Seyedvahid Amirinezhad
email: vahid.amirinezhad@cc.emu.edu.tr

Mustafa K Uyguroğlu
email: mustafa.uyguroglu@emu.edu.tr

KINEMATIC ANALYSIS OF GEARED ROBOTIC MECHANISM USING MATROID AND T-T GRAPH METHODS

In this paper, the kinematic structure of the geared robotic mechanism (GRM) is investigated with the aid of two different methods which are based on directed graphs and the methods are compared. One of the methods is Matroid Method developed by Talpasanu and the other method is Tsai-Tokad (T-T) Graph method developed by Uyguroglu and Demirel. It is shown that the kinematic structure of the geared robotic mechanism can be represented by directed graphs and angular velocity equations of the mechanisms can be systematically obtained from the graphs.

The advantages and disadvantages of both methods are demonstrated relative to each other.

B. INTRODUCTION

Kinematic and dynamic analysis of mechanical systems have been well established by using graph theory in recent years. Non-oriented and oriented graphs were used for this purpose. Nonoriented graph technique is mainly used for the kinematic analysis of robotic bevel-gear trains [1]-[4]. The oriented graph technique has been used for electrical circuits and other types of lumped physical systems including mechanical systems in one-dimensional motion since the early sixties [5]-[8]. Chou et al. [9] used these techniques to three-dimensional systems. Recently, Tokad developed a systematic approach, the so called Network Model Approach, for the formulation of three dimensional mechanical systems [10], and Uyguroglu and Tokad extended this approach for the kinematic and dynamic analysis of spatial robotic bevel-gear trains [11]. Most recently the oriented and non-oriented graph techniques were compared by Uyguroglu and Demirel [12], and the advantages of the oriented graph over the non-oriented graph were demonstrated using the kinematic analysis of bevel-gear trains. In order to overcome the weaknesses of the methods developed by Tsai and Tokad, both method were combined and the so called T-T Graph method was introduced []. On the other hand, Talpasanu et al. [] developed Matroid Method for the kinematic analysis of geared mechanisms based on directed graph as well.

In this paper the kinematic structures of the GRM is investigated with the aid of Matroid Method and T-T Graph method. Since both methods are used directed graph, the similarities and the differences are shown and the advantages of each system are indicated.

C. Geared Robotic Mechanism

 shows functional schematic of the GRM. It has 3 Degree of Freedom (i.e. it has 3 inputs) and the end-effector can have spatial motion (3 Dimensional) because this mechanism can generate two rotations about two intersected axes and one rotation of end-effector about its axis. In this mechanism 4, 5, and 6 are sun gears (input links), 1 and 2 are carriers and 3 , 7 , a n d 7   are planet gears. It is observed that links and joints (gear train) are

used to transmit the rotation of the inputs to the end-effector. The motion of end-effector is produced by links 4, 5, and 6 as inputs. The end-effector is attached to link 3 and carried by link 2. M1, M2 and M3 are actuators. The rotation of link 3 is caused by M3 through 6 and 7 links and the rotation of links 1 and 2 is made by M1 through link 4 and M2 through link 5 respectively.

II. MATROID METHOD

In this section, Matroid method […] is applied to the sample Geared Robotics Mechanism (GRM) to obtain the kinematic equations. First, its digraph is sketched and corresponding matrices are obtained. Then, we calculate relative angular velocities by using these matrices and Screw theory.

Associated digraph and corresponding matrices:

The mechanism in Figure 1 ). The following labeling, which is assigned to links and joints of sample mechanism, is used in Matroid method […]:  0 is assigned to ground link. Here, one could easily obtain the Incidence Node-Edge matrix from Figure 2:

0 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 1                                         Γ (1)
The entries of an Incident Node-Edge Matrix are 1, -1 and 0. Each column represents an directed edge which connects two nodes and contains two nonzero entries. The arrow head side is 1 and the other side is -1. Reduced Incident Node-Edge matrix Γ is obtained by deleting the first row where rows of this matrix are independent. Since k t c  , Γ consists of two sub-matrices: G and * G which correspond to turning and gear pairs respectively.

0 0 1 1 1 0 0 1 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 1                     * Г G G
(2). Now, regarding to associated digraph in Figure 2, we can obtain Path matrix as follows:

                                        Ζ (3)
1 0 1 0 0 0 0 1 0 0 0 0 1 1 0 1 0 0 0 1 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 1 C C C C                 С T U (5).
Actually, one can obtain Cycle-Basis matrix from Figure 2 directly.

1) Independent equations for relative angular velocities:

In this part, we obtain relative angular velocity equations of joints by means of Screw theory and then define output relative velocities in terms of inputs. Let consider the dual vector (Screw) in Eq. ( 6):

  0 0 , , , , 0 ,
ˆT k c k k k k c k c k c k ck L M N P Q R      u u r (6).
This 61  matrix can define spatial geometry of z-axes of local frames attached to k joints. Note that each of these local frames has unit vector

  0 0 1 T  u
with respect to itself. The first vector of Screw is unit vector of So components of 0 k u are:

0 ; s i n ; c o s k k k k k L M N      (9)
where k  are offset angles between z-axis of base and z-axes of turning axes. So, in sample mechanism: 8

9 1 0 1 3 1 1 1 2 1 4 0 a n d 9 0               
° and unit vectors of revolute joints are:

    0 0 0 0 0 0 0 8 9 1 0 1 3 1 1 1 2 1 4 0 0 1 a n d 0 1 0 TT        u u u u u u u
. The second vector is the position vector of with respect to the reference frame:

0 0 0 ,, c k c k k  r I u (10)
where

  0 , , , , T c k c k c k c k x y z  I
is distance vector which orients from c to k and it has the form:

  T k c k c k c x x y y z z    (11).
According to

c k c k k c k k c k c k P z y Q R      (12).
Table 1. Coordinates of turning and gear pairs .

5 7 7 1 4 1 2 1 1 6 3 2 2 1 1 1 2 3 1 1 3 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 0 0 0 0 0 0 0 0 0 0 0 ( ) 0 0 0 2 2 2 2 2 0 0 0 2 2 2 2 k k k x d d d d d d y B B B d d d d z A A A A A A A A           So for
Here, we define twist matrix as a product between screw and relative velocity variables of pairs

[…]: 00 , ˆk c k k  suθ (13)
where

k θ is partitioned into two sub-matrices,   8 9 1 0 1 1 1 2 1 3 1 4 T t         θ and   1 5 1 6 1 7 1 8 T c      θ
relates to velocities of turning and gear pairs respectively. Now, we can obtain relative angular velocity equations by applying Hadamard entry-wise product on Eq.

