Andrés Sarrazola

A Beilinson-Bernstein theorem for twisted arithmetic differential operators on the formal flag variety

Keywords: Formal flag variety, twisted arithmetic differential operators, Beilinson-Bernstein correspondence. 5 The sheaves D

, concerning the D †affinity of the smooth formal flag variety of , of certain sheaves of -adically complete twisted arithmetic differential operators associated to an algebraic character, and the results of [27] concerning the calculation of the global sections.

Introduction

An important theorem in group theory is the so-called Beilinson-Bernstein theorem [START_REF] Beilinson | Localisation de -modules[END_REF]. Let us briefly recall its statement. Let be a semi-simple complex algebraic group with Lie algebra ℂ ∶= Lie( ). Let ℂ ⊂ ℂ be a Cartan subalgebra and ⊂ U ( ℂ ) the center of the universal enveloping algebra of ℂ . For every character ∈ * ℂ we denote by ⊂ the corresponding maximal ideal determined by the Harish-Chandra homomorphism. We put U ∶= U ( ℂ )∕ . The theorem states that if is the flag variety associated to and D , is the sheaf of -twisted differential operators on , then Mod (D , ) ≃ Mod(U ), provided that is a dominant and regular character. Here Mod (D , ) denotes the category of O -quasi-coherent D , -modules. Moreover, under this equivalence of categories coherent D , -modules correspond to finitely generated U -modules [3, théorème principal]. The Beilinson-Bernstein theorem has been proved independently by A. Beilinson and J. Bernstein in [START_REF] Beilinson | Localisation de -modules[END_REF] and, by J-L. Brylinski and M. Kashiwara in [START_REF] Brylinski | Démonstration de la conjecture de Kazhdan-Lusztig sur les modules de Verma[END_REF]. This result is an essential tool in the proof of Kazhdan-Lusztig's multiplicity conjecture [START_REF] Kazhdan | Representations of Coxeter groups and Hecke algebras[END_REF]. In mixed characteristic, an important progress has been achieved by C. Huyghe in [START_REF] Huyghe | D † -affinité de l'espace projectif[END_REF][START_REF] Huyghe | Un théorème de Beilinson-Bernstein pour les D-modules arithmétiques[END_REF] and Huyghe-Schmidt in [START_REF] Huyghe | D-modules arithmétiques sur la variété de drapeaux[END_REF]. They use Berthelot's arithmetic differential operators [START_REF] Berthelot | D-modules arithmétiques I. Opérateurs différentiels de niveau fini[END_REF] to prove an arithmetic version of the Beilinson-Bernstein theorem for the smooth formal flag variety over ℤ1 . In this setting, the global sections of these operators equal a crystalline version of the classical distribution algebra Dist( ) of the group scheme .

Let ℚ be the field of -adic numbers. Throughout this paper will denote a split connected reductive group scheme over ℤ , ⊂ a Borel subgroup and ⊂ a split maximal torus of . We will also denote by = ∕ the smooth flag ℤ -scheme associated to . In this work, we introduce sheaves of twisted differential operators on the formal flag scheme and we show an arithmetic Beilinson-Bernstein correspondence. Here the twist is respect to a morphism of ℤ -algebras ∶ Dist( ) → ℤ , where Dist( ) denotes the distribution algebra in the sense of [START_REF] Demazure | Groupes algébriques[END_REF]. Those sheaves are denoted by D † , . In particular, there exists a basis S of consisting of affine open subsets, such that for every ∈ S we have

D † , | ≃ D † .
In other words, locally we recover the sheaves of Berthelots' differential operators, [START_REF] Berthelot | D-modules arithmétiques I. Opérateurs différentiels de niveau fini[END_REF] (of course, this clarifies why they are called twisted differential operators).

To calculate their global sections, we remark for the reader that actually we dispose of a description of the distribution algebra Dist( ) as an inductive limit of filtered noetherian ℤ -algebras Dist(

) = lim ← ← ← ← ← ← ← ← ← ← → ∈ℕ ( ) ( ),
such that for every ∈ ℕ we have ( ) ( ) ⊗ ℤ ℚ = U (Lie( ) ⊗ ℤ ℚ ) the universal enveloping algebra of Lie( ) ⊗ ℤ ℚ . For instance (0) ( ) = U (Lie( )). In particular, every character ∶ Dist( ) → ℤ induces, via tensor product with ℚ , an infinitesimal character . The last one allows us to define ̂ ( ) ( ) as the -adic completion of the central reduction ( ) ( )∕( ( ) ( ) ∩ ker( + )), and we denote by † ( ) the limit of the inductive system ̂ ( ) ( ) ⊗ ℤ ℚ → ̂ ( ′ ) ( ) ⊗ ℤ ℚ . In this paper we show the following result

Theorem.

Let us suppose that ∶ Dist( ) → ℤ is a character of Dist( ) such that + ∈ * ℚ 2 ( ℚ ∶= Lie( ) ⊗ ℤ ℚ ) is a dominant and regular character of * ℚ . Then the global sections functor induces an equivalence between the categories of coherent D † , -modules and finitely presented † ( ) -modules.

This theorem is based on a refined version for the sheaves (of level m twisted differential operators which we will define later) D( ) , ,ℚ . As in the classical case, the inverse functor is determined by the localization functor

L † , (•) ∶= D † , ⊗ † ( ) (•),
with a completely analogous definition for every ∈ ℕ.

The first section of this work is devoted to fix some important arithmetic definitions. They are necessary to define an integral model (in the sense of definition 2.5.1) of the usual sheaf of twisted differential operators on the smooth flag variety ℚ ∶= × Spec(ℤ ) Spec(ℚ ), associated to the split connected reductive algebraic group ℚ ∶= × Spec(ℤ ) Spec(ℚ ). One of the most important is the algebra of distributions of level , which is denoted by ( ) ( ) and it is introduced in subsection 2.5. This is a filtered noetherian ℤ -algebra which plays a fundamental roll in the rest of our work.

Inspired by the works [START_REF] Ardakov | Representations of compact -adic analytic groups[END_REF], [START_REF] Bezrukavnikov | Localization of modules for a semisimple Lie algebra in prime characteristic[END_REF] and [START_REF] Borho | Differential operators on homogeneous spaces. II: relative enveloping algebras[END_REF], in the first part of the third section we construct our level m twisted arithmetic differential operators on the formal flag scheme . To do this, we denote by the commutative ℤ -Lie algebra of the maximal torus and by ℚ ∶= ⊗ ℤ ℚ . They are Cartan subalgebras of and ℚ , respectively. Let N be the unipotent radical subgroup of the Borel subgroup and let us define

̃ ∶= ∕N, ∶= ∕
the basic affine space and the flag scheme of . These are smooth and separated schemes over ℤ , and ̃ is endow with commuting ( , )-actions making the canonical projection ∶ ̃ → a locally trivial -torsor for the Zariski topology on . Following [START_REF] Borho | Differential operators on homogeneous spaces. II: relative enveloping algebras[END_REF], the right -action on ̃ allows to define the level m relative enveloping algebra of the torsor as the sheaf of -invariants of * D ( ) ̃ :

D( ) ∶= * D ( ) ̃ .
As we will explain later this is a sheaf of ( ) ( )-modules and we will show that over an affine open subset ⊂ that trivialises the torsor, the sheaf D( ) may be described as the tensor product D ( ) | ⊗ ℤ ( ) ( ) (this is the arithmetic analogue of [11, page 180]).

Two fundamental properties of the distribution algebra Dist( ) are: it constitutes an integral model of the universal enveloping algebra U ( ℚ ) and Dist( ) = lim ← ← ← ← ← ← ← ← ← ← → ( ) ( ). These properties allow us to introduce the central reduction as follows. First of all, we say that an ℤ -algebra map ∶ Dist( ) → ℤ is a character of Dist( ) (cf. 3.5.3). By the properties just stated, it induces a character of the Cartan subalgebra ℚ 3

( =) ⊗ 1 ∶ ℚ → U ( ℚ ) = Dist( ) ⊗ ℤ ℚ → ℚ .
Now, let us consider the ring ℤ as a ( ) ( )-module via . We can define the sheaf of level m twisted arithmetic differential operators on the flag scheme by

D ( ) , ∶= D( ) ⊗ ( ) ( ) ℤ . ( 1
)
This is a sheaf of ℤ -algebras which is an integral model of D . 4The second part of the third section is dedicated to explore some finiteness properties of the cohomology of coherent D ( ) , -modules. Notably important is the case when the character + ∈ * ℚ is dominant and regular. Under this assumption, the cohomology groups of every coherent D ( ) , -module have the nice property of being of finite -torsion. This is one of the central results in this work.

In section 4 we will consider the -adic completion of (1). It will be denoted by D( ) , and the sheaf D( ) , ,ℚ ∶= D( ) , ⊗ ℤ ℚ is our sheaf of level m twisted arithmetic differential operators on the formal flag scheme .

Let ( ℚ ) be the center of the universal enveloping algebra U ( ℚ ). From now on, we will assume that + ∈ * ℚ is dominant and regular, and that ∶ ( ℚ ) → ℚ is the central character induced by via the classical Harish-Chandra homomorphism. We will show that if Ker( + ) ℤ ∶= ( ) ( ) ∩ Ker( + ) and ̂ ( ) ( ) denotes the -adic completion of the central reduction ( ) ( ) ∶= ( ) ( )∕ ( ) ( )Ker( + ) ℤ , then we have an isomorphism of complete ℤ -algebras

̂ ( ) ( ) ⊗ ℤ ℚ ≃ ← ← ← ← ← ← ← → 0 , D( ) , ,ℚ .
Finally, in subsection 4.3 we will introduce the localization functor L ( ) , , from the category of finitely generated ̂ ( ) ( ) ⊗ ℤ ℚ -modules to the category of coherent D( ) , ,ℚ -modules as the sheaf associated to the presheaf defined by

U ⊆  → D( ) , ,ℚ (U ) ⊗ ̂ ( ) ( ) ⊗ ℤ ℚ ,
where is a finitely generated ̂ ( ) ( ) ⊗ ℤ ℚ -module. We will show theorem: Let us suppose that ∶ Dist( ) → ℤ is a character of Dist( ) such that + ∈ * ℚ is a dominant and regular character of ℚ .

(i) The functors L ( ) , and 0 ( , •) are quasi-inverse equivalences of categories between the abelian categories of finitely generated (left) ̂ ( ) ( ) ⊗ ℤ ℚ -modules and coherent D( ) , ,ℚ -modules.

(ii) The functor L ( ) , is an exact functor.

Section 5 of this work is devoted to treat the problem of passing to the inductive limit. In fact, if ≤ ′ the sheaves D( ) , ,ℚ and D( ′ ) , ,ℚ are related via a natural map D( ) , ,ℚ → D( ′ ) , ,ℚ and we can define D † , to be the inductive limit. Similarly, we dispose of canonical maps ̂ ( ) ( ) ⊗ ℤ ℚ → ̂ ( ′ ) ( ) ⊗ ℤ ℚ and we denote by † ( ) its limit. Introducing the localization functor L † , exactly as we have made before, we get an analogue of the previous theorem for the sheaves D † , .

The work developed by Huyghe in [START_REF] Huyghe | D † -affinité de l'espace projectif[END_REF] and by D. Patel, T. Schmidt and M. Strauch in [START_REF] Patel | Locally analytic representations and sheaves on the Bruhat-Tits building[END_REF], [START_REF] Patel | Locally analytic representations of GL(2, ) via semistable models of ℙ 1[END_REF] and [START_REF] Huyghe | D † -affinity of formal models of flag varieties[END_REF], shows that the Beilinson-Berstein theorem is an important tool in the following localization theorem [24, theorem 5.3.8]: if denotes the formal flag scheme of a split connected reductive group , then the theorem provides an equivalence of categories between the category of admissible locally analytic (ℚ )-representations (with trivial character!) [START_REF] Patel | Algebras of -adic distribution and admissible representations[END_REF] and a category of coadmissible equivariant arithmetic D-modules (on the family of formal models of the rigid analytic flag variety of ). Our motivation is to study this localization in the twisted case (in a work still in progress, we have proved the affinity of an admissible formal blow-up of for the sheaf of twisted arithmetic differential operators with a congruence level 5 . This is a fundamental result to achieve the previous localization theorem [START_REF] Huyghe | D † -affinity of formal models of flag varieties[END_REF]).

The case of a finite extension of ℚ seems to present several technical problems (the reader can take a look to 3.5.3 to see one of this obstacles). We will treat this case in a future work.

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.

Notation: Throughout this work will denote a prime number and ℤ the ring of -adic integers. Furthermore, if is an arbitrary noetherian scheme over ℤ and ∈ ℕ, then we will denote by ∶= × Spec(ℤ ) Spec(ℤ ∕ +1 ) the reduction modulo +1 , and by

= lim ← ← ← ← ← ← ← ← ← ← → the formal completion of along the special fiber. Moreover, if E is a sheaf of ℤ -modules on then its -adic completion E ∶= lim ← ← ← ← ← ← ← ← ← ← ← E ∕ +1
E will be considered as a sheaf on . Finally, the base change of a sheaf of ℤ -modules on (resp. on ) to ℚ will always be denoted by the subscript ℚ.

