F the respective directed system of definition 6.0.1. By the preceding accord, the sets F and F ′ are disjoint if ≠ ′ . Let

Now, we identify the global sections 0 ( , D † , ( )) with D an ( ( ) • ) and obtain the group homomorphism

where , +1 = ( ) • ( ) denotes the group of -rational points (or -points of ( + 1)). We will follow the same lines of reasoning given in [START_REF] Huyghe | D † -affinity of formal models of flag varieties[END_REF]Proposition 5.3.2] to prove the following proposition.

Proposition 7.0.3. Suppose ( ′ , ′ ) ⪰ ( , ) for pairs ( ′ , ′ ), ( , ) ∈ F with morphism ∶ ′ → . There exists a canonical morphism of sheaves of rings

which is -equivariant in the sense that for every ∈ we have

Proof. Let us denote by pr ′ ∶ ′ → ′ and pr ∶ → the blow-ups morphisms, and let us put pr ∶= pr • . We have the following commutative diagram

Here ′ , , ′ and denote the -scheme of finite type whose completions are ′ , , ′ and , respectively. The morphisms between these schemes will be denoted by the same letters, for instance pr ∶ → . We recall for the reader that the sheaf D ( , ′ ) ′ ( ) is filtered by locally free sheaves of finite rank

and therefore by the projection formula [17, Part II, Section 5, exercise 5.1 (d) ] and given that pr

because the direct image commutes with inductive limits on a noetherian space. By proposition 3.3.8 and the preceding relation we have a canonical map of filtered -algebras

in particular we get a morphism of sheaves of filtered -algebras (this is exactly as we have done in (17))

Applying Sym ( ) (•) • ′ pr ′ * (•) to the surjection ( 18) we obtain a surjection

which equals the associated graded morphism of (55) by proposition 4.1.5. Hence Φ ( , ′ ) ′ is surjective. On the other hand, if we apply pr * to the surjection

we obtain the surjection O ′ ⊗ ( ) ( ( )) → pr * D ( , ) ( ). Let us recall that ( ′ , ′ ) ⪰ ( , ) implies, in particular, that Lie( ′ ( ′ )) ⊆ Lie( ( )) and therefore ′ Lie( ′ ) ⊂ Lie( ). By ( 14), the preceding inclusion gives rise to an injective ring homomorphism ( ) ( ′ ( ′ )) → ( ) ( ( )). Let us see that the composition

Since by lemma 3.3.6 all those sheaves are -torsion free, this can be checked after tensoring with in which case we have that D

is the (push-forward of the) sheaf of algebraic differential operators on the generic fiber of ′ (cf. discussion given at the beginning of subsection 4.1). We thus get the canonical morphism of sheaves (54). Passing to completions we get a canonical morphism D( , ′ ) ′ ( ) → pr * D( , ) ( ). Taking inductive limit over all and inverting gives a canonical morphism D † ′ , ′ ( ) → pr * D † , ( ). Now, let us consider the formal scheme ′ as a blow-up of via pr. Then becomes a morphism of formal schemes over and we consider pr * D † , ( ) as the sheaf of arithmetic differential operators with congruence level defined on ′ via pr * . Using the invariance theorem (theorem 4. (i) For all ℎ, ∈ we have 24

(ii) For all open subsets U ⊆ , all ∈ D † , ( )(U ), and all ∈ M , (U ) one has ,U ( . ) = ,U ( ). ,U ( ). (iii) For all ∈ +1, the map ∶ M , → ( ) * M , = M , is equal to the multiplication by ∈ 0 ( , D † , ( )).

(b) For any two pairs ( ′ , ′ ) ⪰ ( , ) in F with morphism ∶ ′ → there exists a transition morphism

for any ∈ (where we have use the relation

in F , and if ′ , are ,0 -equivariant, then we require additionally that the morphism induced by

is an isomorphism of D † , ( )-modules. As in theorem 6.0.6, the morphisms

A morphism M → N between two coadmissible -equivariant arithmetic D( )-modules consists in a family of morphisms M , → N , of D † , ( )-modules, which respect the extra conditions imposed in ( ) and ( ). We denote the resulting category by C F , .