(5) and Eq. ( 13) as follows: . In […], to two orthogonality conditions, one for relative velocities and another for relative moments, are defined in order to Eq. ( 14) holds true. According to these two conditions, we can express equations of relative velocity variables of meshing and turning pairs in Eq. ( 15) and Eq. ( 16) respectively:

0 c t t      θ
T u θ [START_REF] Freudenstein | Kinematics and Statics of A Coupled Epicyclic Spur-Gear Train[END_REF] and because 0 ,

cc  r0 then 0 , c t t c
   Tr θ0 [START_REF] Hsu | Graph Notation for the Kinematic Analysis of Differential Gear Trains[END_REF] Since ,, 0

c t c t
QR  , we can simplify Eq. ( 16) in the following form:

, c t t c
   P θ0 [START_REF] Hsu | Graph Representation for the Structural Synthesis of Geared Kinematic Chains[END_REF] where , ct P is coefficient matrix:

,, c t c t P    PT [START_REF] Hsu | Automatic Analysis of Kinematic Structure of Planetary Gear Trains[END_REF] In Eq. ( 17), , ct P are coefficients in terms of pitch diameter. These scalar coefficients are used to acquire independent equations of relative angular velocities. For sample mechanism, these independent equations are expressed in Eq. ( 19):

Moreover, we will later rewrite these coefficients in terms of gear ratios in Eq. ( 24): ; ; ;

d d d d i i i i d d d d       (20)
Up to here, we obtain a set of independent equations for relative angular velocities of turning pairs including input pairs and output ones. Now according to Kutzbach criterion […] in Eq. [START_REF] Lang | Graph-Theoretic Modeling of Epicyclic Gear Systems[END_REF] , we can calculate output relative velocities in terms of input relative velocities:

E t r  (21)
where E is the number of inputs (Degree of Freedom), t is the number of turning pairs and r is the number of outputs (rank of Cycle-Basis matrix). So Eq. ( 17) is partitioned in the following form:

        1 r r E E      θ P P θ (23)
For sample mechanism in Figure 1, since

0 0 0 0 0 1 0 i ii ii i                                        (24)
Note that in Eq. ( 24), we changed the order of third and fourth columns in P matrix and third and fourth rows in θ vector compared with Eq. ( 19).

According to Eq. ( 23), we can obtain independent equations for output relative velocities in terms of gear ratio for the sample mechanism:

0 i i i i i i i i i i i i i i i i i i i                                   (25)

III. TSAI-TOKAD (T-T) GRAPH METHOD

On the functional schematic of the GRM shown in Figure 3: is:

1 3 1 3 21 2 3 2 3 21 M n M n w w                    (26)
Where 2 1

2 1 1 2 / 1 / n N N n  and 1
N and 2 N are teeth numbers of gear 1 and gear 2, respectively. The sign of the gear ratio n is determined based on the rotation directions of the gears. If both of the gears rotate in the same direction the sign is (+), otherwise the sign is (-).

In order to obtain the T-T graph representation of the GRM shown in Fig. 3, first, turning pairs are drawn by replacing them with their graph representations. The turning pairs axes labels are inserted into the graph as shown in Fig. 5. As it is seen from Fig. 5, the turning pairs constitute the tree branches.

Then, the graph representations of the gear pairs are drawn for completing the graph. At this point, it is required to determine the transfer vertices representing the carrier arms for the gear pairs (4,1), (5,2), [START_REF] Gibson | Symbolic Notation and Kinematic Equations of Motion of the Twenty-Two Basic Spur, Planetary Gear Trains[END_REF][START_REF] Uicker | Displacement Analysis of Spatial Mechanisms by an Iterative Method Based on 4x4 Matrices[END_REF] and [START_REF] Uicker | Displacement Analysis of Spatial Mechanisms by an Iterative Method Based on 4x4 Matrices[END_REF]3). In order to determine the transfer vertex, we will start from one of the node representing the gear in meshes and go through the tree branches to reach the other node representing the other gear. The vertex on this path, which has different levels on opposite sides is the transfer vertex. For the kinematic analysis of the GRM, we will consider only the angular velocities. Therefore the following terminal equations can be written by using Eq.( 26) for the gear pairs: [START_REF] Mruthyunjaya | Structural Analysis of Kinematic Chains and Mechanisms Based on Matrix Representation[END_REF] From the graph shown in Fig. 5, the following fundamental circuits (f-circuit) equations can be written easily. w can be obtained as:

1 0 1 0 1 4 4 0 4 1 4 0 1/nn w w w w       (39)
Eqs.( 35), [START_REF] Zawislak | Artificial Intelligence Aided Design of Gears[END_REF], and (39) yields:

  2 1 2 1 5 2 5 1 5 2 5 0 1 0 5 2 4 1 4 0 5 2 5 0 n n n n n w w w w w w w            (40)
For 32 w , Eqs.( 38), ( 30), ( 37), ( 36), ( 29), [START_REF] Tsai | Mechanisms design. Enumeration of kinematic structures according to function[END_REF], and (40) are used.

Or in more compact form:

If we use the same notations for gear ratios as shown in Eq.( 20) by using the following relations

Then the following equation is obtained which is the same as Eq.( 25)

Conclusion

In this paper, the kinematic equations of the GRM are obtained by using Matroid and T-T Graph Methods in sequence. Both methods use oriented graphs and represent the links with nodes. In both methods, representations of turning pairs constitute the tree branches and gear pair representations constitute the links are co-tree branches. Matroid method uses the oriented lines in order to obtain the incident and path matrices and cycle-basis matrix is derived from these two matrices. Then using screw theory kinematic equations of the mechanism is obtained. On the other hand, T-T graph carries more information than Matroid. Since each line is a part of terminal graph of either turning of gear pair it carries complementary terminal variables.

Therefore using terminal equations of the gear pairs, fundamental circuit and fundamental cut-set equations kinematic and static equations of the mechanism can be obtained. This example shows that kinematic equations of the GRM can be obtained more easily than Matroid Method by using T-T Graph method. The only advantage of the Matroid method is the derivation of the relative velocities of the gears with respect to carrier arm. Since it uses the diameter of the gears and the diameters are written relative to the reference frame, the sign of the gear ratio is obtained directly without considering the turning direction of the gears.

Figure 1 .

 1 Figure 1. The GRM mechanism.

 1 ,

 1 2, 3, 4, 5, 6, and 7 are assigned to gears and carriers.  8, 9, 10, 11, 12, 13, and 14 are assigned to turning joints.  15, 16, 17, and 18 are assigned to meshing joints.

Figure 2

 2 Figure 2 shows associated digraph of the sample mechanism. In this digraph, nodes present links

Figure 2 .

 2 Figure 2. Mechanism associated digraph.

Figure 3 :

 3 Figure 3: Functioal Schematic of the GRM.T-T Graph Representing of the GRM

Figure 4 :

 4 Figure 4: (a) Turning-pair and graph representation, (b) Gear-Pair and graph representation

Figure 5 :

 5 Figure 5: T-T Graph representation of the turning pairs of the GRM.

: vertex 2 (

 2 pair axes d, e).Therefore the sets of gear pair and corresponding carrier arm are (4,1)(0), (5,2)(1), (6,7)(2) and (7,3)(2).

Figure 6 Figure 6 .

 66 Figure 6 shows the T-T graph representation of the GRM. The thin lines representing the gear

  angular velocities can be determined in terms of input velocities by using the terminal and f-circuit equations. the GRM.Using Eqs.(33), (33), and (27) 10

  .