For instance E ℚ ∶= E ⊗ ℤ ℚ (resp. E ℚ ∶= E ⊗ ℤ ℚ ).

Arithmetic definitions

In this section we will describe the arithmetic objects on which the definitions and constructions of our work are based. We will give their functorial constructions and we will enunciate their most remarkable properties. For a more detailed approach, the reader is invited to take a look to the references [START_REF] Berthelot | Notes on crystalline cohomology[END_REF], [START_REF] Huyghe | D † -affinité de l'espace projectif[END_REF], [START_REF] Huyghe | D-modules arithmétiques, distributions et localisation[END_REF] and [START_REF] Berthelot | D-modules arithmétiques I. Opérateurs différentiels de niveau fini[END_REF].

Partial divided power structures of level m

Let ∈ ℤ be a prime number. In this subsection ℤ ( ) denotes the localization of ℤ with respect to the prime ideal ( ).

We start recalling the following definition [7, Definition 3.1]. Definition 2.1.1. Let be a commutative ring and ⊂ an ideal. By a structure of divided powers on I we mean a collection of maps ∶ → for all integers ≥ 0, such that

(i) For all ∈ , 0 ( ) = 1, 1 ( ) = and ( ) ∈ if ≥ 2.
(ii) For , ∈ and ≥ 1 we have

( + ) = ∑ + = ( ) ( ).
(iii) For ∈ and ∈ we have ( ) = ( ).

(iv) For ∈ we have ( ) ( ) = (( , )) + ( ), where (( ,

)) ∶= ( + )!( !) -1 ( !) -1 . (v) We have ( ( )) = , ( ), where , ∶= ( )!( !) -1 ( !) -.
Throughout this paper we will use the terminology: "( , ) is a PD-ideal", "( , , ) is a PD-ring" and " is a PD-structure on ". Moreover, we say that ∶ ( , , In particular, we dispose of a PD-structure on ( ) ⊂ ℤ . We let [ ] ∶= ( ) and we denote by (( ), [ ]) this PD-ideal.

) → ( , , ) is a PD-homomorphism if ∶ →
Let us fix a positive integer ∈ ℤ. For the next terminology we will always suppose that ( , , ) is a ℤ ( ) -PDalgebra whose PD-structure is compatible (in the sense of [5, subsection 1.2]) with the PD-structure induced by (( ), [ ]) (we recall to the reader that the PD-structure (( ), [ ]) always extends to a PD-structure on any ℤ ( ) -algebra [7, proposition 3.15] ). We will also denote by ( ) the ideal generated by , with ∈ .

Definition 2.1.3. Let be a positive integer. Let be a ℤ ( ) -algebra and ⊂ an ideal. We call a m-PDstructure on a PD-ideal ( , ) ⊂ such that ( ) + ⊂ .

We will say that ( , , ) is a m-PD-ideal of . Moreover, we say that ∶ ( , , , ) → ( ′ , ′ , ′ , ′ ) is a m-PD-morphism if ∶ → ′ is a ring morphism such that ( ) ⊂ ′ , and such that ∶ ( , , ) → ( ′ , ′ , ′ ) is a PD-morphism.

For every ∈ ℕ we denote by = + the Euclidean division of by , and for every ∈ we define { } ( ) ∶= ( ( )). We remark to the reader that the relation ! ( ) = (which is an easy consequence of (i) and (iv) of definition 2.1.1) implies that ! { } ( ) = . On the other side, the m-PD-structure ( , , ) allows us to define an increasing filtration ( { } ) ∈ℕ on the ring which is finer that the -adic filtration and called the m-PD-filtration. It is characterized by the following conditions [6, 1.3]:

(i) {0} = , {1} = .
(ii) For every ≥ 1, ∈ { } and ≥ 0 we have { } ∈ { } .

(iii) For every ≥ 0, ( + ) ∩ { } is a PD-subideal of ( + ). Proposition 2.1.4. [7, proposition 1.4.1] Let us suppose that is a ℤ ( ) -algebra endowed with a m-PD-structure ( , , ). Let be a -algebra and ⊂ an ideal. There exists an -algebra ( ) ( ), an ideal ⊂ ( ) ( ) endowed with a m-PD-structure ( ̃ , [ ]) compatible with ( , ), and a ring homomorphism ∶ → ( ) ( ) such that ( ) ⊂ . Moreover, ( ( ) ( ), , ̃ , [ ], ) satisfies the following universal property: for everyhomomorphism ∶ → ′ sending to an ideal ′ which is endowed with a m-PD-structure ( ′ , ′ ) compatible with ( , ), there exists a unique m-PD-morphism ∶ ( ( ) ( ), , ̃ , [ ]) → ( ′ , ′ , ′ , ′ ) such that • = .

Definition 2.1.5. Under the hypothesis of the preceding proposition, we call the -algebra ( ) ( ), endowed with the m-PD-ideal ( ̄ , ̃ , [ ]), the m-PD-envelope of ( , ). Moreover, there exists an -homomorphism ∶ → ( ) ( ) such that ( ) ⊂ , and universal for the -homomorphisms → ( ′ , ′ , ′ , ′ ) sending into a m-PD-ideal ′ compatible with ( , ) and such that ′{ +1} = 0.

Arithmetic differential operators

Let us suppose that ℤ is endowed with the m-PD-structure defined in example 2.1.2 (cf. [5, Subsection 1.3, example (i)]). Let be a smooth ℤ -scheme, and I ⊂ O a quasi-coherent ideal. The presheaves

 → ( ) (Γ( , I)) and  → ( ) (Γ( , I))
are sheaves of quasi-coherent O -modules which we denote by P ( ) (I) and P ( ) (I). In a completely analogous way, we can define a canonical ideal I of P ( ) (I), a sub-PD-ideal ( Ĩ, [ ]) ⊂ Ī, and the sequence of ideals ( Ī{ } ) ∈ℕ defining the m-PD-filtration. Those are also quasi-coherent sheaves [5,

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 (2)

1 ← ← ← ← ← ← ← ← ← → P ,( ) ← ← ← ← ← ← ← → O .

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]. Let be a ℤ -scheme, 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. Using the notation of section 2.1 we define

Γ ,( ) (L) ∶= P S (L),( ) (I) and Γ ,( ) (L) ∶= Γ ,( ) (L)∕ Ī{ +1} . ( 3
)
Those algebras are graded [22, 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 [7, 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(ℤ ) . In [22, 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-PDalgebra Γ ,( ) (I∕I 2 ) = Γ ,( ) (Ω 1 ). More exactly, we have canonical isomorphisms Γ ,(

) ∶= Γ ,( ) (Ω 1 ) ≃ ← ← ← ← ← ← ← → • (P ,( ) ) which, by definition, induce a graded isomorphism of algebras Sym ( ) (T ) ≃ ← ← ← ← ← ← ← → • D ( ) .
(4)

Arithmetic distribution algebra of finite level

As in the introduction, let us consider a split connected reductive group scheme over ℤ 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 ℤ -algebras ∶ ℤ [ ] → ℤ , given by the identity element of . We know that ∕ 2 is a free ℤ = ℤ [ ]∕ -module of finite rank. Let 1 , . . . , ∈ such that modulo 2 , these elements form a basis of ∕ 2 . The -divided power enveloping of (ℤ [ ], ) (proposition 2.1.4) denoted by ( ) ( ), is a free ℤ -module with basis the elements

{ } = { 1 } 1 . . . { }
, where ! { } = , for every = + and 0 ≤ < . These algebras are endowed with a decreasing filtration by ideals { } (subsection 2.1), such that

{ } = ⊕ | |≥ ℤ { } . The quotients ( ) ( ) ∶= ( ) ( )∕ { +1}
are therefore ℤ -modules generated by the elements { } with | | ≤ [5, Proposition 1.5.3 (ii)]. Moreover, there exists an isomorphism of ℤ -modules

( ) ( ) ≃ ⨁ | |≤ ℤ { } . Corollary 2.1.6 gives us for any two integers , ′ such that ≤ ′ a canonical surjection ′ , ∶ ′ ( ) ( ) → ( ) ( ).
Moreover, for every ′ ≥ , the universal property of the divided powers gives us a unique morphism of filtered ℤ -algebras

, ′ ∶ ( ′ ) ( ) → ( ) ( ) which induces a homomorphism of ℤ -algebras , ′ ∶ ( ′ ) ( ) → ( ) ( ).
The module of distributions of level and order is ( ) ( ) ∶= Hom( ( ) ( ), ℤ ). The algebra of distributions of level m is

( ) ( ) ∶= lim ← ← ← ← ← ← ← ← ← ← → ( ) ( ),
where the limit is formed respect to the maps Hom ℤ ( ′ , , ℤ ). The multiplication is defined as follows. By universal property (Corollary 2.1.6) there exists a canonical application

, ′ ∶ + ′ ( ) ( ) → ( ) ) ⊗ ℤ ′ ( ) ( ).
If ( , ) ∈ ( ) ( ) × ( ) ′ ( ), we define . as the composition

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

< > with | | ≤ [26, proposition 4.1.6].
Remark 2.4.1. 6 Let A be an ℤ -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 -PD-envelope of (S( ∨ ), I( ∨ )). We put

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

{ 1 } 1 ... { } with ∑ ≤ [22, 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 ( ) ( ) ⊗ ℤ ℚ gives the relation

< > = ! ! . ( 5
)
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 ℤ -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

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

Integral models

We start this subsection with the following definition [5, subsection 3.4].

Definition 2.5.1. Let be a ℚ -algebra (resp. a sheaf of ℚ -algebras). We say that a ℤ -subalgebra 0 (resp.

a subsheaf of ℤ -algebras) is an integral model of if 0 ⊗ ℤ ℚ = .
Let us recall that throughout this paper denotes the Lie algebra of a connected reductive group scheme and U ( ) its universal enveloping algebra. As we have remarked in the introduction, if ℚ denotes the Lie algebra of the algebraic group ℚ = × Spec(ℤ ) Spec(ℚ ) and U ( ℚ ) its universal enveloping algebra, then it is known that U ( ) is an integral model of U ( ℚ ). Moreover, the algebra of distributions of level , introduced in the preceding subsection, is also an integral model of U ( ℚ ) [26, subsection 4.1]. This latest example will be specially important in this work.

In the following discussion we will assume that is a smooth ℤ -scheme endowed with a right -action.

Proposition 2.5.2. The right -action induces a canonical homomorphism of filtered ℤ -algebras

Φ ( ) ∶ ( ) ( ) → 0 ( , D ( ) ).
Proof. The reader can find the proof of this proposition in [26, Proposition 4.4.1 (ii)], we will briefly discuss the construction of Φ ( ) . The central idea in the construction is that if ∶ × ℤ → denotes the action, then the comorphism

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

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

Remark 2.5.3.

(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 ℤ -submodule of (left) -invariant global sections (cf. definition 3.2.4). This isomorphism induces a bijection between ( ) ( ) and 0 ( , D ( ) , ) , and it is compatible when varying .

Let us define A ( ) ∶= O ⊗ ℤ ( ) ( ), and let us remark that we can endow this sheaf with the skew ring multiplication (see [START_REF] Berthelot | Notes on crystalline cohomology[END_REF] below) coming from the action of ( ) ( ) on O via the morphism Φ ( ) of proposition 2.5.2. This multiplication defines over A ( ) a structure of a sheaf of associative ℤ -algebras, such that it becomes an integral model of the sheaf of ℚ -algebras

U • ∶= O ℚ ⊗ ℚ U ( ℚ ).
To see this, let us recall how the multiplicative structure of the sheaf U • is defined (cf. [34, subsection 5.1] or [30, section 2]). Differentiating the right action of ℚ on ℚ we get a morphism of ℚ -Lie algebras

∶ ℚ → 0 ( ℚ , T ℚ ). (6)
This implies that ℚ acts on O ℚ by derivations and we can endow U • with the skew ring multiplication

( ⊗ )( ⊗ ) = ( ) ⊗ + ⊗ (7)
for ∈ , ∈ U ( ℚ ) and , ∈ O ℚ . With this product, the sheaf U • becomes a sheaf of associative algebras and given that ( ) ( ) is an integral model of the universal enveloping algebra U ( ℚ ), then A ( ) is also a sheaf of associative ℤ -algebras being a subsheaf of U • . We have the following result from [START_REF] Huyghe | D-modules arithmétiques, distributions et localisation[END_REF]Corollary 4.4.6]. Proposition 2.5.4.

(i) The sheaf A ( ) is a locally free O -module.

