Maxime Bergeat
email: maxime.bergeat@insee.fr

Nora Cuppens-Boulahia

Frédéric Cuppens
email: nora.cuppens@telecom-bretagne.eu

Noémie Jess
email: noemie.jess@sante.gouv.fr

Françoise Dupont
email: francoise.dupont@casd.eu

Saïd Oulmakhzoune
email: said.oulmakhzoune@telecom-bretagne.eu

Gaël De Peretti
email: gael.de-peretti@insee.fr

A French Anonymization Experiment with Health Data

Keywords: case study, health data, k-anonymization, microdata, statistical disclosure control

In this paper, a case study about a microdata anonymization test is presented. The work has been made considering a French administrative health dataset with indirect identifiers and sensitive variables about hospital stays. Two approaches to build a k-anonymized file are described, and software used in the test are compared.

Introduction

Following the "Free Health Data" petition launched by the French Inter-Associative Health Group CISS (Collectif Interassociatif Sur la Santé) in January 2013, the French Minister of Health and Solidarity Marisol Touraine asked P.L. Bras to write a report on health data use and governance. It was released in September 2013. The debate on increasing access to public health data was launched in late November 2013. M. Touraine commissioned F. Von Lennep, head of the statistical department of the Ministry of Health and Solidarity, and P. Burnel, responsible for IT System Strategies in the Ministry of Health and Solidarity, to lead the Open Data committee. The committee issued its conclusions in late April 2014.

A taskforce leaded by André Loth, project manager in the statistical department of the Ministry of Health and Solidarity, was launched to assess data safety and individual reidentification risk in health data. Its work focused on health data specifics, anonymization techniques, and how a wider access to health data could be organized. In order to get practical perspectives, the taskforce launched a test on real data. The aim was to determine how anonymized data files could be built from the original French exhaustive administrative database on hospital stays. The objective of this work is to specify a suitable method for producing anonymized microdata files to be released as public use files. A wide range of different users is expected to use these data files: from citizens to health insurers, including pharmaceutical companies, doctors… Our work fell into a binding timetable due to organizational and legal constraints. The French Commission for Data Protection and Liberties (Commission Nationale de l'Informatique et des Libertés) gave its agreement on January 30 th , 2014, for a three-month renewable period. The test began in early March 2014 because of late availability of data and needed nomenclatures. Test conclusions were given in late April 2014.

This paper aims at presenting and comparing two ways to elaborate a safe microdata file on this case study. This paper is organized as follows. Section 2 gives an overview of the software used for the test, µ-Argus and ARX. In Section 3 test data are introduced and we present objectives of the anonymization process in terms of disclosure risk and allowed technical solutions to reduce it. Section 4 presents two approaches tested to reach anonymization goals. A discussion is led in Section 5 and some conclusions are given in Section 6.

Data Anonymization Tools

The result of de-identification tools market browsing has shown that there is a wide range of solutions developed for structured data anonymization offering a long list of functionalities. But few of them are used or can be smartly used (because they are internal to specific organizations or developed by researchers for specific use).

Moreover, practical applicability of the promising functions needs to be tested, in particular to highlight what has to be refined, updated or further developed to enhance the usability.

Of course these tools are different from those used for masking data that we do not consider in this paper, because they only manipulate direct identifiers by creating pseudonyms or applying randomization techniques, but never deal with indirect identifiers and thus do not provide suitable protection. Some of those de-identification tools and some of their features are summarized in Table A.1 in Appendix.

We can observe that there are few interesting commercial products whereas there are several tools and prototypes developed by academic researchers not broadly used. To further study the usability, the robustness of these tools and privacy protection they can provide, a testing process on real data of significant size needs to be performed. In the two next subsections, we will introduce the software used in the test, µ-Argus and ARX. Further comparison is also given in Section 5. Many protection methods can be used with µ-Argus. The anonymization scheme can be summarized in a few steps:

µ-Argus

-Import microdata into µ-Argus and define metadata before beginning the statistical disclosure control (SDC) process. In this step we must for instance define the indirect identifiers and the sensitive variables of the microdata file. -Estimation of the disclosure risk of the file. It is possible to set a minimum threshold for specified combinations of values of indirect identifiers. In the risk estimation you can also, in case of a survey, use the sampling weights in order to probabilistically estimate the re-identification risk. -Several protection methods to reduce disclosure risk are implemented in the software: they are perturbative (noise addition, microaggregation, postrandomization method (PRAM), rounding, data swapping…) or not (global recoding, local suppression…). For a further description of µ-Argus features see [START_REF] Hundepool | µ-Argus User's Manual[END_REF]. -After residual disclosure risk estimation, safe microdata files can be exported. µ-Argus is an interactive program mostly used by National Statistical Institutes (NSIs). A questionnaire on SDC tools led in the European NSIs in 2013 has shown that µ-Argus is often used for microdata protection in an automated way, even if some institutes are still applying manual procedures or homemade tools.

ARX

ARX Data Anonymization

Tool is an open source data anonymization framework developed in Java. It implements dedicated optimizations for anonymization algorithms. It encodes data in a way that allows evaluating transformations efficiently. ARX is available online1 . See also [START_REF] Kohlmayer | Highly efficient optimal k-anonymity for biomedical datasets[END_REF].

ARX is a flexible and intuitive high-performance data anonymization tool. It implements a wide variety of common privacy criteria such as k-anonymity, l-diversity, t-closeness, δpresence and their combinations. See [START_REF] Sweeney | k-anonymity: A model for protecting privacy[END_REF], [START_REF] Machanavajjhala | l-diversity: privacy beyond k-anonymity[END_REF], [START_REF] Li | t-closeness: privacy beyond k-anonymity and l-diversity[END_REF] and [START_REF] Nergiz | Hiding the presence of individuals from shared databases[END_REF] for more information about these concepts. ARX allows protecting a dataset from a multitude of privacy threats. It allows non-domain-experts to specify privacy guarantees in an intuitive manner and to explore potentially large solution spaces efficiently. It implements also a set of metrics to evaluate information loss.

The ARX graphical tool is composed of three main perspectives (views). The first one allows defining the required privacy guarantees. The second perspective allows browsing the complete solution space according to the privacy criteria defined on the first perspective. Solutions that fulfill the criteria as well as the "optimal" solution are highlighted. The third perspective allows comparing transformed datasets to the original dataset.