								8	9	1 0	1 1	1 2 1 3 1 4 1 5	1 6	1 7 1 8
								15
								16
								17
								18
	Path matrix	tn Z […], is a tn  matrix and comes from the spanning tree. Here, , z	(the entries
	of Path matrix) can be -1, 0, and +1. If edge t belongs to one of Spanning Tree's paths which are
	started from node n toward the ground link and its orientation is the same as path's direction,
	z	, tn	tn becomes +1. If it belongs to the path but the orientation is opposite, , z	becomes -1.
	z	tn ,		0	if edge does not belong to the related path.
	Spanning Tree Matrix is obtained by using the formula:
								T	*TT  G Z	(4).
	Then, Cycle-Basis matrix is obtained as:

Table 1

 1

	, components of position vector are:
	,	,	s i n	,	c o s	;	,	0 ;	,	0

  each cycle, the ,

	Cycle 16 C :	P	1 6 , 9		55 1 6 ,1 0 2 2 ; dd P 	;	P	1 6 ,1 2	 	2 2 d	;	P	1 6 ,1 6		0	.
	Cycle 17 C :	P	1 7 ,1 1		6 2 d	;	P	1 7 ,1 2		6 2 d	;	P	1 7 ,1 3		 	7 2 d 	;	P	1 7 ,1 7		0	.
	Cycle 18 C :	P	1 8 ,1 3	 	dd  73 1 8 ,1 4 22 ; P  	;	P	1 8 ,1 8		0
							P	ck	coefficients are calculated as follows:
	Cycle 15 C :	P	1 5 , 8	 	41 1 5 ,1 0 ; 22 dd P 	;	P	1 5 ,1 5	0  .


Laurent Clozel
email: clozel@math.u-psud.fr

Freydoon Shahidi

À L'

Anniversaire

Globally analytic p-adic representations of the pro-p-Iwahori subgroup of GL(2) and base change, I : Iwasawa algebras and a base change map

Introduction

This is the first part of a study of base change for globally analytic representations of the pro-p-Iwahori subgroup of GL(2) over an unramified extension of Q p . This paper contains an extension to such fields, and to the pro-p-Iwahori, of the presentation given in [START_REF] Clozel | Laurent Presentation of an Iwasawa algebra: the case of Γ 1 SL(2, Z p )[END_REF] of the Iwasawa algebra. We are led naturally ( §3.3) to a base change map; however, this map makes sense only for the algebras of global distributions on the rigid-analytic spaces; it is constructed in Chapter 4.

In the second part, we will follow the implications of this construction for certain p-adic representations of our groups. This Introduction concerns both papers.

1.1

We first recall some well-known facts about admissible complex representations of GL(n, F ) when F is a local field.

Assume first that F is p-adic. Let E/F be an unramified extension of degree r. There exists a base change map which to each admissible irreducible representation π of GL(n, F ) associates an admissible representation π E of GL(n, E) [1, §2.2] The local Langlands conjecture is known [START_REF] Harris | Richard The geometry and cohomology of some simple Shimura varieties[END_REF][START_REF] Henniart | Guy Une preuve simple des conjectures de Langlands pour GL(n) sur un corps p-adique[END_REF] and we know that π is associated to a representation r of degree n of the Weil-Deligne group W D F of F . (Here a systematic normalization of the correspondence must be specified, but this is inessential for us. For instance adopt the normalization of [START_REF] Harris | Richard The geometry and cohomology of some simple Shimura varieties[END_REF]Introduction]). We can restrict r to W D E , to r E , and it is known (indeed part of the proof) that r E is associated to π E .

There are two opposite kinds of representations. We can first consider the unramified principal series. Here π = ind G B (χ 1 , . . . , χ n ) (unitary induction) where G = GL(n, F ), B is a Borel subgroup, and the χ i are unramified characters of F × . Assume the χ i unitary, so the induced representation is irreducible. Then

π E = ind G E B E (η 1 , . . . η n )
where G E = GL(n, E), B E is its Borel subgroup, and

η i = χ i • N E/F .
In this case, base change is naturally associated to a homomorphism of Hecke algebras [START_REF] Langlands | Base change for GL(2)[END_REF]. Let H F , H E be the unramified Hecke algebras of compactly supported functions invariant by GL(n, O F ), GL(n, O E ). There is a homomorphism b : H E -→ H F and the action of ϕ ∈ H E on the spherical vector in π E is given by λ F • b, λ F : H F → C being the character associated to the spherical vector in π F . On the other hand, we can consider a supercuspidal representation π of GL(n, F ), associated to an irreducible representation r of W F (thus also of W D F .) In this case π has a minimal K-type -a type in the terminology of [START_REF] Bushnell | The admissible dual of GL(N) via compact open subgroups[END_REF], which defines a representation τ of GL(n, O F ) [START_REF] Paskunas | Vytautas Unicity of types for supercuspidal representations of GLN[END_REF]. This is uniquely associated to the inertial type of π, i.e., its equivalence class under twist by unramified characters of the determinant. This class corresponds bijectively to the restriction of r to the inertia I F ⊂ W F , an irreducible representation of I F . Thus we can see τ , a representation of GL(n, O F ), as being associated to r| I F .

The restriction of r to I E is not necessarily irreducible -of course, the cases of reducibility are easily understood. When it is, however, we see that the representation τ E occurring in π E is naturally associated to τ .

In this case, however, although the map π π E is compatible, via the base change "transfer" [1, §1.3], to a correspondence of functions ϕ f , there does not seem to exist, in general, a homomorphism of Hecke algebras realising this correspondence.

We end this introductory description with the case when F = R and E = C. In this case the local correspondence, with representations of degree n of W R (resp. W C ), has been proved by Langlands [START_REF] Langlands | On the classification of irreducible representations of real algebraic groups[END_REF] (completed by Speh [START_REF] Speh | Birgit Some Results on Principal Series of GL(n,R)[END_REF]). We can consider the restricted class of the finite-dimensional, algebraic, irreducible representations of GL(n, R).

If π is such a representation, on a complex vector space V , we can extend it holomorphically to GL(n, C), identifying the extension C of R with the complex field of coefficients : we still denote it by π. Let π be the complex conjugate. Then [START_REF] Clozel | Laurent Changement de base pour les représentations tempérées des groupes réductifs réels[END_REF]Prop. 3.2] π ⊗ π = π C is the representation of GL(n, C) associated by base change to π.

Finally, assume F is a finite field, E/F an extension of degree r, and fix an embedding of E into an algebraic closure F of F . Then Steinberg's theorem [START_REF] Steinberg | Robert Representations of algebraic groups[END_REF] tells us that any algebraic, irreducible representation π E of GL(n, E) (seen as an F -group) over F can be written

π E = π 1 ⊗ π ϕ 2 ⊗ • • • ⊗ π ϕ r-1 r
where π i is an irreducible algebraic representation of GL(n, F ), extended E-linearly to GL(n, E), and ϕ is the Frobenius automorphism acting in GL(n, E). If the π i are isomorphic to π, we can see π E as the base change of π. This is the construction that we will extend.

1.2

This paper arose from the natural attempt at extending to a p-adic field L the presentation given in [START_REF] Clozel | Laurent Presentation of an Iwasawa algebra: the case of Γ 1 SL(2, Z p )[END_REF] of the Iwasawa algebra of the level-1 subgroup

Γ 1 SL(2, O L ) = ker(SL(2, O L ) -→ SL(2, k L )) .
We extend this in two ways. First, following a suggestion of Gaëtan Chenevier (whom we heartfully thank) we consider instead the pro-p-Iwahori subgroup, denoted by I in this introduction :

I = {g ∈ GL(2, O L ) : ḡ ∈ N(k L )} where N = 1 * 0 1
. This is a maximal pro-p-subgroup of GL(2, L).

If L is unramified over Q p , we will give (for p > 5, see Chapter 2) a presentation of the Iwasawa algebra entirely similar to that given in [START_REF] Clozel | Laurent Presentation of an Iwasawa algebra: the case of Γ 1 SL(2, Z p )[END_REF].