(ii) There exist a unique structure over A ( ) of filtered O -ring and a canonical isomorphism of graded

O -algebras (A ( ) ) = O ⊗ ℤ Sym ( ) ( ). (iii) The sheaf A ( ) (resp. (A ( ) )) is a coherent sheaf of O -
( ) ( ) 0 ( , D ( ) ) U ( ℚ ) 0 ( ℚ , D ℚ ) Φ ( ) Ψ ℚ
where Ψ ℚ is the morphism induced by [START_REF] Berthelot | Introduction a la theorie arithmetique des D-modules, Cohomologies -adiques et applications arithmetiques[END_REF] and it is called the operator-representation [START_REF] Borho | Differential operators on homogeneous spaces[END_REF].

3 Integral twisted arithmetic differential operators

Torsors

Let us suppose that is a smooth affine algebraic group over ℤ with Lie algebra denoted by , and that ̃ and are smooth separated schemes over ℤ , such that ̃ is endowed with a right -action ∶ ̃ × Spec(ℤ ) → ̃ . We will also assume that acts trivially on . 7 We say that a morphism ∶ ̃ → is a -torsor for the Zariski topology, if is a faithfully flat morphism such that the diagram

̃ × Spec( ) ̃ ̃ 1
is commutative and the map (induced by the previous diagram)

̃ × Spec( ) → ̃ × ̃ ; ( , ℎ)  → ( , ℎ) (8)
is an isomorphism. 7 For example if ⊂ and = ∕ is the flag variety.

Let ⊂ be an affine open subset and pr 1 ∶ × Spec(ℤ ) → the first projection. We say that trivializes the torsor if there is a -equivariant isomorphism

∶ × Spec(ℤ ) ≃ ← ← ← ← ← ← ← → -1 ( ),
where acts on × Spec(ℤ ) by right translations on the second factor, and such that pr 1 = | -1 ( )

 → 1 ( -1 ( ), M).
As is locally trivial, the set S of affine open subsets of that trivializes the torsor is a base for the Zariski topology of . Moreover, if ∈ S then by definition -1 ( ) is an affine open subset of and given that M is a quasi-coherent O ̃ -module, we can conclude that 1 ( -1 ( ), M) = 0.

-equivariant sheaves and sheaves of -invariant sections

Let us denote by ∶ × Spec(ℤ ) → the group law of and by

1 ∶ ̃ × Spec(ℤ ) → ̃ and 1,2 ∶ ̃ × Spec(ℤ ) × Spec(ℤ ) → ̃ × Spec(ℤ )
the respective projections. We will also denote by

1 , 2 , 3 ∶ ̃ × Spec(ℤ ) × Spec(ℤ ) → ̃ the morphisms defined by 1 ( , 1 , 2 ) = , 2 ( , 1 , 2 ) = 1 and 3 ( , 1 , 2 ) = 1 2
. Following [32, chapter 0, section 3], we say that a sheaf M of O ̃ -modules is -equivariant if there exists an isomorphism

* 1 M Ψ ← ← ← ← ← ← ← ← → * M (9)
such that the following diagram is commutative (cocycle condition [20, (9.10.10)])

( ̃ × ) * * 1 M = * 1 M = * 1,2 * 1 M * 1,2 * M = * 2 M = ( × ) * * 1 M ( ̃ × ) * * M = * 3 M = ( × ) * * M. ( ̃ × ) * Ψ * 1,2 Ψ ( × ) * Ψ (10)
We will need the following lemmas. First of all, we recall for the reader that the category of -equivariant quasicoherent O ̃ -modules is an abelian category [19, Lemma 2.17]. In particular it is complete and cocomplete.

Lemma 3.2.1. Let M be an O ̃ -module filtered by a family (M ) ∈ℕ of -equivariant O ̃ -modules such that the inclusions M → M +1 are -equivariant. Then M is also -equivariant.

In fact, under the hypothesis on the lemma, the exactness of the functors * and * 1 allows us to conclude that * (M) and * 1 (M) are endowed with canonical filtrations ( * (M )) ∈ℕ and ( * 1 (M)) ∈ℕ , respectively. Since the components of this filtration have compatible -equivariant structures we can conclude that M is alsoequivariant via a filtered isomorphism. Lemma 3.2.2. Let (L, Ψ) be a -equivariant locally free O ̃ -module of finite rank. Then its dual L ∨ is also -equivariant.

Proof. Given that L is a locally free sheaf of finite rank, we dispose of a canonical and functorial isomorphism

* L ∨ ≃ ← ← ← ← ← ← ← → H O ̃ × ℤ ( * L, O ̃ × ℤ ).
This implies that (Ψ -1 ) ∨ , the dual of Ψ -1 , defines the -equivariant structure on L ∨ .

Since the category of -equivariant quasi-coherent O ̃ -modules is an abelian category, the cokernel of a -equivariant morphism M → N between two -equivariant quasi-coherent O ̃ -modules is again aequivariant quasi-coherent O ̃ -module. As we will demonstrate below, we can give an independent proof of this result. Lemma 3.2.3. Let (M, Φ 1 ) and (N , Φ 2 ) be -equivariant quasi-coherent O ̃ -modules. Let L be a quasicoherent O ̃ -module such that

0 → M ← ← ← ← ← ← ← → N ← ← ← ← ← ← ← ← → L → 0 is an exact sequence which induces a commutative diagram 0 * 1 (M) * 1 (N ) * 1 (L) 0 0 * (M) * (N ) * (L) 0. * 1 ( ) Φ 1 * 1 ( ) Φ 2 * ( ) * ( ) ( 11
)
Then L is also -equivariant.

Proof. To define the right vertical morphism, we consider a basis A of ̃ consisting of affine open subsets, which can be assumed stable under intersection because ̃ is separated. Let us fix ∈ A . As ( 11) is a diagram of quasi-coherent O ̃ × ℤ -modules, the horizontal short exact sequences remains exact when we take local sections on × ℤ . To soft the notation we will assume that

( ) ∶= * (M)( × ℤ ) ( ( ) ∶= * (N )( × ℤ ) and ( ) ∶= * (L)( × ℤ )) and ( 1 ) ∶= * 1 (M)( × ℤ ) ( ( 1 ) ∶= * 1 (N )( × ℤ ) and ( 1 ) ∶= * 1 (L)( × ℤ ))
. Also, we will suppose that

1 ∶= * 1 ( )( × ℤ ) (resp. 1 ∶= * 1 ( )( × ℤ )), 2 ∶= * ( )( × ℤ ) (resp. 2 ∶= * ( )( × ℤ )), Φ 1, ∶= Φ 1 ( × ℤ ) and Φ 2, ∶= Φ 2 ( × ℤ ).
In such a way that we have the following commutative diagram

0 ( 1 ) ( 1 ) ( 1 ) 0 0 ( ) ( ) ( ) 0 1 Φ 1, 1 Φ 2, 2 2
Let ∈ ( 1 ). By surjectivity of 1 we can find 1 ∈ ( 1 ) such that 1 ( 1 ) = . We define then Φ ( ) ∶= 2 (Φ 2, ( 1 )) ∈ ( ). Let us see that Φ is well-defined, this means that it does not depend of the choice of 1 ∈ ( 1 ). Let 2 ∈ ( 1 ) such that 1 ( 2 ) = . We want to see

Φ ( ) ∶= 2 (Φ 2, ( 1 )) = 2 (Φ 2, ( 2 )). Let ∶= 1 -2 ∈ ( 1 )
. By definition ∈ Ker( 1 ) = Im( 1 ), therefore there exists ∈ ( 1 ) such that 1 ( ) = . Let ′ = Φ 1, ( ) ∈ ( ). By commutativity of the diagram we have

2 ( ′ ) = 2 (Φ 1, ( )) = Φ 2, ( 1 ( )) = Φ 2, ( ) = Φ 2, ( 1 ) -Φ 2, ( 2 )
and therefore

0 = 2 ( 2 ( ′ )) = 2 (Φ 2, ( 1 )) -2 (Φ 2, ( 2 )) = Φ ( ) -2 (Φ 2, ( 2 )).
Moreover, it is straightforward to see that Φ is in fact a morphism of O ̃ × ℤ ( × ℤ )-modules, which by the well-known five lemma becomes an isomorphism. From the preceding reasoning, and the quasi-coherence of the sheaves involved, for every ∈ A we get an isomorphism

Φ ∶ * 1 (N )| × → * (L)| × of sheaves of O × -modules.
To complete the construction of the right vertical isomorphism we need to globalize the preceding reasoning. This means that if we chose , ∈ A then we have Φ

| ∩ = Φ | ∩ .
As A is stable under intersections we can construct, in the same way as before, an isomorphism Φ ∩ over ∩ . Let us see that Φ | ∩ = Φ ∩ (the reader can fallow the same reasoning to proof that Φ | ∩ = Φ ∩ ). We consider the following cube *

1 (N )( × ℤ ) * 1 (L)( × ℤ ) * 1 (N )( ∩ × ℤ ) * 1 (L)( ∩ × ℤ )
Except for the right lateral face, all the other faces form, by construction, commutative diagrams which implies that also the right lateral face forms a commutative diagram. This shows that Φ | ∩ = Φ ∩ . We have constructed an isomorphism Φ ∶ * 1 (L) → * (L) of quasi-coherent O ̃ × ℤ -modules. Let us show that Ψ defines a -equivariant structure. To do that we consider the following diagram

( ̃ × ) * * 1 N ( × ) * * 1 N ( ̃ × ) * * N = ( × ) * * N ( ̃ × ) * * 1 L ( × ) * * 1 L ( ̃ × ) * * L = ( × ) * * L ( ̃ × ) * * 1 * 1,2 Φ ( ̃ × ) * Φ ( × ) * Φ
The triangle on the top is commutative and by functoriality the lateral faces of the prism are also commutative.

Its is straightforward to show from this that

( ̃ × ) * Φ • ( ̃ × ) * * 1 = ( × ) * Φ • * 1,2 Φ • ( ̃ × ) * * 1 ,
and as ( ̃ × ) * * 1 is surjective we can conclude that the triangle on the bottom is also commutative. Therefore Ψ defines a -equivariant structure for L.

Let (M, Ψ) be a -equivariant O ̃ -module. By the Künneth formula [16, Theorem 6.7.8] we have a canonical isomorphism

0 ( ̃ × ℤ , * 1 M) ≃ 0 ( ̃ , M) ⊗ ℤ ℤ [ ]
which composing with the application

0 ( ̃ , M) ←→ 0 ( ̃ × ℤ , * M) 0 (Ψ) ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← → 0 ( ̃ × ℤ , * 1 M)
(the first application is induced via the canonical map M → * * M) gives us a morphism

Δ ∶ 0 ( ̃ , M) → 0 ( ̃ , M) ⊗ ℤ ℤ [ ],
defining a structure of -module on 0 ( ̃ , M). The co-module relations are given by the cocycle condition [32, chapter 0, definition 1.6].

Definition 3.2.4. The -invariant elements of 0 ( ̃ , M) are the elements ∈ 0 ( ̃ , M) such that Δ( ) = ⊗ 1. This subspace will be denoted by 0 ( ̃ , M) . Now, let us suppose that ̃ can be covered by a family S of affine open subsets, which are stable under finite intersection and invariant under the right action of . This means that for every ̃ ∈ S the morphism , inducing the right -action on ̃ , induces a right -action

̃ ∶= | ̃ × ℤ ∶ ̃ × ℤ → ̃ on ̃ . By pulling back Ψ under the inclusion ̃ × ℤ → ̃ × ℤ we get an isomorphism Ψ| ̃ ∶ ̃ * M ̃ → * 1 M ̃ which
satisfies the respective cocycle condition [START_REF] Borho | Differential operators on homogeneous spaces[END_REF], and, as before, we obtain a comodule map

Δ ̃ ∶ Γ( ̃ , M) → Γ( ̃ , M) ⊗ ℤ ℤ [ ].
As in definition 3.2.4, we can define the ℤ -submodule of -invariant sections on ̃ by

Γ( ̃ , M) = ∈ Γ( ̃ , M)| Δ ̃ ( ) = ⊗ 1 . ( 12
)
Finally, let us suppose that M is also quasi-coherent. By [18, Chapter II, Corollary 5.5] on every affine open subset ̃ ∈ S we can define a subsheaf

M| ̃ ∶= Γ( ̃ , M) ⊂ Γ( ̃ , M) = M| ̃ .
By definition, and giving that S was supposed to be stable under finite intersections, the preceding sheaves glue together to a subsheaf (M) ⊂ M which does not depend on the covering S. We sum up the preceding construction in the next definition.

Definition 3.2.5. Let ̃ be a smooth separated ℤ -scheme endowed with a right -action, and covered by a

family of affine open subsets S stable under finite intersections and the -action. For every -equivariant quasi-coherent O ̃ -module M, the subsheaf (M) is called the subsheaf of -invariant sections of M.