We recall for the reader that ( 0 , ) is a Fréchet-Stein algebra [START_REF] Schneider | Continuous and locally analytic representation theory[END_REF]Theorem 24.1]. Moreover, a ( , )-module is called coadmissible if it is coadmissible as a ( , )-module for every compact open subgroup ⊆ (cf. [33, Definition subsection 6]). Given that for any two compact open subgroups ⊆ ′ ⊆ the algebra ( ′ , ) is finitely generated free and hence coadmissible as a ( , )-module, it follows from [START_REF] Schneider | Algebras of -adic distributions and admissible representations[END_REF]Lemma 3.8] that the preceding condition needs to be tested only for a single compact open subgroup ⊆ . This motivates the following definition where we will consider the weak Freéchet-Stein structure of ( 0 , ) defined in [START_REF] Waterhouse | One-dimensional affine group schemes[END_REF]. Definition 7.0.5. We say that is a coadmissble ( , )-module if is coadmissisble as a ( 0 , )-module.

Let us construct now the bridge to the category of coadmisible ( , ) -modules. Let be such a coadmissible ( , ) -module and let ∶= ′ . We fix ∈ B a special vertex. Let ( ) • -an 25 be the subspace of ( ) • -analytic vectors and let , be its continuous dual. For any ( , ) ∈ F we have a coherent D † , ( )-module

, 24 Here we use the fact the action of on B is on the right and therefore ( ℎ ) * • ( ℎ ) * = ( ℎ ) * . 25 Here we use the fact that ( ) = .

and we can consider the family

.

On the other hand, given an object M ∈ C F , , we may consider the projective limit

with respect to the transition maps ′ , . Here the projective limit is taken in the sens of abelian groups and over the cofinal family of pairs ( , ) ∈ F with ,0 -equivariant , cf. remark 6.1.2.

Theorem 7.0.6. Let us suppose that ∈ Hom( , ) is an algebraic character such that + ∈ * is a dominant and regular character of . (and therefore, a dominant and regular character on every special vertex of B). The functors L (•) and Γ(•) induce quasi-inverse equivalences between the category of coadmissible ( , ) -modules and C F , .

The proof follows the same lines of reasoning given in [START_REF] Huyghe | D † -affinity of formal models of flag varieties[END_REF]Theorem 5.3.12].

Proof. The proof is an extension of the the proof of theorem 6.0.6, taking into account the additional -action. Let be a coadmissible ( , ) -module and let M ∈ C F , . The proof of the theorem follows the following steps.

Exactly as we have done in theorem 6.0.6, the family ( ) satisfies the requirements ( ), ( ) and ( ). Let us verify now condition ( ). Given ( ′ , ′ ) ⪰ ( , ) in F , we have ′ ( ′ ) • ⊆ ( ) • in rig and we denote by ̃ ′ , ∶ ′ , ′ → , the map dual to the natural inclusion ( ) • -an ⊆ ′ ( ′ ) • -an . Let U ⊆ ′ be an open subset and ∈ * D † ′ , ′ ( )(U ), ∈ ′ , ′ . We then define 27

) is the canonical morphism given by the preceding proposition. This definition extends to a map

which satisfies all the required conditions. The functoriality of L (•) can be verified exactly as we have done for the functor L 0 (•).

Claim 2. Γ(M ) is a coadmissible ( , ) -module. 26 Here we use ( ) • = -1 ( ) • in rig . 27 We avoid the subscript U in order to soft the notation.