To be more specific in this Introduction, assume now that k r is an extension of degree r of F p and that L is the fraction field of O L = W (k r ), the ring of Witt vectors.

Let Λ 1 , Λ r be the Iwasawa algebras for the pro-p-Iwahori subgroups, respectively in GL(2, Q p ) and GL (2, L). Then it appears immediately in the presentation obtained for Λ 1 , Λ r that there is a formal homomorphism b : Λ r -→ Λ 1 , exactly similar to that existing, in the classical situation between unramified Hecke algebras.

Here "formal" is taken in the colloquial, or informal sense. To explain this, note that, just as in [START_REF] Clozel | Laurent Presentation of an Iwasawa algebra: the case of Γ 1 SL(2, Z p )[END_REF], the algebra Λ 1 can be seen as a space of formal series

(1.1) f = n a n Y n 1 H n 2 W n 3 Z n 4
(n = (n 1 , n 2 , n 3 , n 4 ) ∈ N 4 ) with a n ∈ Z p . (See §2.) Similarly, an element ϕ ∈ Λ r can be seen as a formal series, with coefficients in Z p , in 4r variables.

(The variables, in each case, do not commute ! Their relations are given before Theorem 3.1.) Given ϕ we can naturally define an expansion (1.1) for a "distribution" f , but the coefficients a n do not belong to Z p , and are not even integral.

It is then natural to inquire for which spaces of distributions ϕ, f (subspaces of Λ r , Λ 1 ) this map is actually defined. It turns out that these spaces are the spaces D r , D 1 of global distributions on the pro-p-Iwahori subgroups I r , I 1 .

Recall that on a p-adic Lie group such as I r , we can consider the space D loc introduced by Schneider and Teitelbaum (who denote it by D) of distribution, dual to the space C an of locally analytic functions. Here, for p > 5, our pro-p-Iwahori subgroups have a richer structure : they are saturated pro-p-groups in the sense of Lazard, i.e., group objects in the category of rigid-analytic varieties, respectively over Q p and L. Cf. [START_REF] Lazard | Michel Groupes analytiques p-adiques[END_REF]III 1.3.4,III 3.3.2 ]. We can then consider the Q p -vector space A 1 of globally analytic functions on I 1 (• • • seen as a rigid-analytic space) and the Q p -vector space A r of globally analytic function on I r , seen by restriction of scalars as a rigid analytic space over Q p . The map which appeared formally in § 3.3 is then obtained naturally in § 4 as a map between the duals D r , D 1 of A r and A 1 . (Note that the spaces A and D are Banach spaces, unlike the topologically more complicated spaces of Schneider-Teitelbaum.) We must however, for this, extend scalars, so we obtain b : D r -→ D 1 ⊗ Qp L .

.

2 Iwasawa algebra : the case of Q p

2.1

Let L/Q p be a finite extension, and e its ramification index. We will assume, in this paragraph, that L is mildly ramified, i.e.

(2.1) 2e < p -1 .

In particular p ≥ 5. (Note that the condition (2.1) is adequate only because we are writing with GL(2). For GL(n) it would be ne < p -1, cf. [START_REF] Lazard | Michel Groupes analytiques p-adiques[END_REF]III.3.2.7.5]).

Let p, O denote the prime ideal, the integers of L. We denote by G the pro-p-Iwahori subgroup of GL(2, L), i.e.

G = g ∈ GL(2, O L ) : g ≡ 1 * 0 1 [p] .
Proposition 2.1 (Lazard).-G is a saturated p-valued pro-p-group.

We recall Lazard's proof. (Cf. also [START_REF] Schneider | Grundlehren der Mathematischen Wissenschaften[END_REF], p. 172.) Denote by v or v p the canonical p-valuation on any finite extension of Q p , i.e., v(p) = 1.

By (2.1) we can choose a rational number α such that

1 p -1 < α < 1 e - 1 p -1 .
Choose an extension L ′ of L containing an element a of (canonical) valuation

α. Let D ∈ GL(2, L ′ ) be the matrix 1 a ; let w be the natural valuation inf v(x ij ) on M 2 (L ′ ). We set, for g ∈ G, v G (g) = w(Dg D -1 -1) . For g = t x y u ∈ G, DgD -1 = t ax a -1 y u ).
This yields the formula

(2.2) v G (g) = inf(v(x) + α, v(y) -α, v(t -1), v(u -1) . Moreover, v(t -1) ≥ 1 e > 1 p-1 , v(u -1) ≥ 1 p-1 , v(x) + α ≥ α > 1 p -1 , v(y) -α ≥ 1 e -α > 1 p -1 . This implies [11, III 3.2] that g is sent into the subgroup of GL(2, L ′ ) where w(g -1) is a valuation. Moreover, it is saturated (if v G (g) > p p-1 , g is a p-th power) because this is true in GL(2, L ′ ) [11, III 3.2].

2.2

The group G has a triangular decomposition

G = UT N , U = 1 y 1 : y ∈ p T = t v : t, v ∈ O, t, v ≡ 1[p] N = 1 x 1 : x ∈ O .
For g = uhn, we obtain the expression (2.3) t tx ty txy + v .

Since p ≥ 5, the torus T is the direct product of the centre Z = z z (z ≡ 1) and of the torus T S in SL(2) :

T S = t t -1 (t ≡ 1) .
The valuation v G restricts to a valuation v S on the group S = G∩SL(2, L). In this group we have the expression

g = t tx ty txy + t -1 . Lemma 2.2.-For g ∈ S, v S (g) = inf(v(x) + α, v(y) -α, v(t -1)) .
We first prove :

Lemma 2.3.-For x ∈ O, y ∈ p, v(xy) > inf(v(x) + α, v(y) -α) . Assume first v(y) -α ≤ v(x) + α. Since v(xy) = (v(x) + α) + (v(y) -α) it suffices to prove that v(x) + α > 0, which is clear; if v(x) + α ≤ v(y) -α , we must check that v(y) > α ; since y ∈ p, v(y) ≥ 1 e > α .
We can now prove Lemma 2.2. Write z = txy + t -1 . Since |t| = 1, we must check :

(2.4) inf(v(x) + α, v(y) -α, v(z -1)) = inf(v(x) + α, v(y) -α, v(t -1)) . Write v = v(t -1), v 1 = v(z -1) , X = inf(v(x) + α, v(y) -α) . Then (2.4) is equivalent to v 1 ≥ X if v ≥ X v 1 = v if v < X . If v = v(t -1 -1) ≥ X, v(xy) > X by the previous lemma, so v(z -1) ≥ X. If v < X < v(xy), v(z -1) = v,
which proves the Lemma.