The ARX anonymization scheme could be summarized in the following steps:

-Import the original dataset in CSV format.

-Define direct identifiers, indirect identifiers and sensitive variables of the original dataset. -Define the privacy criteria. For instance 10-anonymity for indirect identifiers, 3diversity for the sensitive variable, … -Define generalization hierarchies for each indirect identifier. ARX allows importing generalization hierarchies (nomenclatures) or creating new ones from an interactive menu. -Define the maximum number of outliers that can be tolerated and will be suppressed.

-Start the anonymization process.

-Browse and analyze solutions.

-Export desired transformations.

The Test Framework

The Dataset to Anonymize

The anonymization test is performed with the French dataset PMSI (Programme de Médicalisation des Systèmes d'Information). This exhaustive medical-administrative file contains more than 35 million records corresponding to all hospital stays for one calendar year in the three major hospitalization areas: MCO (médecine, chirurgie, obstétrique or MSO: medicine, surgery and obstetrics) also known as short stays, rehabilitation care and home hospitalization.

For our test, consistently with the desired output file, we experiment on the latest PMSI-MCO file that contains short stays. Besides, the working group decides to focus on "unique" stays, in other words hospital stays except medical sessions (e.g. chemotherapy or dialysis sessions) that generally happen more than once a year. The underlying reason is that there are different ways for encoding doctors to fill in information about medical sessions: they are allowed to create one record for each stay or one for all. As such, interpretation of these stays is hard and it is difficult to deal with them in the anonymization process: for instance the variable "length of stay" is very volatile because of the different possibilities for health workers to fill in data: this can create fake anonymization issues.

All things considered, we deal with the 2012 PMSI-MCO file excluding medical sessions. The dataset contains 20.6 million hospital stays. For each stay detailed medical information (notably major diagnosis, related and accompanying diagnoses, medical procedures, care areas) and administrative information (hospital identification number FINESS 2 , dates of stay, admission and discharge modes, patient birth date, sex and residential area) are provided. The FINESS number is public information since it is released on a dedicated website 3 , and enables one to find the name, location, legal status and type of a hospital. Each record (i.e. each stay) is classified into one GHM4 code (the French version of DRG, Diagnosis Related Group) that is derived from medical data and in a way sums up the pathology for which the patient is hospitalized. The most aggregated level of the GHM classification is the CMD5 (Major category of diagnosis), which is a 28-item nomenclature including one code for medical sessions and one error code.

First of all we asked health experts what the most used classification for health studies currently is, in order to maximize data utility of anonymized files and to ensure consistency with other datasets already disseminated. For example, it is very common to classify age in five-year age groups.

Before beginning the anonymization process, we have analyzed our dataset (distributions of the variables, health experts requirements) in order to see the variables (and their related items) that are rare and as such likely to lie behind anonymization issues. Hospital stays are uncommon among children between 1 and 14 years old in particular (see Figure 1 below). Concerning admission and discharge variables, origins and destinations other than home, unknown or regular transfers are quite unusual and death is in addition particularly sensitive (see Figure 1 below). Likewise, we find a few long hospital stays (more than one week). Lastly, distance between residential and hospitalization areas has a strong power of reidentification: patients whose residential area is not in the same territory as their hospital are in general easy to disclose. We take one random example with the first (by alphabetic order) French department (Ain) with population of about 600 000 inhabitants: for 43% of hospital stays in Ain the patient also lives in the department. Conversely, 86 combinations of Ainother department (out of a total of 100 combinations) consist in less than 10 hospital stays.

Objectives of the Anonymization Process

The anonymization task consists in obtaining a dataset that reaches k-anonymity and ldiversity given that we address exhaustive data. We recall the definitions of these concepts below.

Definition 1 k-anonymity

A file is said to be k-anonymized if at least k records match every combination of values taken by the indirect identifiers. Each combination is called an identification key.

Definition 2 l-diversity

A file is said to be l-diverse if for each identification key, there are at least l "well represented" different values for each sensitive attribute.

The taskforce about the anonymization test decided that neither perturbative methods nor local suppression were suitable in order to obtain the safe file. We have considered that using perturbative methods may increase risk of misuse, especially with non-specialist users. Moreover we agreed on the idea that local suppression may lead to bad interpretation of the data so local suppression was not allowed either. In the ARX scenario, outliers with a high risk of re-identification are computed, and there is post-processing in order to deal with them without deleting these risky records or using local suppression: see also Section 4.2.

We use the following variables as indirect identifiers:

-Sex -Age -Residential area (ZIP code) -Hospitalization area (FINESS number) -Length of hospital stay

For the µ-Argus scenario, we consider two other indirect identifiers to ensure k-anonymity:

-Admission mode -Discharge mode

We also check the variable "Major category of diagnosis" (CMD: Catégorie Majeure de Diagnostic) for l-diversity. This is an aggregated version of the main diagnosis recorded during the patient stay. This variable has 26 modalities for all stays considered in the anonymization test. As previously indicated, repeated medical sessions are excluded from the test, there is a specific CMD code for medical sessions that we do not consider afterwards. There is no error code in the dataset.

The objective of the anonymization process is to obtain at the end a 10-anonymised and 3diverse file. A modality is said "well represented" (for l-diversity) if there is at least one record with the considered modality. Definition of such levels was done after a long discussion: these levels were decided in a way both cautious and reasonable… A value of k= 5 is often used for some health data dissemination; we chose k= 10 to ensure a security margin, for instance if the potential attacker uses auxiliary information to deconstruct the anonymization process. Moreover, it is theoretically easier to get 3-diversity with a 10anonymized file than a 5-anonymized one. For more details about the choice of k in the SDC process see El [START_REF] El Emam | A Globally Optimal k-Anonymity Method for the De-Identification of Health Data[END_REF]. The choice of l= 3 aims at avoiding exact disclosure of the sensitive variable "Major category of diagnosis".

Results of the Experiment

Two ways to create a safe file were proposed during the test. In this section we will present two different approaches to elaborate a file regarded as safe given the anonymization goals introduced in Section 3. We will in the next Section compare the two methods and also software used to build the anonymized files.