As an application of the preceding construction let us point out that if ∶ ̃ → is a locally trivialtorsor, then we actually dispose of a subsheaf of -invariant sections of the direct image sheaf * M, with M a -equivariant quasi-coherent O ̃ -module. In fact, if S denotes the collection of all affine open subsets that trivialises the torsor , then for every ∈ S we know that -1 ( ) is stable under the right -action and, as in [START_REF] Brylinski | Démonstration de la conjecture de Kazhdan-Lusztig sur les modules de Verma[END_REF], we can define

( * M)( ) = M( -1 ( )) ⊂ M( -1 ( )). ( 13
)
As ̃ is noetherian * M is quasi-coherent and therefore from [START_REF] Demazure | Groupes algébriques[END_REF] we have a subsheaf

( * M)| ∶= ̃ ( * M)( ) ⊂ ̃ * M ( ) = * M | .
Since S is stable under finite intersections, those sheaves glue together to define a subsheaf * M ⊂ * M.

For the rest of this subsection we will always suppose that ∶ ̃ → is a locally trivial -torsor.

Lemma 3.2.6. The morphism ∶ ̃ → induces an isomorphism

♮ ∶ O → * O ̃ .
Proof. As this is a local problem, we can take ∈ S and suppose that ∶ -1 ( ) = × Spec(ℤ ) → is the first projection. Since rational cohomolgy commutes with direct limits [28, Part I, Lemma 4.17] and O ( ) is a direct limit of free ℤ -modules, we can conclude that

* O ̃ ( ) = O ( ) ⊗ ℤ ℤ [ ] = O ( ).

Relative enveloping algebras of finite level

Let us fix a positive integer ∈ ℤ. As in the preceding subsections ̃ and will denote smooth separated ℤ -schemes, and a smooth affine commutative group scheme over ℤ . We will also assume that ∶ ̃ → is a locally trivial -torsor. We start this subsection recalling the construction of the -equivariant structures of the sheaf of level differential operators D ( ) ̃ .

Let 1 ∶ ̃ × Spec(ℤ ) → ̃ and 2 ∶ ̃ × Spec(ℤ ) → be the projections. For every ∈ ℕ the universal property of the m-PD-envelopes (proposition 2.1.6) gives us two canonical morphisms

1 ∶ * 1 P ̃ ,( ) → P ̃ × Spec(ℤ ) ,( ) and 2 ∶ * 2 P ,( ) → P ̃ × Spec(ℤ ) ,( ) .
Let J be the m-PD-ideal of the m-PD-algebra P ,( ) . We have a canonical m-PD-morphism

∶ P ̃ × Spec(ℤ ) ,( ) → P ̃ × Spec(ℤ ) ,( ) ∕ * 2 J
and ∶= • 1 is a m-PD-isomorphism. Then we dispose of a canonical section of is -equivariant and the inclusions

D ( ) ̃ , → D ( ) ̃ , +1
are -equivariant morphisms. In particular, by lemma 3.2.1, the sheaf of level differential operators is -equivariant.

Remark 3.3.1. (Notation as at the end of subsection 2.3) Following the preceding lines of reasoning we can also show that, for every ∈ ℕ, there exists a -PD-morphism

′ 1 ∶ Γ ̃ × ,( ) → * 1 Γ ̃ ,( ) which is a section of the canonical -PD-morphism * 1 Γ ̃ ,( ) → Γ ̃ × ,( ) induced by 1 [26, Subsection 2.2.2]. Let Γ ( ) ∶ * Γ ̃ ,( ) → Γ ̃ × ,( ) be the canonical -PD-morphism induced by . Then Φ ′ ∶= ′ 1 •Γ ( ) is a -equivariant structure for Γ ̃ ,( )
. As before, this implies that ( ) (T ̃ ) is -equivariant.

Remark 3.3.2.

Although it is well-known that the tangent sheaf T ̃ is a -equivariant quasi-coherent sheaf, we point out to the reader that this can be proven using the preceding discussion. In fact, as P 0

̃ ,( ) = O ̃ and P 1 ̃ ,( ) = O ̃ ⊕ Ω 1 ̃ [26, Subsection 2.

(18)], we can apply the lemma 3.to the sequence

0 → O ̃ → O ̃ ⊕ Ω 1 ̃ → Ω 1 ̃ → 0
and lemma 3.2.2 gives us the -equivariance of T ̃ . In particular, we dispose of the sheaves (T ) and ( * T ̃ ) .

Let us recall the following discussion from [1, 4.4]. Let us suppose ∈ S and ∈ T ̃ ( -1 ( )) . This assumption in particular implies that is a -invariant vector field on -1 ( ) and therefore a -invariant endomorphism of O ̃ ( -1 ( )). Hence it preserves O ( -1 ( )) and by lemma 3.2.6 it induces a vector field ( ) ∈ T ( ).

We get then a map of O -modules

∶ * T ̃ → T .
On the other hand, differentiating the right -action on ̃ we obtain a ℤ -linear Lie homomorphism → T ̃ , which induces a map of O -modules

⊗ ℤ O → * T ̃ .
We get a complex of O -modules

⊗ ℤ O → * T ̃ ← ← ← ← ← ← → T which is functorial in ̃ [1, subsection 4.4].
Lemma 3.3.3. 8 If ∶ ̃ → is a locally trivial -torsor, then the preceding complex is in fact a short exact sequence

0 → ⊗ ℤ O → * T ̃ ← ← ← ← ← ← → T → 0.
Before starting the proof we recall for the reader the following relations, which come from the -equivariant structure of T [11, Lemma 2 ]

0 ( , T ) = ℤ [ ] ⊗ ℤ and 0 ( , T ) = . ( 15
)
Moreover, by [18, Section II, exercise 8.3] we also dispose of the local description

T × ℤ = T ⊗ ℤ O ⊕ O ⊗ ℤ T . ( 16
)
Proof of lemma 3.3.3. As the sheaves in the sequence are quasi-coherent it is enough to check exactness over an affine open subset ∈ S. First of all, since T ( ) is a locally free O ( )-module and O ( ) is a flat ℤ -algebra, we can conclude that T ( ) is an inductive limit of free ℤ -modules. Therefore

(T ( ) ⊗ ℤ ℤ [ ]) = T ( ) ⊗ ℤ (ℤ [ ]) = T ( ).
This relation, together with ( 15) and ( 16), allow us to conclude that

T ̃ ( -1 ( )) = T ( ) ⊕ O ( ) ⊗ ℤ . ( 17
)
Remark 3.3.4. The preceding lemma shows that * T ̃ is a locally free O -module of finite rank. In particular ( ) (( * T ̃ ) ) is well-defined.

Definition 3.3.5. Let ∶ ̃ → be a locally trivial -torsor. Following [11, page 180] we define the level relative enveloping algebra of the torsor to be the sheaf of -invariants of * D ( ) ̃ :

D( ) ∶= * D ( ) ̃ .
The preceding sheaf is endowed with a canonical filtration

Fil D( ) = * D ̃ , . (18)
Tensor product filtration. Let A be a filtered sheaf of commutative rings on a topological space [9, A: III. 2]. Let M and N be filtered A-modules [9, A: III. 2.5]. The sheaf of A-modules M ⊗ A N carries a natural filtration called the tensor product filtration and it is defined as follows. Let ∈ ℕ fix. For every ⊂ we let (M( ) ⊗ A( ) N ( )) be the abelian subgroup of M( ) ⊗ A( ) N ( ) generated by elements of type ⊗ with ∈ M ( ), ∈ N ( ), and such that + ≤ . This process defines a presheaf on and we let (M ⊗ A N ) be its sheafification. The sheaf M ⊗ A N becomes therefore a filtered sheaf of A-modules

0 (M ⊗ A N ) ⊆ ... ⊆ (M ⊗ A N ) ⊆ ... ⊆ M ⊗ A N .
Furthermore, for every open subset ⊂ we have a canonical map

• (M( )) ⊗ • (A( )) • (N ( )) → • (M( ) ⊗ A( ) N ( )) by putting ( ) ⊗ ( ) → ( ⊗ ) + , where ∈ M( ) --1 M( ), ∈ N ( ) --1 N ( ), and
( ) ∶= + -1 M( ) and ( ) ∶= + -1 N ( ).
Moreover, these morphisms are compatible under restrictions and therefore, by the universal property of the sheafification, we get a morphism of graded sheaves

• (M) ⊗ • (A) • (N ) → • M ⊗ A N (19)
which is surjective by [START_REF] Hushi | Zariskian filtrations[END_REF]Section I,6.13].

Proposition 3.3.6. For any ∈ S there exists an isomorphism of sheaves of filtered ℤ -algebras

D( ) | ≃ ← ← ← ← ← ← ← → D ( ) | ⊗ ℤ ( ) ( ).
Before starting the proof of the proposition let us consider the following facts. Let ∈ ℕ fix and ≤ . For the next few lines we will suppose that and are smooth ℤ -schemes and that = × Spec(ℤ ) . Let 1 and 2 be the projections. By following [START_REF] Huyghe | D-modules arithmétiques, distributions et localisation[END_REF] we have defined in page 18 two canonical applications

1 ∶ P ,( ) → * 1 P ,( ) and -1 2 ∶ P - ,( ) → * 2 P - ,( ) .
Locally, if ( 1 , ..., ) and ( ′ 1 , ..., ′ ′ ) are coordinated systems on and , respectively, then we obtain a coordinated system on by putting ( * 1 ( 1 ), ..., * 1 ( ), * 2 ( ′ 1 ), ..., * 2 ( ′ ′ ). We have

P ,( ) ≃ ⨁ | 1 |+| 2 |≤ O * 1 ( { 1 } ) * 2 ( ′ { 2 } ) ( ∶= 1 ⊗ -⊗ 1 and ′ ∶= 1 ⊗ ′ 1 -′ ⊗ 1).
In this case [26, subsection 2.2.2]

1 ⎛ ⎜ ⎜ ⎝ ∑ 1 , 2 1 , 2 * 1 ( { 1 } ) * 2 ( ′ { 2 } ) ⎞ ⎟ ⎟ ⎠ = ∑ 1 1 ,0 * 1 ( { 1 } )
(with a similar description for - 2 ) and we have an isomorphism

P ,( ) ≃ ← ← ← ← ← ← ← → ⨁ 0≤ ≤ * 1 P ,( ) ⊗ O * 2 P - ,( ) . ( 20
)
Moreover, since P ,( ) and P - ,( ) are locally free O-modules of finite rank, taking duals in [START_REF] Hotta | D-modules, perverse sheaves and representation theory[END_REF] we get a canonical isomorphism

D ( ) , ≃ ← ← ← ← ← ← ← → ⨁ 0≤ ≤ * 1 D ( ) , ⊗ O * 2 D ( ) , - (21)
Proof of proposition 3.3.6. Let ∈ S and let -1 ( ) ≃ × Spec(ℤ ) be a trivialization of over . We obtain the following isomorphisms of filtered ℤ -algebras * D ( )

̃ ( ) = D ( ) ̃ ( -1 ( )) ≃ D ( ) × ( × ) ≃ D ( ) ( ) ⊗ ℤ 0 ( , D ( ) ) ≃ D ( ) ( ) ⊗ ℤ ( ) ( )
where the first isomorphism follows from the fact that trivializes the -torsor , the second isomorphism becomes from ( 21) and the Kunneth formula [16, Theorem 6.7.8]), and the third isomorphism is given by ( ) in remark 2.5.3. Since the previous isomorphisms are compatible with restrictions to open affine subsets contained in , we obtain the desired isomorphism of sheaves of filtered ℤ -algebras.

Proposition 3.3.7.

If is a locally trivial -torsor, then there exists a canonical and graded isomorphism

( ) * T ̃ ≃ ← ← ← ← ← ← ← → • D( ) .
Proof. We will divide the proof into two cases. We will first consider the case = 0 and then we will generalize for all ∈ ℤ >0 .

Case 1. Let us suppose that = 0. By the remark given after the proposition 1.2.2 in [START_REF] Huyghe | D † -affinité de l'espace projectif[END_REF] we know that if L is a locally free O ̃ -module of finite rank, then Sym (0) (L) = S(L) is the symmetric algebra of L. By (4), which is true for every ∈ ℕ (cf. [22, Proposition 1.3.7.3]), we have a canonical isomorphism of graded O ̃ -algebras

S(T ̃ ) ≃ ← ← ← ← ← ← ← → gr • D (0) ̃ .
Applying the direct image functor * to the preceding isomorphism and then taking -invariants sections (both functors being exact by lemma 3.1.3 and the fact that is diagonalisable [28, Part I, Lemma 4.

3 (b)]) we get an isomorphism * S T ̃ ≃ ← ← ← ← ← ← ← → gr • * D (0) ̃ .
We remark for the reader that the left-hand side of the previous isomorphism is well defined by remark 3.3.1.