2.3

From now on we assume L unramified over Q p . Let e 1 , . . . e r be a basis of

O L over Z p . If x = x i e i ∈ O L (x i ∈ Z p ), we then have (2.5) v(x) = inf v(x i ) .
The elements e i define generators

y i = 1 pe i 1
of U, and

x i = 1 e i 1
of N. We can write the torus T as T = ZT S , the product being direct since L contains no square root of 1. We have generators for T S :

t i = (t ′ i ) -1 (t ′ i ) , t ′ i = (1 + pe i ) 1/2
and for Z :

z i = 1 + pe i 1 + pe i .
From Lemma 2.2 and (2.5) we immediately deduce

(note that v G (y i ) = 1 -α): Proposition 2.4.-The elements y i , t i , z i , x i (1 ≤ i ≤ r) form

an ordered basis of G, in the sense of Lazard : any element of G can be written uniquely

g = y η i i • t τ i i • z ζ i i • x ξ i i for exponents η i , . . . , ξ i ∈ Z p . Moreover (2.6) v G (g) = inf(v(η i ) + 1 -α, v(τ i ), v(ζ i ), v(ξ i ) + α) .
Remark.-Assume L = Q p and G is replaced by the group of [START_REF] Clozel | Laurent Presentation of an Iwasawa algebra: the case of Γ 1 SL(2, Z p )[END_REF] :

{g ∈ SL(2, Z p ) : g ≡ 1[p]}.
The same argument shows that the subgroups U, N, T S (with now x ∈ pZ p ) form an ordered basis ; furthermore, the formula (2.6) (omitting z) is true with α = 1. (This was omitted in [START_REF] Clozel | Laurent Presentation of an Iwasawa algebra: the case of Γ 1 SL(2, Z p )[END_REF].)

2.4

We now consider first the case where L = Q p . Thus

g = y η t τ z ζ x ξ
with obvious notations. Let Λ G be the Iwasawa algebra of G (with coefficients in Z p ). For simplicity, we will consider the group S : this is harmless as G is a direct product. We set, δ denoting a Dirac measure :

Y = δ(y) -1, H = δ(t) -1, X = δ(x) -1 .
The (ordered) decomposition of S as a product :

g = uhn (g ∈ S)
then, as in [START_REF] Clozel | Laurent Presentation of an Iwasawa algebra: the case of Γ 1 SL(2, Z p )[END_REF], implies that we can consider Y , H, X as distributions on S. Moreover, as a space Λ = Λ S is the completed tensor product

Λ U ⊗ Λ T ⊗ Λ N with basis over Z p the elements Y n 1 H n 2 X n 3 (n i ≥ 0). This is equal to the product Y n 1 • T n 2 • X n 3 in the Iwasawa algebra.
We determine the commutation relation between these elements. The computation is the same as in [START_REF] Clozel | Laurent Presentation of an Iwasawa algebra: the case of Γ 1 SL(2, Z p )[END_REF] ; we have changed the generator t in order to simplify the formulas. We have, with q = 1 + p :

(a) (1 + X)(1 + H) = (1 + H)(1 + X) q (b) (1 + H)(1 + Y ) = (1 + Y ) q (1 + H) . Moreover xy = 1 1 1 1 p 1 = 1 + p 1 p 1 .
Given the expression

g = t tx ty txg + t -1 , we have t = 1 + p = ((1 + p) -1/2 ) -2 , tpy ′ = p so y ′ = t -1 = (1 + p) -1 and likewise x = (1 + p) -1 . Thus (c) (1 + X)(1 + Y ) = (1 + Y ) Q (1 + H) -2 (1 + X) Q (Q = (1 + p) -1 ) .
The elements X, H, Y therefore verify in Λ the relations (a,b,c). We now consider the universal non-commutative p-adic algebra A = Z p {{Y, H, X}} over Y , H, X (cf. [START_REF] Clozel | Laurent Presentation of an Iwasawa algebra: the case of Γ 1 SL(2, Z p )[END_REF]), with maximal ideal M A generated by (p, Y, H, X) and a prime ideal P A generated by (Y, H, X). It contains the dense subalgebra A = Z p {Y, H, X} of non-commutative polynomials. Let R ⊂ A be the closed two-sided ideal generated by the relations (a, b, c); let Ā = F p {{X, Y, Z}} and R the image of R in Ā. As in [START_REF] Clozel | Laurent Presentation of an Iwasawa algebra: the case of Γ 1 SL(2, Z p )[END_REF] we have:

Lemma 2.5.-R is the closed two-sided in Ā generated by the relations (a, b, c)
There is a natural map A → Λ.

Lemma 2.6.-This extends to a continuous map A → Λ.

It suffices to show that if a sequence in A converges to 0 in the topology of A, the image converges to 0 in Λ.

The topology of A is given by the valuation v A ; write

F = n λ i x i ,
n ≥ a, i ranging over the applications maps {1, . . . , n} → {1, 2, 3}, and

(2.7)

x i = x i 1 x i 2 • • • x in ; we set x 1 = Y , x 2 = H, x 3 = X. For F = 0, v A (F ) = inf n,i (v p (λ i ) + n) .
Lazard [START_REF] Lazard | Michel Groupes analytiques p-adiques[END_REF]III 2.3] shows that Λ has an additive valuation (v(λµ

) = v(λ) + v(µ)) given by v(λ) = v p (λ)(λ ∈ Z p ), v(Y ) = 1 -α, v(H) = 1, v(X) = α. For F ∈ A, with image µ ∈ Λ, we have v(µ) = v n λ i x i ) ≥ inf n,i (v p (λ i ) + (1 -α)n(1) + n(2) + αn(3))
where n(1) is the number of elements in [1 . . . n] with image 1, etc. . .

≥ inf(α, 1 -α) v A (F ) .
This proves the continuity, whence the existence of the map A → Λ G . Note that this is sujective, since the ordered monomials Y n 1 H n 2 Z n 3 are obtained. The previous proof shows that it extends naturally to a map Ā → Ω, whose kernel contains R. We will show:

Theorem 2.8.-The map Ā → Ω gives an isomorphism B = Ā/ R ∼ -→ Ω.
Note that Ā has again an natural valuation, given on F = 0 by inf n n, the infimum being taken on the integers n such that there exists a non-zero monomial of degree n in the expression of F .

Exactly as in [START_REF] Clozel | Laurent Presentation of an Iwasawa algebra: the case of Γ 1 SL(2, Z p )[END_REF], Theorem 2.7 follows from Theorem 2.8. We now begin the proof of this last result. Note that

1 p -1 < 1 2 < p -2 p -1 since p > 3.
We can therefore take α = 1/2. Then

v(Y ) = v(X) = 1 2 , v(H) = 1 .
The Iwasawa algebra Λ is filtered by 1 2 N. As an F p -vector space, gr ν Λ is generated by the independent elements

p ℓ Y n 1 H n 2 X n 3 , ℓ + 1 2 n 1 + n 2 + 1 2 n 3 = ν .
(Cf. [14, p. 199].) The filtration of Λ defines a filtration of Ω by F il ν Ω = F il ν Λ ⊗ F p ; we see that gr ν Ω is then generated by the elements

Y n 1 H n 2 X n 3 , 1 2 n 1 + n 2 + 1 2 n 3 = ν ,
and these elements are linearly independent. We do not change the filtration by replacing ν ∈ 1 2 N by n = 2ν ∈ N, the valuations of (Y, H, X) being now (1,2,1).

In particular we now have (for n ∈ N) Lemma 2.9.-The dimension of gr n Ω (n ≥ 0) over F p is equal to the dimension of the space of homogeneous symmetric polynomials of degree n in (Y, H, X), the variables having degrees (1, 2, 1).

Note that this dimension is equal to (µ + 1)

(n + 1 -µ) where µ = [ n 2 ]
. We denote it by d n .