Given we want to obtain a safe file without any local suppression or use of perturbative methods, the only approach to reach k-anonymization and l-diversity is global recoding. The first step in preliminary work was to discuss with users of the PMSI file and domain experts in order to have predefined ways of recoding. The goal was to have an overview of "smart" plausible recoding if an expert in health data wants to make statistical studies with the protected file. For instance, studying children under 1 year old is a classical subject in pediatrics. The definition of possible ways of recoding is completed with the analysis of distributions of the indirect identifiers.

We also have reached the conclusion in the preliminary work that sampling should give additional protection in order to avoid residual disclosure risk. Yet the completeness of hospital stays is a major asset of the French PMSI file.

µ-Argus: Using Global Recoding with an Iterative Approach

In the first test scenario we produced safe files with µ-Argus. Here are the different steps:

-After discussion with domain experts, definition of multiple predefined ways of recoding. -Calculation of frequencies for all identification keys. In order to limit the number of identification keys to compute some variables have to be discretized (e.g. age has been recoded to larger bands with a top coding, and ZIP codes are replaced by department -NUTS3 level). -For each modality of each variable, µ-Argus computes the number of identification keys with insufficient frequency where this modality is implied. According to these results we use a step-by-step approach: -While the file is not 10-anonymized, do:

o Detect modalities that are implied in a big number of identification keys with less than 10 stays. Use the ratio between number of concerned identification keys and frequency of occurrence (the two numbers are given by µ-Argus) to detect "risky" modalities. o If it is possible from an analytic point of view given conclusions of discussion with health experts, combine this "risky" modality with an other one. o If not possible (for instance there is no sense in combining two geographical places that are not in the same area!), make another recoding: go back to the detection step.

-When you obtain a 10-anonymized file, check for 3-diversity using a SAS algorithm.

We have constructed two files using this empirical approach. Two files that are compatible with anonymization goals are described in Table 1 andTable 2. In the file of Table 1 there is no geographical dimension: we have tried to introduce the variable "Residential area" and we have obtained the file summarized in Table 2. 2 One file that meets anonymization criteria with one geographical dimension The variable "hospitalization area" is not included in the two proposed files: it is hard to build such l-diverse file because some hospitals are specialized in the treatment of a particular disease, for instance cancer centers. Moreover a file with two geographical levels will a priori be very risky because hospital stays where residence and hospitalization areas are different are extremely rare: like already described in Section 3, this can easily lead to reidentification. However we didn't try to consider instead "hospitalization area" a variable that takes the modalities "hospitalization area is close to residential area" and "hospitalization area is far away from residential area". This binary variable should reach a quite good trade-off between data utility and disclosure risk; it has not been tested here because of hard work needed to compute this variable.

Name of variable Type of variable Number of modalities

ARX: Two Levels of Detail Depending on Disclosure Risk of the Stay

The second test has been implemented with the ARX Data Anonymization Tool. Privacy criteria are also 10-anonymity and 3-diversity. In ARX, we define first possible generalizations (i.e. ways of recoding) for each indirect identifier. Then ARX chooses a possible combination of those generalizations that fulfills the given privacy criteria.

We use the following five variables as indirect identifiers:

-Sex -Age -ZIP code of residential area -FINESS number -Length of hospital stay Note that admission and discharge modes are not studied in this anonymization scenario.

The different steps of the test scenario could be summarized as follows:

-Creation and loading of the dataset input. The ARX input file should be in CSV format.

-After loading the original dataset, indirect identifiers and sensitive attributes are defined. Indirect identifiers will be k-anonymized, and sensitive attributes will be ldiversified. -For each indirect identifier attribute we import its corresponding generalization hierarchy, which defines a way of iteratively generalizing the values of an attribute.

The generalization hierarchy should be monotonic, i.e. within one hierarchy the groups at level n+1 are built by merging groups from level n (see for instance Table A Firstly we calculate possible solutions for the given anonymization criteria with no suppression (threshold= 0%). We obtain 127 solutions among 1000 combinations of generalization hierarchies (2 x 5 x 5 x 5 x 4). There is no solution where at least one indirect identifier is not completely anonymized (the variable is suppressed). Like for the first test scenario, solutions with no suppression suffer from a high loss of utility. Table 4 gives an overview of a possible solution. Information about "hospitalization area" and "length of hospital stay" are not disseminated in order to fulfill anonymization criteria. 4 One combination that meets anonymization criteria in the ARX scenario It is therefore interesting to create files with consideration of the outliers. To do this, we apply the following methodology in order to obtain a 10-anonymized and 3-diverse file:

Name of variable

-Set the maximum threshold of (globally suppressed) outliers to a positive value (4% in this test), and then select one solution. -Extract outliers, i.e. a set of records that do not fulfill the anonymization criteria -Reprocessing deleted records. It corresponds to replay the anonymization process on deleted records. -Construct the final solution. We can integrate deleted records after anonymization, and then we obtain a new file with two levels of detail depending of the rareness of each stay.

In this test, maximum threshold of outliers is fixed to 4%. Table 5 illustrates an example of solution corresponding to the threshold 3.8% (650 388 records are outliers). The first step is to extract outliers corresponding to one solution from the original dataset. Then we anonymize outliers by applying the same process. It is obvious that information provided for outliers is less detailed. This approach differs slightly from a method with local suppression because here the information provided is the same for all outliers. 5 One file that meets anonymization goals with two levels of detail depending on the rareness of a stay

Name of variable

The file described in Table 5 has strictly more utility than the previous one because information provided for non-outliers is more detailed: data about place of hospitalization and duration of the stay may be disseminated for records with a low risk of re-identification.

Discussion

Given the non-uniform distribution of indirect identifiers shown in Section 3, building kanonymized files is very hard without using any perturbative method because of the rareness of some modalities of indirect identifiers. See [START_REF] Sweeney | Simple Demographics Often Identify People Uniquely, Data Privacy working paper[END_REF] for another example with demographic data. The loss of utility is very high when making global recoding. In particular, we didn't manage to obtain safe files with two geographical dimensions (residence and hospitalization areas). Even in the file described in Table 5, there is no information about place of hospitalization for the outliers.