To complete the proof of the first case, we need to show that ( * S(T ̃ )) = S ( * T ̃ ) . To do that, we start by considering the canonical map of O -modules * T ̃ → * S T ̃ which induces, by universal property of S(•), a canonical morphism of graded O -algebras

S * T ̃ ← ← ← ← ← ← ← → * S T ̃ .
Let us see that is indeed an isomorphism. Let us take ∈ S. We have a commutative diagram

̃ ∶= -1 ( ) × ℤ ≃ p 1
which tells us that (cf. [18, Section II, exercise 8.3])

T ̃ = * T ⊕ p * 2 T = * T ⊕ O ̃ ⊗ ℤ . (22)
By ( 17), we have

S * T ̃ ( ) = S * T ̃ ( ) = S T ( ) ⊕ O ( ) ⊗ ℤ .
On the other hand, by [START_REF] Huyghe | D † -affinité de l'espace projectif[END_REF] we have the following relation

S(T ̃ ) = S * T ⊗ O ̃ S O ̃ ⊗ ℤ = * S T ⊗ O ̃ S O ̃ ⊗ ℤ (23)
which implies, by the projection formula [18, Chapter II, section 5, exercise 5.1 (d)] that Summing up, we have the following commutative diagram

S * T ̃ ( ) * S(T ̃ ) ( ) S T ( ) ⊕ O ( ) ⊗ ℤ S T ( ) ⊗ O ( ) S(O ( ) ⊗ ℤ ) ≃ ≃ ≃
which ends the proof of the first case because S is a base for the Zariski topology of .

Case 2. Let us suppose now that ∈ ℤ >0 . Exactly as we have done at the beginning of case 1, applying * to the isomorphism (4) and then taking -invariant sections, we get a canonical isomorphism of graded

O -algebras * Sym ( ) T ̃ ≃ ← ← ← ← ← ← ← → gr • * D ( ) ̃ .
We want to see that the map , built in case 1, induces an isomorphism

Sym ( ) * T ̃ ≃ * Sym ( ) T ̃ .
To do that, we take ∈ S and we begin by noticing that analogously to case 1 the relation [START_REF] Grothendieck | Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné): IV. Étude locale des schémas et des morphismes de schémas[END_REF] gives us

Sym ( ) * T ̃ ( ) = Sym ( ) T ( ) ⊕ O ( ) ⊗ ℤ . (25)
Moreover, the relation ( 17) and [START_REF] Huyghe | D † -affinité de l'espace projectif[END_REF]Proposition 1.3.5] give us

Sym ( ) T ̃ = Sym ( ) * T ⊗ O ̃ Sym ( ) O ̃ ⊗ ℤ
which, following the same arguments that in [START_REF] Huyghe | Un théorème de Beilinson-Bernstein pour les D-modules arithmétiques[END_REF] and [START_REF] Huyghe | D † -affinity of formal models of flag varieties[END_REF], implies that * Sym ( )

T ̃ ( ) = Sym ( ) T ( ) ⊗ O ( ) Sym ( ) O ( ) ⊗ ℤ . ( 26
)
Again, by [22, Proposition 1.3.5], we have that ( 25) and ( 26) are canonically isomorphic, so in order to globalize this map, which we denote by ( ) , we need to check that the following diagram is commutative

Sym ( ) T ( ) ⊕ O ( ) ⊗ ℤ S T ( ) ⊕ O ( ) ⊗ ℤ ⊗ ℤ ℚ Sym ( ) T ( ) ⊗ O ( ) Sym ( ) O ( ) ⊗ ℤ S T ( ) ⊗ O ( ) S O ( ) ⊗ ℤ ⊗ ℤ ℚ . ( ) ⊗ ℤ 1 ℚ
Shrinking if necessary, we can suppose that is endowed with a set of local coordinates 1 , ..., , in such a way that if T ( ) is generated as O ( )-module by the derivations 1 , ..., , and if 1 , ..., denotes a ℤ -basis of , then Sym ( ) T ( ) ⊕ O ( ) ⊗ ℤ is generated (as O ( )-module) by all the elements of the form < > ⋅ < >9 . In particular, ( )

< > ⋅ < > = ⊗ ℤ 1 ℚ < > ⋅ < > = ! ! ! ! ⊗ ℚ .
which shows that the preceding diagram is commutative. This ends the proof of the proposition.

Affine algebraic groups and homogeneous spaces

Let us suppose that is a split connected reductive group scheme over ℤ and is a split maximal torus in . As we know, the Lie algebra = Lie( ) is a -module via the adjoint representation [28, I, 7.18] and the decomposition into weight spaces has the form

Lie( ) = Lie( ) ⊕ ⨁ ∈Λ (Lie( )) .
Here Λ is the subset of ( ) = Hom( , ) of non-zero weights of Lie( ), this means the roots of with respect to .

For each ∈ Λ there exists a homomorphism ∶ → satisfying

( ) -1 = ( ( ) ), (27)
for any ℤ -algebra and all ∈ ( ), and such that the tangent map ∶ Lie( ) → (Lie( )) is an isomorphism [START_REF] Jantzen | Representation of algebraic groups[END_REF]II,1.2]. This homomorphism defines a functor  → ( ( )) which is a closed subgroup of and it is denoted by . By definition we have Lie( ) = (Lie( )) and by [START_REF] Huyghe | D-modules arithmétiques sur la variété de drapeaux[END_REF] it is clear that normalises . Now, let us choose a positive system Λ + ⊂ Λ. It is known that Λ + and -Λ + are unipotent and closed subsets of

Λ 10 [28, II, 1.7].
Let N be the closed subgroup of generated by all with ∈ Λ + . As we have remarked normalises N. We set

= N ⋊ (28)
a Borel subgroup of . With this terminology N is called the unipotent radical of . We put

̃ ∶= ∕N, ∶= ∕
for the corresponding quotients (the basic affine space and the flag scheme of [1, subsection 4.7]). Since ℤ is in particular a Dedekind domain, these are smooth and separated schemes over ℤ [1, Lemma 4.7 (a)].

Remark 3.4.1. For technical reasons (cf. Proposition 2.5.2) in this work we will suppose that the group , and the schemes ̃ and are endowed with the right regular -action. This means that for any ℤ -algebra and 0 , ∈ ( ) we have

0 • = -1 0 , 0 N( ) • = -1 0 N( ) and 0 ( ) • = -1 0 ( ).
Under this actions, the canonical projections → ̃ and → are clearly -equivariant.

Now, as normalises N we have N = N and therefore

( N( )). ⊂ ( )N( ),
for any ℤ -algebra , ∈ ( ) and ∈ ( ). This defines a right -action on ̃ which clearly commutes with the right regular -action (cf. Remark 3.4.1). Moreover, this right -action makes the canonical projection

∶ ̃ → a -
]. This map gives a map

∶ → ̃ such that • = . The map ( , N)  → ( ) N is the required - invariant isomorphism × ≃ ← ← ← ← ← ← ← → × H ≃ ← ← ← ← ← ← ← → -1 ( ).

Relative enveloping algebras of finite level on homogeneous spaces

In this section we adopt the notation of the preceding section. In particular, we recall for the reader that the set S, of all affine open subsets of that trivialise the torsor forms a base for the Zariski topology of . Let us recall that by proposition 2.5.2 and remark 3.4.1 the right regular -action on ̃ (introduced in remark 3.4.1) induces a homomorphism Φ ( ) ∶ ( ) ( ) → 0 ̃ , D ( ) ̃ which equals the operator-representation (notation in subsection 2.5)

Ψ ̃ ℚ ∶ U ( ℚ ) → 0 ̃ ℚ , D ̃ ℚ
if we tensor with ℚ (D ̃ ℚ denotes the usual sheaf of differential operators on ̃ ℚ ). Let us consider the base change ℚ ∶= × Spec(ℤ ) Spec(ℚ ). We know by [START_REF] Jantzen | Representation of algebraic groups[END_REF]Part I,2.10 (3)] that

0 ̃ ℚ , D ̃ ℚ ℚ = 0 ̃ , D ( ) ̃ ⊗ ℤ ℚ . ( 29
)
Given that the right regular -action on ̃ commutes with the right action of the torus , the vector fields by which ℚ acts on ̃ ℚ must be invariant under the ℚ -action [START_REF] Borho | Differential operators on homogeneous spaces[END_REF]Lemma 4.5]. This means that the operator-

representation Ψ ̃ ℚ satisfies Ψ ̃ ℚ ( ℚ ) ⊂ 0 ( ̃ ℚ , D ̃ ℚ ) ℚ .
The relation [START_REF] Kazhdan | Representations of Coxeter groups and Hecke algebras[END_REF] tells us that for every ∈ ( ) ( ) there exists ( ) ∈ ℕ (a natural number that depends of ) such that

( ) Φ ( ) ( ) ⊂ 0 ( ̃ , D ( ) ̃ ) .
Since the -action on 0 ( ̃ , D ( ) ̃ ) is ℤ -linear, for every ℤ -algebra and every ∈ ( ) we have

( ) Φ ( ) ( ) = . ( ) Φ ( ) ( ) = ( ) .Φ ( ) ( ) . ( 30
)
Since ̃ is a smooth ℤ -scheme, the local description (2) tells us that the sheaf D ( ) ̃ is -torsion free. In particular, 0 ( ̃ , D ( ) ̃ ) is also -torsion free and therefore, from [START_REF] Miličić | Localization and representation theory of reductive Lie groups[END_REF], we have that Φ ( ) induces the filtered morphism

Φ ( ) ∶ ( ) ( ) → 0 ̃ , D ( ) ̃ = 0 , * D ( ) ̃ .
From the preceding reasoning we have an O -morphism of sheaves of filtered ℤ -algebras

Φ ( ) ∶ A ( ) → D( ) . ( 31
)
The sheaf A ( ) ∶= O ⊗ ℤ ( ) ( ) of associative ℤ -algebras has been introduced in the subsection 2.5. We recall for the reader that this is an integral model of the sheaf U • ∶= O ℚ ⊗ ℚ U ( ℚ ). To twist the sheaves D( ) introduced in the subsection 3.3 we consider the classical distribution algebra as in [START_REF] Demazure | Groupes algébriques[END_REF]Chapter II,4.6.1]. To define it, we suppose that ∶ Spec(ℤ ) → is the identity of and we take

∶= { ∈ ℤ [ ]| ( ) = 0}. Then ℤ [ ] = ℤ ⊕ . We put Dist ( ) ∶= [ ]∕ +1 * = Hom ℤ (ℤ [ ]∕ +1 , ℤ ) ⊂ ℤ [ ] *
the space of distributions of order , and then Dist( (i) The applications ℤ ( , ′ , ℤ ) ∶ ( ) ( ) → ( ′ ) ( ), with , ′ as in subsection 2.4, induce an

) ∶= lim ← ← ← ← ← ← ← ← ← ← → ∈ℕ Dist ( ). Moreover, if Δ ∶ ℤ [ ] → ℤ [ ] ⊗ ℤ ℤ [ ]
∶ ℤ [ ] Δ ← ← ← ← ← ← ← ← ← ← ← ← → ℤ [ ] ⊗ ℤ ℤ [ ] ⊗ ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← → ℤ , ∈ ℤ [ ] *
isomorphism of filtered ℤ -algebras lim ← ← ← ← ← ← ← ← ← ← → ∈ℕ ( ) ( ) ≃ ← ← ← ← ← ← ← → ( ).
(ii) The distribution algebra Dist( ) is an integral model of U ( ℚ ), this means that Dist( ) ⊗ ℤ ℚ = U ( ℚ ).

Example 3.5.2. Let us suppose that = = Spec(ℤ [ , -1 ]). In this case is generated by -1 and the residue classes of 1, -1, ..., ( -1) form a basis of ℤ [ ]∕ +1 . Let ∈ Dist( ) such that (( -1) ) = , (the Kronecker delta). By [START_REF] Jantzen | Representation of algebraic groups[END_REF]Part I,7.8] all with ∈ ℕ form a basis of Dist( ) and they satisfy the relation

! = 1 ( 1 -1)...( 1 -+ 1). ( 32
)
Therefore Dist( )

⊗ ℤ ℚ = ℚ [ 1 ]
. Since = ( ∕ 2 ) * we can conclude that Dist( )

⊗ ℤ ℚ = U ( ℚ ).
The preceding proposition in particular implies that every morphism of ℤ -algebras ∶ Dist( ) → induces for every ∈ ℕ a morphism of ℤ -algebras ∶ ( ) ( ) → ℤ .

Let us clarify the mysterious characters

∶ Dist( ) → ℤ . Let us suppose first that = = Spec(ℤ [ , -1 ]
). By the preceding example we know that the set of distributions { } ∈ℕ , where (( -1) ) = 0 if < and (( -1) ) = 1, is a basis for Dist( ). Moreover, Dist( )

⊗ ℤ ℚ = ℚ [ 1 ]
. Now, let us take ∈ * , which induces a morphism of algebras ∶ U ( ) → ℤ . Taking the tensor product with ℚ and using the canonical isomorphism Dist( ) ℚ ≃ U ( ℚ ) we obtain a character ∶ Dist( ) → ℚ (of course, here we assume Dist( ) ⊂ Dist( ) ℚ ). To see that its image is contained in ℤ , we need to check that ( ) ∈ ℤ . By [START_REF] Mumford | Geometric invariant theory[END_REF] we have

( ) = 1 =
( 1 ) ∈ ℤ .