We must now consider on Ā the filtration of Ā obtained by assigning degrees (1,2,1) to the (formal) variables Y , H, X. Then gr n Ā is isomorphic to the space of non-commutative polynomials of degree n. We endow B = Ā/ R with the induced filtration, so

gr n B = F il n Ā/F il n+1 Ā + (F il n Ā ∩ R) .
The map gr n B → gr n Ω is surjective by construction. We now have:

Lemma 2.10.-dim gr n B ≤ d n (n ≥ 0).
The Lemma is obvious for n = 0, 1. We also note that the generators of R given by the relations (a, b, c) belong to F il 2 Ā. This is obvious for (a, b) since q ≡ 1 [p]; for (c) we obtain modulo terms of ordinary degree 2 in Y , H, X:

(1 + X)(1 + Y ) -(1 + Y )(1 + H) -2 (1 + X) = 2H
and H ∈ F il 2 Ā. Consider now gr 2 B, a quotient of the space

F il 2 Ā/F il 3 Ā with basis XY , Y X, H, X 2 , Y 2 . Since Q ≡ 1 [p], Q -1 ≡ 0 [p] and H is of degree 2, the relation (c) yields 1 + X + Y + XY = 1 + Y + X + Y X -2H modulo terms of degree ≥ 3. Thus dim gr 2 B ≤ d 2 = 4.
In order to prove the general case, we state the necessary relations in B:

Lemma 2.11.- (i) XY -Y X + 2H = 0 ( mod F il 3 ) (ii) XY -HX = 0 ( mod F il 4 ) (iii) HY -Y H = 0 ( mod F il 4 )
Part (i) has just been proven. For (ii) consider the identity (b). We have

(1 + X) q = 1 + qX + q(q -1) 2 X 2 + q(q -1)(q -2) 6 X 3 + • • • = 1 + X ( mod X 4 ) since q = 1 + p, p > 3. Thus (1 + X)(1 + H) = (1 + H)(1 + X) mod F il 4
since the filtration is multiplicative on Ā. Whence (ii), and similarly (iii).

We can now prove Lemma 2.10. Consider a non-commutative monomial -cf. (2.7):

(2.8)

x i = x i 1 x i 2 • • • x it .
Assume the homogeneous degree of x i is equal to n (H being of course of degree 2). As in [5, Lemma 3.2] we can change x i into a well-ordered monomial (α → i α increasing) by a sequence of transpositions. Consider a move (α, α + 1) → (α + 1, α) and assume i α > i α+1 . Write

x i = x j x α x α+1 x k write deg(i) = n, deg(j) = r, deg(k) = s.
There are three possibilities (2.9) (x α , x α+1 ) = (X, H)

Note that the degree of HX is 3. By Lemma 1.11 (ii), XH = HX mod F il 4 . Multiplying by x j and x k , we obtain x j HXx k (mod F il r+4+5 = F il n+1 ).

(2.10)

The case (H, Y ) is similar.

(2.11) (x α , x α+1 ) = (X, Y ). Now XY = Y X + 2H (mod F il 3 ), all elements being of degree 2. The first term yields again (mod F il n+1 ) a monomial with fewer inversions.

The second term yields a monomial with the same homogeneous degree, but with trivial degree -the number t in (2.8) -lowered by 1. We have to show that we have decreased the number of inversions in the monomial. For

x i , it was inv = β<γ i β >iγ 1 + γ>α+1 iα=3>iγ 1 + β<α i β >i α+1 =1 1 + 1
where the indices β, γ are different from α, α + 1. After contraction of XH into (2 times) H, we obtain

inv ′ = β<γ i β >iγ 1 + γ>α+1 iα=2>iγ 1 + β<α i β >2
1 Clearly inv ′ < inv. By performing the induction on the number of inversions (independently of the trivial degree), the proof of Lemma 1.10 is now completed.

Since gr n B → gr n Ω was surjective, the proof of Theorem 1.8 is complete.

Corollary 2.12.-The Iwasawa algebra Λ G is a quotient A/R, with A = Z p {{Z, Y, H, X}} and R defined by the relations (a, b, c) and (Comm) Z commutes with Y , H, X.

3 Iwasawa algebras in the unramified case

3.1

We return to the unramified extension L of Q p , of degree r; notations are as in § 2.3. In particular, the Lazard basis of G is given by Proposition 2.4. This depends on the choice of a basis (e i ) of W over Z p 1

As in §2.4, we will in fact compute the Iwasawa algebra Λ S (with Z pcoefficients) of S = G ∩ SL(2, L), since G is a direct product. Associated to the (ordered) decomposition g = uhn, we have a decomposition Λ S = Λ U ⊗Λ T ⊗Λ N (completed tensor product.) Each factor is a commutative Iwasawa algebra in r variables, respectively Y i , H i and X i , defined as in § 2.4. In particular Λ = Λ S has a topological basis.

(3.1) Y ℓ H m X n with ℓ = (ℓ 1 , . . . ℓ r ), Y ℓ = Y ℓ 1 1 • • • Y ℓn r , etc. As recalled, the elements Y ℓ (resp. H m , X n ) commute.
We want to write the other commutation relations. We have

h i y j h -1 i = 1 p(1 + pe i )e j 1 . Write (3.2) (1 + pe i )e j = k q ijk e k (q ijk ∈ Z p ). Then (3.3) q ijj ≡ 1 [p] q ijk ≡ 0 [p] (k = j).
The commutation relation implies

(A) (1 + X i )(1 + H i ) = (1 + H i ) r k=1 (1 + X k ) q ijk (B) (1 + H i )(1 + Y j ) = r k=1 (1 + Y k ) q ijk (1 + H i )
with the same exponents. We must now compute the "triangular" relation, the analogue of (c) in § 2.4. We have

x i y j = 1 e i 1 1 pe j 1 = 1 + pe i e j e i pe j 1 .
From the expression of the triangular decomposition (before Lemma 2.2):

g = t tx ty txy + t -1 we see that t = 1 + pe i e j = k (1 + pe k ) -1/2P ijk , e j = t k Q ijk e k e i = t k Q jik e k .
The exponents P ijk ∈ Z p are defined by

log(1 + pe i e j ) = - 1 2 k P ijk log(1 + pe k ).
Such an expression exists since exp and log(1 + X) are inverse diffeomorphisms between pW and 1 + pW . Furthermore we see that (3.4) e i e j ≡ -

1 2 k P ijk e k [p]
as p > 2. Moreover

(1 + pe i e j ) -1 e j = k Q ijk e k (3.5) (1 + pe i e j ) -1 e i = k Q jik e k
and the coefficients verify

Q ijj ≡ 1[p] (3.6) Q ijk ≡ 0[p] (k = j)
We now have the relation:

(C) (1 + X i )(1 + Y j ) = α,β,γ (1 + Y α ) Q ijα (1 + H β ) P ijβ (1 + X γ ) Q jiγ .
Now, as in §2.4, we consider in the non-commutative algebra A = Z p {{Y i , H j , X k }} the ideal R generated by the relations (A, B, C) and (Comm 0 ). T he variables Y i , H j , X k (separately) commute. If we consider, as in § 2.4, Ā and Ω, we have likewise:

We have again

Theorem 3.2.-The natural homomorphism Ā/ R -→ Ω is an isomorphism.
proof of Lemma 1.10 shows that the dimension of gr n B is smaller than the dimension of the space of symmetric polynomials of degree n in the Y i H i , X k (with the correct degrees.) This proves Theorem 3.2, and Theorem 3.1 follows.

Changing notations, we now have:

Corollary 3.5.-The Iwasawa algebra Λ G of G is the quotient of A = Z p {{Y i , H i , X i , Z i }}
by the ideal R generated by relations (A), (B), (C), (Comm 0 ) and (Comm 1 ) The variables Z i commute, and commute with Y i , H i , X i .