Both software of this test are quite easy to use with an intuitive interface and handle very large datasets like the French PMSI file -more than 20 million records. After generation of the microdata file and computation of frequencies for all identification keys that are timeconsuming, other steps of the anonymization process are done instantly with µ-Argus. The anonymization process (comparison of all possible combinations of generalization hierarchies) consists in less than 2 minutes with ARX. µ-Argus is a flexible program where all parameters of the SDC process can be controlled, there is no "black-box" effect and the documentation about the software is detailed. ARX has apparently more functionalities and anonymization criteria than µ-Argus like l-diversity (that can be checked directly during the SDC process), t-closeness and δ-presence. However documentation is limited and it is sometimes hard to use some functionalities: in particular, there is only little documentation about metrics used to choose one generalization hierarchy by minimizing information loss. Some steps are not optimized in µ-Argus and ARX. For instance there is no way to compute optimal global recoding in order to minimize information loss under the constraint of getting a k-anonymized file. Some algorithms are described in the literature (see [START_REF] Lefevre | Mondrian Multidimensional k-Anonymity[END_REF] about the Mondrian algorithm) but they are not implemented in the software used in this test.

Moreover it is not easy to add utility constraints when using this kind of algorithm. For instance, we do not want to recode a geographical variable by regrouping some areas that are very far away from each other. That's why all methods presented in this paper consist in iterative empirical approaches in order to reach k-anonymization and l-diversity.

Concluding remarks

We can see in this study that it is hard to reach a good trade-off between disclosure risk and data utility when protecting a microdata file using neither perturbative methods nor local suppression. All files proposed in this test for dissemination suffer from a huge loss of data utility, despite the large number of records in the original PMSI file of hospital stays in 2012. Two methods to obtain a 10-anonymized and 3-diverse file were presented: the former with "pure" global recoding, the latter with different levels of generalization according to the power of re-identification of each record.

In further work it should be interesting to analyze disclosure risk and associated loss of information when perturbing data, for instance with qualitative microaggregation (see also [START_REF] Hundepool | Statistical disclosure control[END_REF] for a definition of microaggregation). Using geographical information for perturbation seems to be a good idea because it is easier with this kind of variable to control the perturbation and the associated loss of information compared to a non-ordinal variable like patient sex. Lastly, values of k and l were fixed in this study: it might be of interest to perform tests with other values and study resulting data utility and residual disclosure risk.

  µ-Argus is a software package developed to help statisticians to anonymize microdata files. It is a deliverable of the European CASC (Computational Aspects of Statistical Confidentiality) project that took place between 2000 and 2003, and additional work has been made since 2003 thanks to other European projects: the Centre of Excellence SDC (CENEX-SDC), ESSNet-harmonization and ESSNet-SDC. The next major release of µ-Argus will be an open-source version; development is still in progress (see de Wolf (2013) for more details).

Figure 1

 1 Figure 1 Univariate analyses about indirect identifiers

  One file that meets anonymization criteria without geographical dimension

	Sex Age Length of hospital stay Admission mode Discharge mode Major category of diagnosis Sensitive variable 26 Indirect identifier 2 Indirect identifier 18 Indirect identifier 12 Indirect identifier 2 Indirect identifier 2 Type of variable Sex Indirect identifier Age Indirect identifier Residential area Indirect identifier Length of hospital stay Indirect identifier Admission mode Indirect identifier Discharge mode Indirect identifier Major category of diagnosis Sensitive variable Table 1 Name of variable Table	Number of modalities 2 6 22 (NUTS2 level grouping overseas departments and combining Corsica and PACA 6 regions) 2 2 2 26

Table 3

 3 .2 and TableA.3 in Appendix). All levels of generalization are represented in Table3.-Definition of anonymization criteria:o 10-anonymity for the five indirect identifiers o 3-diversity for the sensitive attribute "Major category of diagnosis" o Suppression threshold (percentage of outliers that can be globally suppressed) -Production of anonymized solutions. ARX searches all possible solutions by applying systematically all possible combinations of generalization hierarchies of indirect identifiers to check for anonymization criteria. Generalization levels in the ARX scenario ARX allows defining tolerated suppression threshold i.e. the percentage of outliers (records allowed to be suppressed in order to fulfill the anonymization criteria) that are (globally) suppressed.

	Indirect identifier	Level 0	Level 1	Level 2	Level 3	Level 4
	Sex	Cleartext Not disseminated Not disseminated	Not disseminated	Not disseminated
	Age (years)	Cleartext	Five-year age groups (see Figure A.2)	Level 2 (see Figure A.2)	Level 3 A.2) (see Figure	Not disseminated
	ZIP code of residence	Cleartext	NUTS3 (101 departments in France)	NUTS2 (22 regions in Metropolitan departments) overseas France and	NUTS2 level grouping overseas PACA regions Corsica and and combining departments	Not disseminated
	FINESS number (hospitalization area)	Cleartext	NUTS3 (101 departments in France)	NUTS2 (22 regions in Metropolitan departments) overseas France and	NUTS2 level grouping overseas PACA regions Corsica and and combining departments	Not disseminated
	Length of hospital stay (days)	Cleartext	Level 1 (see Figure A.3)	Level 2 (see Figure A.3)	Not disseminated	Not disseminated

Type of variable Level of generalization For the outliers For other stays

	Sex	Indirect identifier Cleartext	Cleartext
	Age	Indirect identifier Five-year age groups	Five-year age groups
		(level 1)	(level 1)
	Residential area	Indirect identifier NUTS3 level	NUTS3 level
	FINESS number	Indirect identifier Not disseminated	NUTS2 level
	Length of hospital stay	Indirect identifier Not disseminated	Level 1
	Major category of diagnosis Sensitive variable Cleartext	Cleartext
	Table

It can be downloaded from http://arx.deidentifier.org

FIchier National des Établissements Sanitaires et Sociaux or National file of the sanitory and social establishments

See http://finess.sante.gouv.fr/jsp/index.jsp

Groupe Homogène de Maladie

Catégorie Majeure de Diagnostic

Provence-Alpes-Côte d'Azur is a French region located in the south and close to Corsica.