Proof. We already know that Γ(M ) is a coadmissible ( ,0 , ) -module for any (theorem 6.0.6). So it suffices to exhibit a compatible -action on Γ(M ). Let ∈ . The isomorphisms ∶ M , → ( ) * M , , which are compatibles with transitions maps, induce isomorphisms at the level of global sections (which we denote again by to soft the notation)

Let us identify

Where we have abused of the notation and we have denoted by ′ , the morphism obtained by taking global sections on the morphism ′ , ∶ ( .

We want to see that .

and that this assignment defines a left -action on Γ(M ).

Taking global sections on (56) we get the relation

• , which implies that ′ , ( . ′ , ′ ) = ′ , ( ( ′ , ′ )) = ( ′ , ( ′ , ′ )) = ( , ) = .

, .

We obtain an isomorphism

According to ( ) in ( ) we have the sequence

which tells us that ℎ.( . ) = (ℎ ). , for ℎ, ∈ and ∈ Γ(M ). This gives a -action on Γ(M ) which, by construction, is compatible with its various ( ,0 , )-module structures.

Proof. By theorem 6.0.6 we know that this holds as a coadmissible ( 0 , ) -module, so we need to identify the -action on both sides. Let be a special vertex. According to (58), the action

we obtain the identification

and therefore we get back the original action of on .

Proof. We know that L (Γ(M )) , = M , as D † , ( )-modules for any ( , ) ∈ F , cf. theorem 6.0.6. It remains to verify that these isomorphisms are compatible with the maps and ′ , on both sides. To do that, let us see that the maps on the left-hand side are induced by the maps of the right-hand side. Given

the corresponding map

equals the map

, M , ). The isomorphism L (Γ(M )) , ≃ M , are induced (locally) by ⊗  → .( | U ). Condition ( ) tells us that these morphisms interchange the maps , as desired. The compatibility with transitions maps ′ , for two models ( ′ , ′ ) ⪰ ( , ) in F is deduced in a entirely similar manner as we have done in theorem 6.0.6 and the fact that ′ , is linear relative to the canonical morphism Ψ ∶ * D † ′ , ′ ( ) → D † , .

This ends the proof of the theorem.

As in the case of the group 0 , we now indicate how objects from C F , can be realized as -equivariant sheaves on the -space ∞ . The following discussion is an adaptation of the discussion given in [31, 5. implies that if, by cofinality, we identify D( )( ) = lim ← ← ← ← ← ← ← ← ← ← ←( , )∈F D † , ( ) pr ( ) and we take projective limits in (60), then we get a ring homomorphism , ∶ D( )( ) → ( ) * D( )( ) which implies that the sheaf D( ) is -equivariant. Furthermore, from construction this -quivariant structure extends the 0 -structure defined in the subsection 6.1.

Finally, let us recall the faithful functor M ⇝ M ∞ from coadmissible 0 -equivariant arithmetic D( )-modules on F to 0 -equivariant D( )-modules on ∞ . If M comes from a coadmissible -equivariant D( )-module on F , then M ∞ is in fact -equivariant (as in (52), this can be proved by using the family of -linear isomorphisms ( ) ∈ . As in proposition 6.1.3, the preceding theorem gives us Theorem 7.0.8. Let us suppose that ∈ Hom( , ) is an algebraic character such that + ∈ * is a dominant and regular character of . The functor M ⇝ M ∞ from the category C F , to -equivariant D( )-modules on ∞ is a faithful functor. 7.0.9. Final remark. We end this work by remarking to the reader that the functors in proposition 6.1.3 and theorem 7.0.8 become fully faithful functors if we required that the objects in the target category carry a structure of locally convex topological D( )-modules (cf. [START_REF] Huyghe | D † -affinity of formal models of flag varieties[END_REF]Propositions 5.2.31 and 5.3.16]). In fact, following [25, 5.2.30] we can see that D( ) carries a natural structure of a sheaf of locally convex topological -algebras and more generally, if M ∈ C 0 , (resp. M ∈ C , ) then M ∞ becomes a 0 -equivariant (resp. a -equivariant) sheaf of locally convex topological -vector spaces, endowed with the structure of a topological D( )-module.