3.3

We now show that, at least formally (in the colloquial sense) there exists a map Λ Gr → Λ G 1 where G r is the group associated to L and G 1 the group associated to Q p . Denote by A r the algebra of non-commutative series in 4r variables of Corollary 3.5, and A the algebra of non-commutative series in 4 variables. We would like to define a map b: A r → A sending the ideal R r of relations to R. We work with the Iwasawa algebras Λ r , Λ of the group S.

Recall the relations defining A:

(a) (1 + X)(1 + H) = (1 + H)(1 + X) q (b) (1 + H)(1 + Y ) = (1 + Y ) q (1 + H) (c) (1 + X)(1 + Y ) = (1 + Y ) Q (1 + H) -2 (1 + X) Q where q = 1 + p, Q = (1 + p) -1
and A r :

(A) (1 + X j )(1 + H i ) = (1 + H i ) r k=1 (1 + X k ) q ijk (B) (1 + H i )(1 + Y j ) = r k=1 (1 + Y k ) q ijk (1 + H i ) (C) (1 + X i )(1 + Y j ) = α,β,γ (1 + Y α ) Q ijα (1 + H β ) p i β (1 + X γ ) Q jiγ ) where (1 + pe i )e j = k q ijk e k , ( 1
+ pe i e j ) -1 e j = k Q ijk e k , log(1 + pe i e j ) = -1 2 k P ijk log(1 + pe k ).
The map b is defined if we give the images of 1 + X i , 1 + H i , 1 + Y i . We first try to satisfy the relations (a, b).

The set A × q of elements of the form 1 + F , where F belongs to the prime ideal (Y, H, X) of A, is a multiplicative subgroup of A × ; it is a compact pro-p-group. In particular g x is well-defined if x ∈ Z p and g belongs to A × 1 . It is natural to consider a map b given by

1 + H i → (1 + H) α i 1 + X j → (1 + X) β i 1 + Y j → (1 + Y ) β j . If h = 1 + H, x = 1 + X, the relation (a) can be written h -1 xh = x q which implies, for α, β ∈ Z p : (3.7) h -α x β h α = x βq α .
Note that q α is well-defined since q = 1 + p.

The relation (a) will follow from (A) if

(1 + X) β j (1 + H) α i = (1 + H) α i (1 + X) k q i,k β k which is true, according to (3.7), if (3.8)
β j q α i = k q ijk β k .
Comparing with the expression of the exponents in relations (A, B, C), we see that this will be verified if (3.9) β j = e j , q α i = 1 + pe i , or equivalently (3.10)

β j = e j , α i = log(1 + pe i ) log(1 + p) .
Furthermore, it can be checked that there is no other solution to the equations (3.8) for α i , β j ∈ O L . In particular, there is no solution with exponents in Z p . This implies that the images of our generators 1 + H i , etc. will have coefficients in L. However, even if we extend coefficients, their images are not in Λ ⊗ L.

The reductions ēj of e j in k L form a basis of k L over F p . We may assume that ē1 = 1, and then |e j -r| = 1 (j > 1) for any r ∈ Z. The formal series (3.11) (1 + X)

e j = α m=0 e i m X m = a m X m
then verifies, for the canonical valuation on O L :

v p (a m ) = -v p (m!) = - 1 p -1 (m -Schiff m) (Lazard's notation, [11, III.1.1.2.5]
) where Schiff m is the sum of the digits of m in base p. In particular the coefficients tend to infinity (in absolute value). We will return to the growth of the images at the end of this section. We neglect this for now, and (computing formally) consider the other relations. By the usual symmetry , (b) is automatically satisfied. We must consider relation (c). Applying our map to (C), we must investigate in A the relation

(3.12) (1 + X) β i (1 + Y ) β j = (1 + Y ) Σβ k Q ijk (1 + H) Σα k P i,k • (1 + X) Σβ k Q jik ,
where β i , α i are given by (3.10). This is to be compared with (c).

Unlike the two previous cases, then is no formal fashion to deduce (3.12) from (c). We must return to the way (c) was obtained. It was equivalent to the relation (in S):

xy = 1 1 1 1 p 1 = 1 Qp 1 (1 + p) -1/2 (1 + p) -1/2 -2 1 Q 1 .
In the additive groups U and N, it is natural to assume that x β i = x e i = 1 e i 1 , and y e j = 1 pe j 1 (elements now of the L-points of the groups.)

Now

x e i y e i = 1 e i 1 1 pe j 1 = 1 + pe i e j e i pe j 1 .
We are reduced to the computation of § 3.1, and this yields relation (C) which, after applying the map b, implies (3.12).

This argument is admittedly formal, even more than the previous ones. It shows, however, that we will have to use the extension of scalars to make sense of the map b between (some algebras replacing) Λ r and Λ.

3.4

We end this section by noting that the apparent problem posed by the growth of the series (3.11) naturally presents its own solution. Consider simply the additive group N (over Z p ). Clearly the series (3.11) does not belong to the Iwasawa algebra, dual to the continuous functions.

However we see that its coefficients a m always satisfy

v p (a m ) ≥ - 1 p -1 (m -Schiff m) = -v p (m!).
On the other hand, an analytic function on Z p (seen as a rigid-analytic space, or, in Lazard's terminology, a saturated pro-p group) admits a Mahler expansion In this section we will make sense of the "formal" base change map of § 3.3 by restricting it to the algebra of global distributions. We will in fact work rationally; in the previous sections this would amount to replacing an Iwasawa algebra Λ by Λ ⊗ Q p . We first check that our groups S, S r are naturally group objects in the category of (affinoid) rigid-analytic spaces (in fact, all isomorphic to B N , B being the closed ball of radius 1.)

f (x) = m c m x m with v(c m ) -v(m!) → ∞.
Consider for simplicity the group S of Z p -points. It is in bijection with Z 3

p by the variables y, x and -2 log(t) log(1+p) . We will denote this last variable by z in the next computations. Let, as before Lemma 2.2, g = t tx tpy tpxy + t -1 and let γ = τ τ ξ τ pη τ pξη + τ -1 .

Then their product tτ + ptτ xη tτ ξ + tτ pxξη + tτ -1 x tτ py + tτ p2 xyη + τ t -1 pη * has parameters given by

T = tτ (1 + pxη) T Y = tτ (y + pxyη + t -2 η) T X = tτ (ξ + pxξη + τ -2 x).
With parameters z, x, y, we see that the coordinates of the product are then in the Tate algebra in the six variables. Note that t = exp(-1 2 z log(1 + p)) is, an a function of z, in the Tate algebra in one variable, and is invertible. Then T an invertible element in the Tate algebra T 6 (Q p ) of the product. A similar argument applies to the map g → g -1 . Thus we can see S as a rigid analytic group, isomorphic as a space to the closed 3-ball B 3 . (This was already proved by Lazard, cf. [START_REF] Lazard | Michel Groupes analytiques p-adiques[END_REF]III.3.3.2].)

The same formulas, and the same argument, apply to S r ∼ = B 3 (L), a rigid analytic space of dimension 3 over L.