See http://www.drees.sante.gouv.fr/rapport-sur-la-gouvernance-et-l-utilisation-des-donnees-de,11202.html

Available online at http://cran.r-project.org/web/packages/sdcMicro

Available online at http://www.himss.org/News/NewsDetail.aspx?ItemNumber=29971

Available online at http://sourceforge.net/projects/anony-toolkit

Available online at http://cs.utdallas.edu/dspl/cgi-bin/toolbox


Jean-Yves Charbonnel

AND Anne Moreau
email: anne.moreau@math.univ-poitiers.fr

THE INDEX OF CENTRALIZERS OF ELEMENTS OF REDUCTIVE LIE ALGEBRAS

Keywords: Mathematics Subject Classification. 22E46, 17B80, 17B20, 14L24 reductive Lie algebra, index, centralizer, argument shift method, Poisson-commutative family of polynomials, rigid nilpotent orbit, Slodowy slice

For a finite dimensional complex Lie algebra, its index is the minimal dimension of stabilizers for the coadjoint action. A famous conjecture due to A.G. Elashvili says that the index of the centralizer of an element of a reductive Lie algebra is equal to the rank. That conjecture caught attention of several Lie theorists for years. It reduces to the case of nilpotent elements. In [Pa03a] and [Pa03b], D.I. Panyushev proved the conjecture for some classes of nilpotent elements (e.g. regular, subregular and spherical nilpotent elements). Then the conjecture has been proven for the classical Lie algebras in [Y06a] and checked with a computer programme for the exceptional ones [deG08]. In this paper we give an almost general proof of that conjecture.

Introduction

In this note k is an algebraically closed field of characteristic 0.

1.1. Let g be a finite dimensional Lie algebra over k and consider the coadjoint representation of g. By definition, the index of g is the minimal dimension of stabilizers g x , x ∈ g * , for the coadjoint representation: ind g := min{dim g x ; x ∈ g * }.

The definition of the index goes back to Dixmier [START_REF] Dixmier | Algèbres enveloppantes[END_REF]. It is a very important notion in representation theory and in invariant theory. By Rosenlicht's theorem [START_REF] Rosenlicht | A remark on quotient spaces[END_REF], generic orbits of an arbitrary algebraic action of a linear algebraic group on an irreducible algebraic variety are separated by rational invariants; in particular, if g is an algebraic Lie algebra, indg = deg tr k(g * ) g ,

where k(g * ) g is the field of g-invariant rational functions on g * . The index of a reductive algebra equals its rank. For an arbitrary Lie algebra, computing its index seems to be a wild problem. However, there is a large number of interesting results for several classes of nonreductive subalgebras of reductive Lie algebras. For instance, parabolic subalgebras and their relatives as nilpotent radicals, seaweeds, are considered in [START_REF] Panyushev | The index of a Lie algebra, the centraliser of a nilpotent element, and the normaliser of the centraliser[END_REF], [START_REF] Tauvel | Sur l'indice de certaines algèbres de Lie[END_REF], [J07]. The centralizers, or normalizers of centralizers, of elements form another interesting class of such subalgebras, [E85a], [START_REF] Panyushev | The index of a Lie algebra, the centraliser of a nilpotent element, and the normaliser of the centraliser[END_REF], [START_REF] Moreau | Indice et décomposition de Cartan d'une algèbre de Lie semi-simple réelle[END_REF]. The last topic is closely related to the theory of integrable Hamiltonian systems [START_REF] Bolsinov | Commutative families of functions related to consistent Poisson brackets[END_REF]. Let us precise this link.

From now on, g is supposed to be reductive. Denote by G the adjoint group of g. The symmetric algebra S(g) carries a natural Poisson structure. By the so-called argument shift method, for x in g * , we can construct a Poisson-commutative family F x in S(g) = k[g * ]; see [START_REF] Mishchenko | Euler equations on Lie groups[END_REF] or Remark 1.4. It is generated by the derivatives of all orders in the direction x ∈ g * of all elements of the algebra S(g) g of g-invariants of S(g). Moreover, if G.x denotes the coadjoint orbit of x ∈ g * : Theorem 1.1 ([Bol91], Theorems 2.1 and 3.2). There is a Poisson-commutative family of polynomial functions on g * , constructed by the argument shift method, such that its restriction to G.x contains 1 2 dim (G.x) algebraically independent functions if and only if indg x = ind g.

Denote by rk g the rank of g. Motivated by the preceding result of Bolsinov, A.G. Elashvili formulated a conjecture: Conjecture 1.2 (Elashvili). Let g be a reductive Lie algebra. Then indg x = rk g for all x ∈ g * .

Elashvili's conjecture also appears in the following problem: Is the algebra S(g x ) g x of invariants in S(g x ) under the adjoint action a polynomial algebra? This question was formulated by A. Premet in [PPY07, Conjecture 0.1]. After that, O. Yakimova discovered a counterexample [Y07], but the question remains very interesting. As an example, under certain hypothesis and under the condition that Elashvili's conjecture holds, the algebra of invariants S(g x ) g x is polynomial in rk g variables, [PPY07, Theorem 0.3].

During the last decade, Elashvili's conjecture caught attention of many invariant theorists [START_REF] Panyushev | The index of a Lie algebra, the centraliser of a nilpotent element, and the normaliser of the centraliser[END_REF], [START_REF] Charbonnel | Propriétés (Q) et (C). Variété commutante[END_REF], [Y06a], [deG08]. To begin with, describe some easy but useful reductions. Since the g-modules g and g * are isomorphic, it is equivalent to prove Conjecture 1.2 for centralizers of elements of g. On the other hand, by a result due to E.B. Vinberg [START_REF] Panyushev | The index of a Lie algebra, the centraliser of a nilpotent element, and the normaliser of the centraliser[END_REF], the inequality ind g x ≥ rk g holds for all x ∈ g. So it only remains to prove the opposite one. Given x ∈ g, let x = x s + x n be its Jordan decomposition. Then g x = (g xs ) xn . The subalgebra g xs is reductive of rank rk g. Thus, the verification of Conjecture 1.2 reduces to the case of nilpotent elements. At last, one can clearly restrict oneself to the case of simple g.

Review now the main results obtained so far on Elashvili's conjecture. If x is regular, then g x is a commutative Lie algebra of dimension rkg. So, Conjecture 1.2 is obviously true in that case. Further, the conjecture is known for subregular nilpotent elements and nilpotent elements of height 2 and 3, [START_REF] Panyushev | The index of a Lie algebra, the centraliser of a nilpotent element, and the normaliser of the centraliser[END_REF], [START_REF] Panyushev | Some amazing properties of spherical nilpotent orbits[END_REF]. Remind that the height of a nilpotent element e is the maximal integer m such that (ad e) m = 0. More recently, O. Yakimova proved the conjecture in the classical case [Y06a]. To valid the conjecture in the exceptional types, W. de Graaf used the computer programme GAP, see [deG08]. Since there are many nilpotent orbits in the Lie algebras of exceptional type, it is difficult to present the results of such computations in a concise way. In 2004, the first author published a case-free proof of Conjecture 1.2 applicable to all simple Lie algebras; see [START_REF] Charbonnel | Propriétés (Q) et (C). Variété commutante[END_REF]. Unfortunately, the argument in [START_REF] Charbonnel | Propriétés (Q) et (C). Variété commutante[END_REF] has a gap in the final part of the proof which was pointed out by L. Rybnikov.