Where we have used the fact that binomial coefficients extend to functions from ℤ to ℤ and the fact that 1 ∈ . In the case of an arbitrary split maximal torus = × Spec(ℤ ) ... × Spec(ℤ ) (n-times), the reader can follow the same reasoning using the canonical isomorphism Dist( ) = Dist( ) ⊗ ℤ ... ⊗ ℤ Dist( ) (n-times) [START_REF] Jantzen | Representation of algebraic groups[END_REF]Part I,7.9 (3)]. We have therefore a correspondence between the characters of (the Lie algebra of a split maximal torus ⊂ ) and the characters of the distribution algebra studied in this text

Hom ℤ -mods , ℤ ≃ ← ← ← ← ← ← ← → Hom ℤ -alg Dist( ), ℤ . ( 33

∑

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

We recall for the reader that ( ) ( ) is an integral model of the universal enveloping algebra U ( ℚ ). The relation given in [START_REF] Patel | Algebras of -adic distribution and admissible representations[END_REF] tells us that, as in the classical case (see for example [START_REF] Beilinson | A proof of Jantzen conjectures. A proof of Jantzen conjectures[END_REF] or [START_REF] Borho | Differential operators on homogeneous spaces[END_REF]), the sheaf D( ) ∶= * D ( ) ̃ can be regarded as a family of twisted differential operators on parametrized by * ∶= Hom( , ℤ ).

Before investigating the finiteness properties of these sheaves, let us study the particular case when

∈ * is an algebraic character. This means, is obtained by differentiating a character of 12 . As is shown in [START_REF] Jantzen | Representation of algebraic groups[END_REF]Part I,chapter 5,(5.8)], the character ∈ Hom( , ) induces an invertible sheaf L( ) on and we can consider the sheaf of (integral) differential operators acting on L( )

D ( ) ( ) ∶= L( ) ⊗ O D ( ) ⊗ O L( ) ∨ .
In fact the (left) action of D ( ) ( ) on L( ) is defined by

⊗ ⊗ ∨ • ∶= • ⟨ ∨ , ⟩ ( , ∈ L( ), and ∨ ∈ L( ) ∨ ).
Let us denote by D( ) ( ) its -adic completion (which is considered as a sheaf on ) and by D † ( ) its inductive limit tensored with ℚ 13 . These sheaves have been studied in [START_REF] Huyghe | Un théorème de Beilinson-Bernstein pour les D-modules arithmétiques[END_REF] and [START_REF] Huyghe | D-modules arithmétiques sur la variété de drapeaux[END_REF]. As before, we put D( ) ,ℚ ( ) ∶= D( ) ( ) ⊗ ℤ ℚ .

For the next result, we will abuse of the notation and we will suppose that ∈ Hom( , ) is an algebraic character of and that ′ ∶ Dist( ) → ℤ denotes the character of the distribution algebra Dist( ) induced by the bijection [START_REF] Patel | Algebras of -adic distribution and admissible representations[END_REF]. Before starting the proof let us recall the following facts 14 . First of all, the order filtration of D ( ) (definition 11 Here the on the right is the character of the ℚ -Lie algebra ℚ induced via the correspondence (33). 12 By abusing notation, we can rephrase this by = d ∈ * , with ∈ Hom( , ). 13 The inductive limit is formed by tensoring the maps [5, (2.2.1.5)] on the left with L( ) and on the right with L( ) ∨ . 14 This facts are detailed in [START_REF] Huyghe | Un théorème de Beilinson-Bernstein pour les D-modules arithmétiques[END_REF].

2.2.1) induces a filtration D ( ) , ( ) ∈ℕ of D ( ) ( ) by D ( ) , ( ) ∶= L( ) ⊗ O D ( ) , ⊗ O L( ) ∨ , such that gr • D ( ) ( ) ≃ Sym ( ) T .
It is easy to see that in order to achieve the preceding isomorphism we only need to use (4) and the commutativity of the symmetric algebra. On the other hand, by the work developed by Huyghe-Schmidt in [START_REF] Huyghe | D-modules arithmétiques sur la variété de drapeaux[END_REF] we have a canonical morphism of filtered ℤ -algebras

Φ ( ) ∶ A ( ) ∶= O ⊗ ℤ ( ) ( ) → D ( ) ( )
which is in fact surjective thanks to the preceding isomorphism and the fact that is an homogeneous space (cf. [START_REF] Huyghe | Un théorème de Beilinson-Bernstein pour les D-modules arithmétiques[END_REF]Section 1.6]).

Proof of proposition 3.5. [START_REF] Grothendieck | Élements de géométrie algébrique: III. Étude cohomologique des faisceaux cohérents (seconde partie)[END_REF]. By the preceding discussion and proposition 3.5.13 we have two canonical surjective morphisms of filtered ℤ -algebras

Φ ( ) ∶ A ( ) → D ( ) ( ) and Φ ( ) , ′ ∶ A ( ) → D ( ) , ′ .
In particular, there exist two-sided ideals J ( ) 1 and J ( )

2 of A ( ) such that D ( ) , ′ = A ( ) J ( ) 1 and D ( ) ( ) = A ( ) J ( ) 2 .
Moreover, since

D ( ) , ′ ⊗ ℤ ℚ = D ( ) ( ) ⊗ ℤ ℚ = D we have Φ ( ) ⊗ ℤ ℚ = Φ ( ) , ′ ⊗ ℤ ℚ and therefore J ( ) 2,ℚ = Ker(Φ ( ) ⊗ ℤ ℚ ) = Ker(Φ ( ) , ′ ⊗ ℤ ℚ ) = J ( ) 1,ℚ . ( 37
) Claim: There exist 1 , 2 ∈ ℕ such that 1 J ( ) 1 ⊆ J ( ) 2 and 2 J ( ) 2 ⊆ J ( ) 1 .
Proof. Let ⊆ be an affine open subset. Since A ( ) has noetherian sections over all affine open subsets (proposition 2.5.4) we can find 1 , ...,

∈ J ( ) 2 ( ) such that 1 1 , ..., 1 ∈ J ( ) 2 ( ) ⊗ ℤ ℚ generate J ( ) 2 ( ) ⊗ ℤ ℚ as A ( ) ( ) ⊗ ℤ ℚ -module. Let 1 , ..., ∈ J ( ) 1
( ) a set of generators of J ( ) 1 ( ) as A ( ) ( )-module. By (37) we can find 11 , ...,

∈ A ( ) ( ) ⊗ ℤ ℚ such that ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1 1 ⋅ ⋅ ⋅ 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 11 ⋅ ⋅ ⋅ 1 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 1 ⋅ ⋅ ⋅ ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1 1 ⋅ ⋅ ⋅ 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , ∶= ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 11 ⋅ ⋅ ⋅ 1 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 1 ⋅ ⋅ ⋅ ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ∈ M × A ( ) ( ) ⊗ ℤ ℚ .
In particular, we can find ∈ ℤ × and ∈ ℕ such that ∈ M × (A ( ) ( )), and therefore ∈ J ( ) 2 ( ), for all 1 ≤ ≤ . Now, as is compact, we can find a finite affine covering = ∪ 1≤ ≤ and we take

1 ∶= max | 1 ≤ ≤ , such that 1 J ( ) 1 ⊆ J ( ) 2
. By applying the same reasoning, we can see that there exists 2 ∈ ℕ such that 2 J ( ) 2 ⊆ J ( ) 1 .

Let us denote by Â( ) , Ĵ ( )

1
and Ĵ ( )

2
the respective -adic completions (considered as sheaves on ). By the preceding claim, we have the following diagram

Â( ) ,ℚ Ĵ ( ) 1,ℚ = Â( ) ,ℚ 1 Ĵ ( ) 1,ℚ Â( ) ,ℚ Ĵ ( ) 2,ℚ = Â( ) ,ℚ 2 Ĵ ( ) 2,ℚ Â( ) ,ℚ Ĵ ( ) 1,ℚ .
Given that A ( ) has noetherian sections over all open affine subsets, the -adic completion is in this case an exact functor and therefore

D( ) , ′ ,ℚ ≃ Â( ) ,ℚ Ĵ ( ) 1,ℚ ≃ Â( ) ,ℚ Ĵ ( ) 2,ℚ = D( ) ,ℚ ( ).

Finiteness properties

Let ∶ Dist( ) → ℤ be a character. In this section we start the study of the cohomological properties of coherent D ( ) , -modules. We follow the arguments of [START_REF] Huyghe | Un théorème de Beilinson-Bernstein pour les D-modules arithmétiques[END_REF] to show a technical important finiteness property about the -torsion of the cohomology groups of coherent D ( ) , -modules, when the character + ∈ * ℚ is dominant and regular (proposition 3.6.4). To start with, let us recall the twist by the sheaf O(1). As is a projective ℤ -scheme, there exists a very ample invertible sheaf O(1) on [18, chapter II, remark 5.16.1]. Therefore, for any arbitrary O -module E we can consider the twist

E ( ) ∶= E ⊗ O O( ),
where ∈ ℤ and O( ) 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 [18, chapter II, theorem 5.2 (b)].

We start the results of this section with the following proposition which states three important properties of coherent A ( ) -modules [26, proposition A.2.6.1]. This is a key result in this work. Let E be a coherent A ( )module. Proposition 3.6.1.

(i) 0 ( , A ( ) ) = ( ) ( ) is a noetherian ℤ -algebra.

(ii) There exists a surjection of 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 3.5.13, in a first time we will be concentrated on coherent A ( ) -modules. The next two results will play an important role when we consider formal completions.

Lemma 3.6.2. For every coherent A ( ) -module E , there exists = (E ) ∈ ℤ such that ( , E ( )) = 0 for every ≥ . (i) The functors L ( ) , and 0 ( , •) are quasi-inverse equivalence of categories between the abelian categories of finitely generated (left) ̂ ( ) ( ) ⊗ ℤ ℚ -modules and coherent D( ) , ,ℚ -modules.

(ii) The functor L ( ) , is an exact functor.

5 The sheaves D †

,

In this section we will study the problem of passing to the inductive limit when varies. Let us recall that ∶ ̃ ∶= ∕N → ∶= ∕ is a locally trivial -torsor (subsection 3.4). For every couple of positive integers ≤ ′ there exists a canonical homomorphism of sheaves of filtered rings [5, (2.

Moreover, by [5, (1.4.7.1)] we dispose of a canonical morphism P ̃ ,( ′ ) → P ̃ ,( ) .

In section 3.3 we have defined a -equivariant structure Φ ( ) ∶ * P ̃ , → * 1 P ̃ , on P ̃ , (w recall for the reader that denotes the right action of on ̃ and 1 the first projection). By universal property of P ̃ ,( ) the preceding -equivariant structures fit into a commutative diagram * P ̃ ,( ′ ) * This implies that the morphisms P ̃ ,( ′ ) → P ̃ ,( ) are -equivariant and therefore by lemma 3.2.1 and lemma 3.2.2, we can conclude that the canonical maps in (39) are -equivariant. In this way, we dispose of morphisms translations [26, theorem 4.4.8.3] and is compatible with variable. This means that if and ′ denote those isomorphisms for ≤ ′ , then we have a commutative diagram The flatness theorem [5, theorem 3.5.3] states that the lower morphism is flat and so is the morphism on the top.

From now on, we suppose that ∶ Dist( ) → ℤ is a character of Dist( ) such that + ∈ * ℚ is a dominant and regular character of ℚ . (ii) For all > 0 one has ( , E ) = 0.

Proof. 18 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( ) , ,ℚ E ≃ ← ← ← ← ← ← ← → E .
Now we use theorem 4.1.5 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

( , E ) = lim ← ← ← ← ← ← ← ← ← ← → ≥ , D( ) , ,ℚ ⊗ D( ) , ,ℚ E = 0,
for every > 0.

Proposition 5.2.7. 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. 19

Finally, if weCorollary 2 . 1 . 6 . [ 5 ,

 2165 endow ( ) ( ) ∶= ( ) ( )∕ { +1} with the quotient m-PD-structure [5, 1.3.4] we have Corollary 1.4.2] Under the hypothesis of the preceding proposition, there exists analgebra ( ) ( ) endowed with a m-PD-structure ( , ̃ , [ ]) compatible with ( , ) and such that

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 ∈ O ( ) and ∈ ℕ .

  Now we can apply [31, Chaper III, Proposition 4.1 (b)]. As in definition 3.1.2 we denote by S the set of all affine open subsets of that trivialise the torsor . This forms a base for the Zariski topology of .

  denotes the comorphism associated to the multiplication of , ∶ ℤ [ ] → ℤ is the counit associated to the identity element and * ∶ ℤ [ ] → ℤ [ ] is the coinverse (these maps defining a structure of Hopf algebra on ℤ [ ]), then the product

  defines a structure of algebra on ℤ [ ] * and Dist( ) is a subalgebra with Dist ( ).Dist ( ) ⊂ Dist + ( ) [28, Part I, 7.7]. Furthermore, Dist ( ) ≃ ℤ ⊕ ( ∕ +1 ) * . Proposition 3.5.1. [26, Subsection 4.1]

) 3 . 5 . 4 .