4.2

Now let A 1 ∼ = T 3 (Q p ) be the space of globally defined analytic functions on S. 2 We assume that the coefficients are in L. If f ∈ A 1 , f defines naturally a function in T 3 (L) (with the same coefficients). We denote it by b * 0 (f ). There is a comultiplication m

* : A 1 → A 1 ⊗A 1 , f (g) → f (g 1 g 2 ).
It is obviously compatible with the map b * 0 : A 1 → A ′ r = T 3 (L) which we have defined. (E.g., use that Z p is dense in O L for the canonical topology.)

A function in A 1 can be paired with an element of the Iwasawa algebra of S (but we obtain a small space of the space of test functions). Similarly, a function in A ′ 1 can be integrated against a distribution in the Iwasawa algebra of S r . Now the Iwahori subgroup S r , a rigid-analytic space of dimension 3 over L, defines by restriction of scalars a rigid-analytic space Res(S r ) of dimension 3r over Q p , but it then has a larger space A r of analytic functions. See [START_REF] Bosch | Michel Néron models[END_REF] for the restriction of scalars in the formal case; by the results in [START_REF] Raynaud | Michel Géométrie analytique rigide d'après Tate[END_REF] this gives the restriction of scalars in the rigid-analytic case. All that we will need is that for functions the corresponding map A ′ 1 → A r is given by the explicit formula (4.1) below. Thus we get a map b * : A 1 → A r ; the image is comprised of the functions (in A r ) which are "L-holomorphic"; again, this is compatible with the comultiplication. Note that all these spaces are Banach spaces, with natural norms.

The construction of Res(S r ) yields the following formula for functions. (Cf. [START_REF] Bosch | Michel Néron models[END_REF]). 3

Let O L = W = Z p e i , so v = v i e i (v ∈ O L , v i ∈ Z p .
). We can apply this to the variables x, y, z.

If f = a(m, n, q)x m z n y q ∈ A 1 , the corresponding function on Res(S r ) is (4.1) F = a(m, n, q)( x i e i ) m ( z i e i ) n ( y i e i ) q = m i ,n i ,q i x m i i z n i i y q i i a( m i , n i , q i ) m m i n n i q q i e m e n e q
where e m = e m i i , etc., obviously in the Tate algebra in the 3r variables. The map f → F is continuous if the Tate algebras are provided with the sup norm; it is compatible with the comultiplication.

Consider now the spaces of (global) distributions D, D r on the rigidanalytic spaces S and Res(S r ) (with coefficients in L.) These are simply the dual spaces of A 1 and A r .

By duality we obtain a map b : D r -→ D .

Since the convolution can be defined by means of the coproduct on the spaces A, we see that b is a homomorphism of algebras for the convolution.

In fact we can say more. Return to the formulas in § 4.1 giving the product map in coordinates, with t = exp(- This obviously applies, with the same proof, to Res(S r ). Since the restriction of scalars preserves integral functions, we finally have the following result. We write D(L), D r (L) to emphasise the fact that the coefficients are in L. By adding the center ( a direct factor for p ≥ 3), Theorem 4.1 obviously extends to the full Iwahori subgroups G, G r .

4.3

It is now easy that the map b : D r → D is the map that we had formally defined in § 3.4. Consider for instance the effect of b on 1 + X i :

1 + X i → (1 + X) e i .

The distribution 1 + X i belongs to D r , and is simply the evaluation of F at the point (x i = 1, 0, . . . , 0) of the ball Z 3r

p . If we compose with the map f → F (cf. (4.1)), we obtain the map f → f (e i , 0, 0) where f is seen as an analytic function on the L-adic group. This distribution does not belong to the Iwasawa algebra: it has no meaning on the continuous functions. However, (1 + X) v is the map f → f (v, 0, 0) if v ∈ Z p ; for v ∈ O L , the power (1 + X) e i which appeared in the computations of § 3.4 is then naturally identified with f → f (e i , 0, 0). The same argument applies to the other variables.

Theorem 2 . 7 .

 27 -The map A → Λ gives, by passing to the quotient, an isomorphism B = A/R ∼ -→ Λ As in our previous paper, we will use for the proof the corresponding algebras with finite coefficients. Thus let Ω = Λ Zp F p be the algebra of distributions with values in F p . We now have a natural map Ā = A⊗F p → Ω.

Theorem 3 . 1 .

 31 -The natural homomorphism A/R -→ Λ is an isomorphism.

  (Amice's theorem, cf[START_REF] Lazard | Michel Groupes analytiques p-adiques[END_REF] III.1.3.8].) This implies that the Iwasawa series(3.11) is convergent on such functions. Thus we are naturally led to replace the Iwasawa algebra by the dual of the space of globally analytic functions (possibly with coefficients in L.)[START_REF] Bushnell | The admissible dual of GL(N) via compact open subgroups[END_REF] The holomorphic base change map 4.1

  where π = log(1 + p).The local parameter Z associated to T is then, with obvious notation,z + ζ -2 1 log(1 + p) log(1 + pxη),a convergent series in (z, ζ, x, η) with integral coefficients. Furthermore, (1 + pxη) -1 is such a series, as is t -2 :t -2 = exp(zπ) = a n z n with v(a n ) = p -2 p -1 n + Schiff(n) > 0for n ≥ 1.Therefore the coordinates Z, X, Y of the product belong to the Tate algebra in the six variables with integral coefficients.4 Let δ : A = A S → A S ⊗A S be the comultiplication. Then convolution in D is defined by < S * T, f >=< S ⊗ T, δ(f ) > while the norm in D is given byT = Sup f ∈A 0 < T, f > .where A 0 is the unit ball in A. For f ∈ A 0 , the previous computation show that δ(f ) is a series with integral coefficients. By [2, §2.1, Prop.1], we then have | < S ⊗ T, δ(f ) > | ≤ 1 if S ≤ 1 and T ≤ 1. Thus S * T ≤ S . T .

Theorem 4 .

 4 1.-(i) The map b defines a morphism of convolution algebras D r (L) → D(L). (ii) D(L), D r (L) are Banach algebras for convolution, the product verifying in fact S * T ≤ S . T . (iii) b(f ) ≤ f (f ∈ D r (L).

We do not see how to obtain a natural basis of W . In particular, the structure constants q ijk , Q ijk of Λ G do not seem to have a natural expression.

We apologise for the conflict of notations with the previous paragraphs. The spaces of non-commutative polynomials will no longer occur in this section.

We will review the restriction of scalars more precisely in part II of this paper.

Note that in this rigid-analytic computation we do not have to assume that p > 3. We did not check which of the other arguments apply.

3.2

We sketch the proof, which is very similar to that given for Q p . We choose the coefficient α of § 2.1 equal to 1 2 . Again, Λ is filtered by 1 2 N, gr ν Λ having a basis given by the independent elements

give a basis of gr ν Ω. The foregoing argument shows that the natural maps A → Λ, Ā → Ω are well-defined. We give weights (1,2,1) to the elements Y i , H j , X 1 and replace in Ω the filtration by 1 2 N by the filtration by n = 2ν ∈ N. We have the analogue of Lemma 2.11; first we have

The same relations are verified by the Y k .

(ii) Same relations, with q ijk replaced by Q ijk .

Part (i) follows from (A) and (i) of the previous Lemma; similarly (ii) follows from (B) and (i). Now from (C) we have, using (ii) of that Lemma:

By Lemma 3.4, we see first that dim gr 1 B = 2r (linear terms in the X i , Y i ), then that dim gr 2 B = 2r(2r+1) 2 + r (symmetric polynomials of degree 2 in the X i , Y j and linear forms in the H k ). Finally, the argument given for the