To summarize, so far, there is no conceptual proof of Conjecture 1.2. Nevertheless, according to Yakimova's works and de Graaf's works, we can claim: [deG08]). Let g be a reductive Lie algebra. Then indg x = rk g for all x ∈ g * .

Theorem 1.3 ([Y06a],
Because of the importance of Elashvili's conjecture in invariant theory, it would be very appreciated to find a general proof of Theorem 1.3 applicable to all finite-dimensional simple Lie algebras. The proof we propose in this paper is fresh and almost general. More precisely, it remains 7 isolated cases; one nilpotent orbit in type E 7 and six nilpotent orbits in type E 8 have to be considered separately. For these 7 orbits, the use of GAP is unfortunately necessary. In order to provide a complete proof of Theorem 1.3, we include in this paper the computations using GAP we made to deal with these remaining seven cases.

1.2. Description of the paper. Let us briefly explain our approach. Denote by N(g) the nilpotent cone of g. As noticed previously, it suffices to prove indg e = rkg for all e in N(g). If the equality holds for e, it does for all elements of G.e; we shortly say that G.e satisfies Elashvili's conjecture.

From a nilpotent orbit O l of a reductive factor l of a parabolic subalgebra of g, we can construct a nilpotent orbit of g having the same codimension in g as O l in l and having other remarkable properties. The nilpotent orbits obtained in such a way are called induced; the other ones are called rigid. We refer the reader to Subsection 2.3 for more precisions about this topic. Using Bolsinov's criterion of Theorem 1.1, we first prove Theorem 1.3 for all induced nilpotent orbits and so the conjecture reduces to the case of rigid nilpotent orbits. To deal with rigid nilpotent orbits, we use methods developed in [START_REF] Charbonnel | Propriétés (Q) et (C). Variété commutante[END_REF] by the first author, and resumed in [START_REF] Moreau | Indice du normalisateur du centralisateur d'un élément nilpotent dans une algèbre de Lie semi-simple[END_REF] by the second author, based on nice properties of Slodowy slices of nilpotent orbits.

In more details, the paper is organized as follows:

We state in Section 2 the necessary preliminary results. In particular, we investigate in Subsection 2.2 extensions of Bolsinov's criterion and we establish an important result (Theorem 2.7) which will be used repeatedly in the sequel. We prove in Section 3 the conjecture for all induced nilpotent orbits (Theorem 3.1) so that Elashvili's conjecture reduces to the case of rigid nilpotent orbits (Theorem 3.1). From Section 4, we handle the rigid nilpotent orbits: we introduce and study in Section 4 a property (P) given by Definition 4.2. Then, in Section 5, we are able to deal with almost all rigid nilpotent orbits. Still in Section 5, the remaining cases are dealt with set-apart by using a different approach.

Notations.

• If E is a subset of a vector space V , we denote by span(E) the vector subspace of V generated by E. The grassmanian of all d-dimensional subspaces of V is denoted by Gr d (V ). By a cone of V , we mean a subset of V invariant under the natural action of k * := k \ {0} and by a bicone of V × V we mean a subset of V × V invariant under the natural action of k * × k * on V × V .

• From now on, we assume that g is semisimple of rank ℓ and we denote by ., . the Killing form of g. We identify g to g * through ., . . Unless otherwise specified, the notion of orthogonality refers to the bilinear form ., . .

• Denote by S(g) g the algebra of g-invariant elements of S(g). Let f 1 , . . . , f ℓ be homogeneous generators of S(g) g of degrees d 1 , . . . ,d ℓ respectively. We choose the polynomials f 1 , . . . ,f ℓ so that d 1 ≤ • • • ≤d ℓ . For i = 1, . . . , ℓ and (x, y) ∈ g × g, we may consider a shift of f i in direction y: f i (x + ty) where t ∈ k. Expanding f i (x + ty) as a polynomial in t, we obtain

f i (x + ty) = d i m=0 f (m) i (x, y)t m ; ∀(t, x, y) ∈ k × g × g (1)
where y → (m!)f (m) i (x, y) is the differential at x of f i of the order m in the direction y. The elements f (m) i as defined by (1) are invariant elements of S(g) ⊗ k S(g) under the diagonal action of G on g × g. Note that f

(0) i (x, y) = f i (x) while f (d i ) i (x, y) = f i (y) for all (x, y) ∈ g × g.
Remark 1.4. The family . One says that the family F x is constructed by the argument shift method.

F x := {f (m) i (x, .); 1 ≤ i ≤ ℓ, 1 ≤ m ≤ d i } for x ∈ g, is a Poisson- commutative family of S(g) by
• Let i ∈ {1, . . . , ℓ}. For x in g, we denote by ϕ i (x) the element of g satisfying (df i ) x (y) = f

(1) i (x, y) = ϕ i (x), y , for all y in g. Thereby, ϕ i is an invariant element of S(g) ⊗ k g under the canonical action of G. We denote by ϕ (m) i , for 0 ≤ m ≤ d i -1, the elements of S(g) ⊗ k S(g) ⊗ k g defined by the equality:

ϕ i (x + ty) = d i -1 m=0 ϕ (m) i (x, y)t m , ∀(t, x, y) ∈ k × g × g. (2)
• For x ∈ g, we denote by g x = {y ∈ g | [y, x] = 0} the centralizer of x in g and by z(g x ) the center of g x . The set of regular elements of g is

g reg := {x ∈ g | dim g x = ℓ}
and we denote by g reg,ss the set of regular semisimple elements of g. Both g reg and g reg,ss are G-invariant dense open subsets of g.

We denote by C(x) the G-invariant cone generated by x and we denote by x s and x n the semisimple and nilpotent components of x respectively.