 354 Let us consider the positive system Λ + ⊂ Λ ⊂ ( ) ( ( ) the group of algebraic characters) associated to the Borel subgroup scheme ⊂ defined in the preceding subsection. The Weyl subgroup = ( )∕ acts naturally on the space * ℚ ∶= Hom ℚ -mod ( ⊗ ℤ ℚ , ℚ ), and via differentiation ∶ ( ) → * we may view ( ) as a subgroup of * in such a way that * ( ) ⊗ ℤ ℚ = * ℚ . Let = 1 2

Definition 3 . 5 . 5 .

 355 We say that a morphism of ℤ -algebras ∶ Dist( ) → ℤ (resp. the induced morphism ∶ ( ) ( ) → ℤ ) is a character of the distribution algebra ( ) (resp. a character of the level distribution Remark 3.5.14. (a) By construction D ( ) , ,ℚ = D is the sheaf of usual + 11 -twisted differential operators on the flag variety ℚ [11, page 170]. (b) Let us recall that the regular right action of on ̃ (cf. 3.4.1) induces a natural map Φ ∶ U ( ℚ ) → 0 ( ℚ , D ). This implies that if Φ ( ) denotes the canonical map induced by Φ ( ) , by taking global sections, then Φ ( ) ⊗ ℤ ℚ = Φ [11, Page 170 and 186].

Proposition 3 . 5 . 16 .

 3516 The sheaves D( ) ,ℚ ( ) and D( ) , ′ ,ℚ are canonically isomorphic.

Theorem 4 . 4 . 2 .Theorem 4 . 4 . 3 .

 442443 Let us suppose that ∶ Dist( ) → ℤ is a character of Dist( ) such that + ∈ * ℚ is a dominant and regular character of ℚ . The functors L ( ) , and 0 ( , •) are quasi-inverse equivalence of categories between the abelian categories of finitely generated D( ) , ,ℚ -modules and coherent D( ) , ,ℚ -modules. Proof. The proof of [22, Proposition 5.2.1] carries over word by word. Given that any equivalence between abelian categroies is exact, theorems 4.2.2 and 4.4.2 clearly imply (Principal theorem) Let us suppose that ∶ Dist( ) → ℤ is a character of Dist( ) such that + ∈ * ℚ is a dominant and regular character of ℚ .

  2.1.5)] ′ , ∶ D ( ) ̃ → D ( ′ ) ̃ . (39)Let us fix a character ∶ Dist( ) → ℤ . As we have remarked for ≤ ′ we have a commutative diagram

1 P

 1 ̃ ,( ′ ) * P ̃ ,(

  morphisms ′ , are obtained by dualizing the canonical morphisms ′ , in subsection 2.4 and the morphisms ′ , are defined in (39).

(Proposition 5 . 2 . 4 .

 524 ii) Again by remark 2.5.3 the isomorphism of proposition 3.5.8 are compatible for varying .Let us recall the following proposition. Proposition 5.2.3.[START_REF] Berthelot | D-modules arithmétiques I. Opérateurs différentiels de niveau fini[END_REF] 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.Proof. The previous proposition tells us that we only need to show that the morphisms ̂ * ( ′ , ) ℚ are flat. As this is a local property we can take ∈ S and to verify this property over the formal completion . In this case, remark 5.2.2 and the argument used in the proof of the first part of proposition 3.5.12 give us, by functoriality, the following commutative diagram

Lemma 5 . 2 . 5 .Proposition 5 . 2 . 6 .

 525526 For every coherent D † , -module E there exists ≥ 0, a coherent D( ) , ,ℚ -module E and an isomorphism of D † , -modules∶ D † , ⊗ D( ) ( ′ , E ′ , ′) is another such triple, then there exists ≥ max{ , ′ } and an isomorphism of D( ) This is [5, proposition 3.6.2 (ii)]. We remark that is quasi-compact and separated, and the sheaf D( ) , ,ℚ satisfies the conditions in[5, 3.4.1]. 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 † , -modulesD † , (-) ⊕ → E → 0.

E

  Theorem 4.1.5 gives us a coherent D( ) , ,ℚ -module E such that E ≃ D † , ⊗ D( ) has a resolution by finite free D( ) , ,ℚ -modules ( proposition 4.4.1). 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 ( , •).

Example 2.1.2. [

		is a
	homorphism of rings such that ( ) ⊂ and • | = • , for every ≥ 0.
	7, Section 3, Examples 3.2 (3)] Let |ℚ be a finite extension of ℚ , such that is its valuation
	ring and is a uniformizing parameter. Let us write =	, with a unit and a positive integer (called the
	absolute ramification index of ). Then ( ) ∶= ∕ ! defines a PD-structure on ( ) if and only if ≤ -1.

  , 2 ∶ O → P ,( ) endowing P ,( ) of a left and a right structure of O -algebra, respectively.

	P ,( ) ∶= P ( ) (I) is quasi-coherent and its support is contained in . In particular, it is independent of the
	open subset	[5, 2.1]. Moreover, by proposition 2.1.4 the projections 1 , 2 ∶	× ℤ	→	induce two
	morphisms 1 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 ≤ ′ corollary 2.1.6 gives us a canonical surjection P	′ ,( ) → P ,( ) which induces the injection D ( ) , →
	D ( ) , ′ and the sheaf of differential operators of level is defined by
						subsection 1.4].
	Now, let us consider the diagonal embedding Δ ∶	→ × ℤ	and let	⊂ × ℤ	be an open subset
	such that ⊂	is a closed subset, defined by a quasi-coherent sheaf I ⊂ O . For every ∈ ℕ, the algebra

  By[START_REF] Huyghe | D † -affinité de l'espace projectif[END_REF] Propositions 1.3.1, 1.3.3 and 1.3.6] we know that Sym ( ) (L) = ⊕ ∈ℕ Sym ( ) (L) is a commutative graded algebra with noetherian sections over any open affine subset. Moreover, locally over a basis 1 , ..., we have the following description Sym

O ,

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

	Remark 2.5.5. If we take the tensor product with ℚ in proposition 2.5.2, then by functoriality we get the
	following commutative diagram

  • . As is separated, the set S of open affine subschemes of that trivialises the torsor and such that O ( ) is a finitely generated ℤ -algebra, it is stable under intersections. Moreover, if ∈ S and is an open affine subscheme of , then ∈ S. We say that is locally trivially for the Zariski topology if can be covered by opens in S.Proof. We recall for the reader that R 1 * M is the sheaf associated to the presheaf[18, chapter III, prop. 8.1]

	Remark 3.1.1. Definition 3.1.2. Lemma 3.1.3. Let ∶ ̃ → be a locally trivial -torsor and let M be a quasi-coherent O ̃ -module. Then
	R 1	* M = 0.

  1 , named 1 ∶= -1 • [26,[START_REF] Dixmier | Enveloping algebras[END_REF]]. On the other hand, by functoriality, we obtain a morphism ∶

* P ̃ ,( ) → P ̃ × ℤ ,( )

and the -equivariant structure for P ̃ is defined by Φ ( ) ∶= 1

• [26, Proposition 3.4.1]. Definition 2.2.1 and lemma 3.2.2 allow us to conclude that for every ∈ ℕ the sheaf D ( ) ̃ ,

  torsor for the Zariski topology of . To see this we recall first that from[START_REF] Jantzen | Representation of algebraic groups[END_REF] the abstract Cartan group H ∶= ∕N is canonical isomorphic to . Let us consider the covering of given by the open

	subschemes	, ∈	∶=	( )∕ (the Weyl group) where
						∶= image of N
	under the canonical projection	→	[28, Part II, chapter 1, 1.9 (7)]. For every	∈	we can find a
	morphism	∶	→ splitting the projection map →	[28, Part II, chapter

In reality over a discrete valuation ring.

Here the character is induced via tensor product with ℚ and the correspondence[START_REF] Patel | Algebras of -adic distribution and admissible representations[END_REF].

In order to soft the notation through this work we will always denote by the character ⊗1 ℚ . This should not cause any confusion to the reader.

Tensoring with ℚ and restricting to the generic fiber ℚ → equals D ,[11, page 170].

The point here is that an admissible formal blow-up is not necessarily smooth and therefore, we can not use immediately Berthelot's differential operators. For the definition of these differential operators the reader is invited to visit[START_REF] Huyghe | Arithmetic structures for differential operators on formal schemes[END_REF].

This remark exemplifies the local situation when = Spec( ) with a ℤ -algebra [23, Subsection 1.3.1].

The reasoning is as in[START_REF] Ardakov | Representations of compact -adic analytic groups[END_REF] Lemma 4.4].

Here we use the multi-index notation introduced in sections 2.2 and 2.4.

Λ + ∩ (-Λ + ) = ∅ and (ℕ + ℕ ) ∩ Λ ⊂ Λ + for any , ∈ Λ + .

We follow the same argument given in[START_REF] Huyghe | D † -affinité de l'espace projectif[END_REF].

The proof as in[27, lemme 3.3].

This proof is exactly as in[24, proposition 4.3.1].

This is exactly as in[24, theorem 4.2.8]

This is exactly as in[22, theorem 5.1]

algebra ( ) ( )). We say that a character ∶ Dist( ) → ℤ (resp. a character ∶ ( ) ( ) → ℤ ) is a dominant and regular character if the ℚ -linear map induced by tensoring with ℚ is a dominant and regular character of ℚ .

Let ∶ Dist( ) → ℤ be a character of the distribution algebra of . We can consider the ring ℤ as a ( ) ( )-module via .

The reader can easily verify the following elementary lemma.

Lemma 3.5.6. Let be a ℚ -algebra and 0 ⊂ a ℤ -subalgebra such that 0 ⊗ ℤ ℚ = . If ( ) denotes the center of (resp. ( 0 ) denotes the center of 0 ), then ( ) = ( 0 ) ⊗ ℤ ℚ .

Let us consider D ̃ ℚ the usual sheaf of differential operators [START_REF] Grothendieck | Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné): IV. Étude locale des schémas et des morphismes de schémas[END_REF][START_REF] Berthelot | Notes on crystalline cohomology[END_REF] on ̃ ℚ ∶= ̃ × Spec(ℤ ) Spec(ℚ ) (resp.

ℚ ∶= × Spec(ℤ ) Spec(ℚ ) and ℚ = × Spec(ℤ ) Spec(ℚ )). By [28, Part I, 2.10 (3)] we have

On the other hand, we know by 2.5.2 that the right -action on ̃ induces a canonical morphism of filtered ℤ -algebras

and by [4, page 7] Φ ( ) ⊗ ℤ ℚ factors through the center of 0 ( ℚ , ( × ℤ ℚ ) * D ̃ ℚ ) ℚ . By [START_REF] Patel | Locally analytic representations and sheaves on the Bruhat-Tits building[END_REF] and the preceding lemma we have the following morphism

(we recall for the reader that ℚ ∶= Lie( ) ⊗ ℤ ℚ and that ( ) ( ) ⊗ ℤ ℚ = U ( ℚ ), for every ∈ ℕ).

Following the same lines of reasoning that in page 26 we can conclude that Φ ( ) induces a morphism of filtered ℤ -algebras Φ ( ) ∶ ( ) ( ) → 0 ( , ( D( ) )).

Here ( D( ) ) is the center of D( ) and its filtration is the one induced by (18). We have the following definition.

Definition 3.5.7. Let ∶ ( ) ( ) → ℤ be an integral character. We define the sheaf of level integral twisted arithmetic differential operators D ( ) , on the flag scheme by D ( ) , ∶= D( ) ⊗ ( ) ( ) ℤ .

If we endow ℤ with the trivial filtration as a ( ) ( )-module, this means 0 =∶ F -1 ℤ and F ℤ ∶= ℤ for all ≥ 0, then using (18) we can view D ( ) , as a sheaf of filtered ℤ -algebras, equipped with the tensor product filtration. Proposition 3.5.8. Let ∈ S. Then D ( ) , | is isomorphic to D ( ) | as a sheaf of filtered ℤ -algebras. Proof. Let us recall that by proposition 3.3.6 for every ∈ S we have an isomorphism of filtered ℤ -algebras

Remark 3.5.9. This proposition justifies the name of "twisted arithmetic differential operators".

Let us recall that, as is a smooth ℤ -scheme, the sheaf of Berthelot's differential operators 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 m twisted arithmetic differential operators on the formal flag scheme . Proposition 3.5.12.