• The nilpotent cone of g is N(g). As a rule, for e ∈ N(g), we choose an sl 2 -triple (e, h, f ) in g given by the Jacobson-Morozov theorem [CMa93, Theorem 3.3.1]. In particular, it satisfies the equalities:

[h, e] = 2e, [e, f ] = h, [h, f ] = -2f
The action of ad h on g induces a Z-grading:

g = i∈Z g(i) , g(i) = {x ∈ g | [h, x] = ix}.
Recall that e, or G.e, is said to be even if g(i) = 0 for odd i. Note that e ∈ g(2), f ∈ g(-2) and that g e , z(g e ) and g f are all ad h-stable.

• All topological terms refer to the Zariski topology. If Y is a subset of a topological space X, we denote by Y the closure of Y in X.

1.4. Acknowledgments. We would like to thank O. Yakimova for her interest and useful discussions and more particularly for bringing Bolsinov's paper to our attention. We also thank A.G. Elashvili for suggesting Lawther-Testerman's paper [START_REF] Lawther | Centres of centralizers of unipotent elements in simple algebraic groups[END_REF] about the centers of centralizers of nilpotent elements. We are grateful to Ilya Zakharevich for pointing out a mistake in Lemma 3.1 in a previous version.

Contents

Preliminary results

We start in this section by reviewing some facts about the differentials of generators of S(g) g . Then, the goal of Subsection 2.2 is Theorem 2.7. We collect in Subsection 2.3 basic facts about induced nilpotent orbits.

2.1. Differentials of generators of S(g) g . According to subsection 1.3, the elements ϕ 1 , . . . ,ϕ ℓ of S(g) ⊗ k g are the differentials of f 1 , . . . ,f ℓ respectively. Since f i (g(x)) = f i (x) for all (x, g) ∈ g × G, the element ϕ i (x) centralizes x for all x ∈ g. Moreover:

Lemma 2.1. (i)[Ri87, Lemma 2.1] The elements ϕ 1 (x), . . . ,ϕ ℓ (x) belong to z(g e ).

(ii)[Ko63, Theorem 9] The elements ϕ 1 (x), . . . ,ϕ ℓ (x) are linearly independent elements of g if and only if x is regular. Moreover, if so, ϕ 1 (x), . . . ,ϕ ℓ (x) is a basis of g x .

We turn now to the elements ϕ (m) i , for i = 1, . . . , ℓ and 0 ≤ m ≤ d i -1, defined in Subsection 1.3 by (2). Recall that d i is the degree of the homogeneous polynomial f i , for i = 1, . . . , ℓ. The integers d 1 -1, . . . , d ℓ -1 are thus the exponents of g. By a classical result [Bou02, Ch. V, §5, Proposition 3], we have d i = b g where b g is the dimension of Borel subalgebras of g. For (x, y) in g × g, we set:

V x,y := span{ϕ (m) i (x, y) ; 1 ≤ i ≤ ℓ, 0 ≤ m ≤ d i -1}. (3)
The subspaces V x,y will play a central role throughout the note.

Remark 2.2. (1) For (x, y) ∈ g × g, the dimension of V x,y is at most b g since d i = b g . Moreover, for all (x, y) in a nonempty open subset of g × g, the equality holds [START_REF] Bolsinov | Commutative families of functions related to consistent Poisson brackets[END_REF]. Actually, in this note, we do not need this observation.

(2) By Lemma 2.1(ii), if x is regular, then g x is contained in V x,y for all y ∈ g. In particular, if so, dim [x, V x,y ] = dim V x,yℓ.

The subspaces V x,y were introduced and studied by Bolsinov in [START_REF] Bolsinov | Commutative families of functions related to consistent Poisson brackets[END_REF], motivated by the maximality of Poisson-commutative families in S(g). These subspaces have been recently exploited in [START_REF] Panyushev | The argument shift method and maximal commutative subalgebras of Poisson algebras[END_REF] and [START_REF] Charbonnel | Nilpotent bicone and characteristic submodule of a reductive Lie algebra[END_REF]. The following results are mostly due to Bosinov, [START_REF] Bolsinov | Commutative families of functions related to consistent Poisson brackets[END_REF]. We refer to [START_REF] Panyushev | The argument shift method and maximal commutative subalgebras of Poisson algebras[END_REF] for a more recent account about this topic. We present them in a slightly different way:

Lemma 2.3. Let (x, y) be in g reg × g.

(i) The subspace V x,y of g is the sum of the subspaces g x+ty where t runs through any nonempty open subset of k such that x + ty is regular for all t in this subset.

(ii) The subspace g y + V x,y is a totally isotropic subspace of g with respect to the Kirillov form

K y on g × g, (v, w) → y, [v, w] . Furthermore, dim (g y + V x,y ) ⊥ ≥ 1 2 dim G.y. (iii) The subspaces [x, V x,y ] and [y, V x,y ] are equal.
Proof. (i) Let O be a nonempty open subset of k such that x + ty is regular for all t in O. Such an open subset does exist since x is regular. Denote by V O the sum of all the subspaces g x+ty where t runs through O. For all t in O, g x+ty is generated by

ϕ 1 (x + ty), . . . ,ϕ ℓ (x + ty), cf. Lemma 2.1(ii). As a consequence, V O is contained in V x,y . Conversely, for i = 1, . . . , ℓ and for t 1 , . . . ,t d i pairwise different elements of O, ϕ (m) i (x, y) is a linear combination of ϕ i (x + t 1 y), . . . , ϕ i (x + t d i y); hence ϕ (m) i (x, y) belongs to V O . Thus V x,y is equal to V O , whence the assertion. (ii) results from [PY08, Proposition A4].
Notice that in (ii) the inequality is an easy consequence of the first statement.

At last, [PY08, Lemma A2] gives us (iii).