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

The first and third isomorphism are given by [15, (0 , 3.2.6)] and the second one arises from the preceding proposition. By [5, 3.2.3 (iv)] the ring 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 [5, proposition 3.3.4] we can conclude that (ii) is an immediately consequence of (i) because Using the morphism Φ ( ) defined in [START_REF] Milne | Etale cohomology[END_REF] and the canonical projection from D( ) onto D ( ) , ( is ℤ -linear) we can define a canonical map

Proposition 3.5.13.

(i) There exists a canonical isomorphism ( ) (T ) ≃ • (D ( ) , ).

(ii) The canonical morphism Φ ( ) , is surjective.

(iii) The sheaf D ( ) , is a coherent A ( ) -module.

Proof. By [START_REF] Hashimoto | Equivariant sheaves and their applications to invariant theory[END_REF] we have a canonical map

By proposition 3.3.7 we know that • ( D( ) ) ≃ ( ) (( * T ̃ ) ). Moreover, by definition, we know that

2)-module. We obtain a morphism of sheaves of graded ℤ -algebras

(the structure of Sym ( ) ( )-module is guaranteed by ( 25)). Using the short exact sequence 0

is an isomorphism and we get a canonical morphism of ℤ -algebras

, By proposition 3.5.8, we have a commutative diagram for any ∈ S Sym ( ) T ( ) gr • D ( ) , ( )

here the left diagonal arrow is given by (4). As S is a basis for the Zariski topology of we can conclude that is an isomorphism.

For the second claim we can calculate • (Φ ( ) , ). By the first part of the proof and proposition 2.5.4 this morphism is identified with

which is surjective by [START_REF] Huyghe | Un théorème de Beilinson-Bernstein pour les D-modules arithmétiques[END_REF]Proposition 1.6.1]. Finally, item ( ) follows from ( ) and proposition 2.5.4.

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

Now, by the second part of proposition 3.6.1 there exist 0 ∈ ℕ and 0 ∈ ℤ together with an epimorphism of A ( ) -modules

If ≥ 0 -0 we see that ( , E 0 ( )) = 0. Now we way use this relation and the preceding proposition to follow word by word the reasoning given in [23, Proposition 2.2.1] to conclude the lemma. Lemma 3.6.3. For every coherent D ( ) , -module E , there exist = (E ) ∈ ℤ, a natural number ∈ ℕ and an epimorphism of D ( ) , -modules

Proof. Using the epimorphism in proposition 3.5.13 we can suppose that E is also a coherent A ( ) -module. In this case, by the second part of proposition 3.6.1, there exist = (E ) ∈ ℤ, a natural number ∈ ℕ and an epimorphism of A ( ) -modules

Taking the tensor product with D ( ) , we get the desired epimorphism of D ( ) , -modules

We recall to the reader that the distribution algebra of level , which has been denoted by ( ) ( ) in subsection 2.4, it is a filtered noetherian ℤ -algebra. This finiteness property is essential in the following proposition which we (personal) consider as the heart of this paper 15 .

NOTATION:

In the sequel we will refer to a character ∈ * ℚ as an ℚ -linear application induced, via base change, by a character ∶ Dist( ) → ℤ of the distribution algebra of the torus . (i) Let us fix ∈ ℤ. For every positive integer ∈ ℤ >0 , the cohomology group ( , D ( ) , ( )) has bounded -torsion.

(ii) For every coherent D ( ) , -module E , the cohomology group ( , E ) has bounded -torsion for all > 0.

Proof. To show ( ), we recall that D ( ) , ,ℚ = D is the usual sheaf of twisted differential operators on the flag variety ℚ (remark 3.5.14). 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 ( ) , ( )) ⊗ ℤ ℚ = 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 3.5.13, we know that D ( ) , ( ) is in particular a coherent A ( ) -module and hence, by the third part of the proposition 3.6.1 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.

To show ( ) the reader can follow the same arguments exhibited in [

Passing to formal completions

Let us start by recalling the formal completion of the sheaves introduced in the subsection 3.5. Let ∈ * ℚ be a character of the Cartan subalgebra ℚ (we recall for the reader that this means an ℚ -linear map induced by a character of Dist( )). We have introduced the following sheaves of -adically complete ℤ -algebras on the formal -adic completion of

The sheaf D( ) , ,ℚ is our sheaf of level m twisted arithmetic differential operators on the smooth formal flag scheme .

Cohomological properties

From now on, we will suppose that + ∈ * ℚ is a dominant and regular character of ℚ (to see 3.5.4 and definition 3.5.5). Our objective in this subsection is to prove an analogue of proposition 3.6.4 for coherent D( ) , -modules and to conclude that 0 ( , •) is an exact functor over the category of coherent D( ) , ,ℚ -modules.

Proposition 4.1.1. Let E be a coherent D ( ) , -module and Ê ∶= lim

E its -adic completion, which we consider as a sheaf on .

(i) For all ≥ 0 one has

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

(iii) The global section functor 0 ( ,

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 3.5.10), 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 [22, proposition 3.2].

The next proposition is a natural result from lemmas 3.6.2 and 3.6.3. Except for some technical details, the proof is exactly the same that in [24, proposition 4.2.2]. Therefore, we will only give a brief sketch of it. (i) There exists 2 = 2 (E ) ∈ ℤ such that, for all ≥ 2 there is ∈ ℤ and an epimorphism of D( ) , -modules

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

Proof. We start the proof of the part ( ) by remarking that the torsion subsheaf E of E is a coherent D( ) ,module. As is quasi-compact, there exists ∈ ℕ such that E = 0. Defining G ∶= E ∕E , G 0 ∶= G ∕G and E ∶= E ∕ +1 E , we have for every ≥ an exact sequence

Viewing as a closed subset of and denoting by this topological embedding, we can suppose that * G 0 is a coherent D ( ) , -module via the canonical isomorphism of sheaves of rings D ( )

(similarly, we can consider E as a coherent D ( ) , -module). By using the fact G 0 is also a coherent A ( ) -module, lemma 3.6.2 gives us an integers ′ 2 (G 0 ) such that the canonical maps 0 ( , E +1 ( ′ )) → 0 ( , E ( ′ ))

are surjective for ′ ≥ ′ 2 (G 0 ) and ≥ . Moreover, lemma 3.6.3 gives another integer ′ 3 (E ) such that, for every ′′ ≥ ′ 3 (E ) there exists ∈ ℕ and a surjection

From the previous morphisms and proceeding as in [24, proposition 4.2. (i)], we can define for every

To show the part ( ), we remark that if we fix 0 ∈ ℤ such that ( , O( )) = 0 for every > 0 and ≥ 0 , exactly as we have done in lemma 3.6.2, then via the second part of proposition 4.1.2 we also have that ( , D( ) , ( )) = 0 and the rest of the proof can be deduced exactly as in the proof of lemma 3.6.2.

The same inductive argument exhibited in the second part of proposition 3.6. The proof of the next theorem follows exactly the same lines than in [24, theorem 4.2.8]. We will reproduce the proof because it is a central result for our goal. (i) There is (E ) ∈ ℤ such that, for every ≥ (E ) there exists ∈ ℕ and an epimorphism of D( ) , ,ℚ -modules

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

Proof. By the preceding proposition, there exists a coherent D( ) , -module F such that F ⊗ ℤ ℚ ≃ E . Therefore, applying proposition 4.1.2 to F gives ( ). Moreover, as is a noetherian space, corollary 4.1.3 allows us to conclude that

for every > 0 [5, (3.4.0.1)].

Calculation of global sections

We recall for the reader that throughout this section + ∈ * ℚ denotes a dominant and regular character, which is induced by ∶ Dist( ) → ℤ via the correspondence [START_REF] Patel | Algebras of -adic distribution and admissible representations[END_REF]. In this subsection we propose to calculate the global sections of the sheaf D( ) , ,ℚ . Inspired in the arguments exhibited in [START_REF] Huyghe | D-modules arithmétiques sur la variété de drapeaux[END_REF], we will need the following lemma ( [27, lemma 3.3]) whose conclusion is an essential tool for our goal. Lemma 4.2.1. 16 Let be a noetherian ℤ -algebra, , two -modules of finite type, ∶ → an -lineal application and ̂ ∶ ̂ → ̂ the morphism obtained after -adic completion.

Proof. Let be the kernel (resp. the cokernel) of . Since the -adic completion is an exact functor over the finitely generated -modules [5, 3.2.3 (ii)], the -completion ̂ is the kernel (resp. the cokernel) of ̂ . But ̂ = because is of -torsion, and therefore ̂ ⊗ ℤ ℚ = ⊗ ℤ ℚ = 0.

Let us identify the universal enveloping algebra U ( ℚ ) of the Cartan subalgebra ℚ with the symmetric algebra ( ℚ ), and let ( ℚ ) denote the center of the universal enveloping algebra U ( ℚ ) of ℚ . The classical Harish-Chandra isomorphism ( ℚ ) ≃ ( ℚ ) (the subalgebra of Weyl invariants) [14, theorem 7.4.5], allows us to define for every linear form ∈ * ℚ a central character [14, 7.4.6]

we can consider the central redaction

, defined by taking global sections in [START_REF] Schmidt | On locally analytic Beilinson-Bernstein localization and the canonical dimension[END_REF], induces an isomorphism of ℤ -algebras

Proof. The key of the proof of the theorem is the following commutative diagram, which is an immediate consequence of remark 3.5.14

By the classical Beilinson-Bernsein theorem [START_REF] Beilinson | Localisation de -modules[END_REF] and the preceding commutative diagram, we have that Φ ( )

which becomes an isomorphism after tensoring with ℚ . The preceding lemma implies therefore that Φ ( ) gives rise to an isomorphism

and proposition 4.1.1 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. For this, we will first fix the following notation which will make more pleasant the reading of the proof of our principal theorem. We will consider D( ) , ,ℚ ∶= 0 , D( ) , ,ℚ . Now, let be a finitely generated D( ) , ,ℚ -module. We define L ( ) , ( ) as the associated sheaf to the presheaf on defined by

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

The arithmetic Beilinson-Bernstein theorem

We are finally ready to prove the main result of this paper. To start with, we will enunciate the following proposition. 17 Proof. By theorem 4.1.5 we know that E is a quotient of a module D( ) , ,ℚ (-) for some ∈ ℤ and some ∈ ℕ. We can therefore assume that E = D( ) , ,ℚ (-) for some ∈ ℤ. Let ∶= 0 ( , D ( ) , (-)), a finitely generated ( ) ( )-module by proposition 3.6.1. Let us consider the linear map of D ( ) , -modules equals to the composite

where the first map is the surjection induced by the map Φ ( ) of theorem 4.2.2. Let F be the cokernel of the composite map. Since ( ) ( ) is noetherian, the source of this map is a coherent D ( ) , -module and so is F . Moreover, this module is of -torsion because D ( ) , (-) ⊗ ℤ ℚ is generated by its global sections [START_REF] Beilinson | Localisation de -modules[END_REF]. Now, let us take a linear surjection ( ) ( ) ⊕ → . By tensoring with D ( ) , we obtain the exact sequence of coherent modules

Passing to -adic completions (which is exact in our situation [18, chapter II, proposition 9.1]) and inverting yields the linear surjection.

D( ) → D( ′ ) . The diagram (40) implies that we also have maps D ( ) , → D ( ′ ) , and therefore an inductive system

Definition 5.0.1. We will denote by D † , the limit of the inductive system (41), tensored with ℚ In a completely analogous way as we have done in the subsection referenced above, we define the localization functor L † , from the category of finitely presented D † , -modules to the category of coherent D † , -modules. This is, if denotes a finitely presented D † , -module, then L † , ( ) denotes the associated sheaf to the presheaf on defined by

It is clear that L † , is a functor from the category of finitely presented D † , -modules to the category of coherent D † , -modules. (ii) The functor L † , is an exact functor.

The arithmetic

To do this, we recall the following facts. which tells us that → 0 ( , L † , ( )) is an isomorphism. To show that if E is coherent D † , -module then the canonical morphism D † , ⊗ 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.

Calculation of global sections

Let us recall that in the subsection 4.2 we have used the fact that associated to the linear form ∈ * ℚ there exists a central character ∶ ( ℚ ) → ℚ , where ( ℚ ) denotes the center of the universal enveloping algebra U ( ℚ ). In this case, if Ker( + ) ℤ ∶= ( ) ( ) ∩ Ker( + ), we can consider the central redaction ( ) ( ) ∶= ( ) ( )∕ ( ) ( )Ker( + ) ℤ and its -adic completion ̂ ( ) ( ) . We have ( ) ( ) ⊗ ℤ ℚ = U ( ℚ ) . We have shown that there exists a canonical isomorphism of ℤ -algebras ̂ ( ) ( )

, D( ) , ,ℚ . Taking the inductive limit we can conclude that if † ( ) ∶= lim ← ← ← ← ← ← ← ← ← ← → ̂ ( ) ( ) , then we also have a canonical isomorphism of ℤ -algebras † ( ) ≃ ← ← ← ← ← ← ← → 0 ( , D † , ). Theorem 5.2.1 and this calculation complete the Beilinson-Bernstein correspondence.