Let σ and σ i , for i = 1, . . . , ℓ, be the maps

g × g σ -→ k bg+ℓ (x, y) -→ (f (m) i (x, y))1≤i≤ℓ, 0≤m≤d i , g × g σ i -→ k d i +1 (x, y) -→ (f (m) i (x, y)) 0≤m≤d i
respectively, and denote by σ ′ (x, y) and σ ′ i (x, y) the tangent map at (x, y) of σ and σ i respectively. Then σ ′ i (x, y) is given by the differentials of the f (m) i 's at (x, y) and σ ′ (x, y) is given by the elements

σ ′ i (x, y). Lemma 2.4. Let (x, y) and (v, w) be in g × g. (i) For i = 1, . . . , ℓ, σ ′ i (x, y) maps (v, w) to ( ϕ i (x), v , ϕ (1)
i (x, y), v + ϕ (0) i (x, y), w , . . . , ϕ (d i -1) i (x, y), v + ϕ (d i -2) i (x, y), w , ϕ i (y), w ). (ii) Suppose that σ ′ (x, y)(v, w) = 0. Then, for w ′ in g, σ ′ (x, y)(v, w ′ ) = 0 if and only if w -w ′ is orthogonal to V x,y . (iii) For x ∈ g reg , σ ′ (x, y)(v, w ′ ) = 0 for some w ′ ∈ g if and only if v ∈ [x, g].
Proof. (i) The verifications are easy and left to the reader.

(ii) Since σ ′ (x, y)(v, w) = 0, σ ′ (x, y)(v, w ′ ) = 0 if and only if σ ′ (x, y)(v, ww ′ ) = 0 whence the statement by (i).

(iii) Suppose that x is regular and suppose that σ ′ (x, y)(v, w ′ ) = 0 for some w ′ ∈ g. Then by (i), v is orthogonal to the elements ϕ 1 (x), . . . ,ϕ ℓ (x). So by Lemma 2.1(ii), v is orthogonal to g x . Since g x is the orthogonal complement of [x, g] in g, we deduce that v lies in [x, g]. Conversely, since σ(x, y) = σ(g(x), g(y)) for all g in G, the element ([u, x], [u, y]) belongs to the kernel of σ ′ (x, y) for all u ∈ g. So, the converse implication follows.

2.2. On Bolsinov's criterion. Let a be in g and denote by π the map

g × G.a π -→ g × k bg+ℓ (x, y) -→ (x, σ(x, y)).
Remark 2.5. Recall that the family (F x ) x∈g constructed by the argument shift method consists of all elements f (m) i (x, .) for i = 1, . . . , ℓ and 1 ≤ m ≤ d i , see Remark 1.4. By definition of the morphism π, there is a family constructed by the argument shift method whose restriction to G.a contains 1 2 dim G.a algebraically independent functions if and only if π has a fiber of dimension

1 2 dim G.a.
In view of Theorem 1.1 and the above remark, we now concentrate on the fibers of π. For (x, y) ∈ g × G.a, denote by F x,y the fiber of π at π(x, y):

F x,y := {x} × {y ′ ∈ G.a | σ(x, y ′ ) = σ(x, y)}.
Lemma 2.6. Let (x, y) be in g × G.a.

(i) The irreducible components of F x,y have dimension at least 1 2 dim G.a. (ii) The fiber F x,y has dimension 1 2 dim G.a if and only if any irreducible component of F x,y contains an element (x, y ′ ) such that (g y ′ + V x,y ′ ) ⊥ has dimension 1 2 dim G.a. Proof. We prove (i) and (ii) all together. The tangent space T x,y ′ (F x,y ) of F x,y at (x, y ′ ) in F x,y identifies to the subspace of elements w of [y ′ , g] such that σ ′ (x, y ′ )(0, w) = 0. Hence, by Lemma 2.4(ii),

T x,y ′ (F x,y ) = [y ′ , g] ∩ V ⊥ x,y ′ = (g y ′ + V x,y ′ ) ⊥ , since [y ′ , g] = (g y ′ ) ⊥ . But by Lemma 2.3(ii), (g y ′ + V x,y ′ ) ⊥ has dimension at least 1 2 dim G.a; so does T x,y ′ (F x,

y ). This proves (i). Moreover, the equality holds if and only if (g y

′ + V x,y ′ ) ⊥ has dimension 1
2 dim G.a, whence the statement (ii). Theorem 2.7. The following conditions are equivalent:

(1) ind g a = ℓ;

(2) π has a fiber of dimension 1 2 dim G.a;

(3) there exists (x, y) ∈ g × G.a such that (g y + V x,y ) ⊥ has dimension 1 2 dim G.a; (4) there exists x in g reg such that dim (g a + V x,a ) = 1 2 (dim g + dim g a ); (5) there exists x in g reg such that dim V x,a = 1 2 dim G.a + ℓ; (6) σ(g × {a}) has dimension 1 2 dim G.a + ℓ.
Proof. By Theorem 1.1 and Remark 2.5, we have (1)⇔(2). Moreover, by Lemma 2.6(ii), we have (2)⇔(3).

(3)⇔(4): If (4) holds, so does (3). Indeed, if so,

dim g - 1 2 dim G.a = 1 2 (dim g + dim g a ) = dim (g a + V x,a ).
Conversely, suppose that (3) holds. By Lemma 2.3(ii), g y +V x,y has maximal dimension 1 2 (dim g+ dim g y ). So the same goes for all (x, y) in a G-invariant nonempty open subset of g × G.a. Hence, since the map (x, y) → V x,y is G-equivariant, there exists x in g reg such that

dim (V x,a + g a ) = 1 2 (dim g + dim g a ).
(4)⇔(5

): Let x be in g reg . By Lemma 2.3(iii), [x, V x,a ] = [a, V x,a ]. Hence g a ∩ V x,a has dimension ℓ by Remark 2.2(2). As a consequence, dim (g a + V x,a ) = dim g a + dim V x,a -ℓ,
whence the equivalence.

(2)⇔(6): Suppose that (2) holds. By Lemma 2.6, 1 2 dim G.a is the minimal dimension of the fibers of π. So, π(g × G.a) has dimension

dim g + dim G.a - 1 2 dim G.a = dim g + 1 2 dim G.a.
Denote by τ the restriction to π(g × G.a) of the projection map g × k bg+ℓ → k bg+ℓ . Then τ •π is the restriction of σ to g × G.a. Since σ is a G-invariant map, σ(g × {a}) = σ(g × G.a). Let (x, y) ∈ g reg,ss × G.a. The fiber of τ at z = σ(x, y) is G.x since x is a regular semisimple element of g. Hence,

dim σ(g × {a}) = dim π(g × G.a) -(dim g -ℓ) = 1 2 dim G.a + ℓ
and we obtain (6). Conversely, suppose that (6) holds. Then π(g × G.a) has dimension dim g + 1 2 dim G.a by the above equality. So the minimal dimension of the fibers of π is equal to

dim g + dim G.a -(dim g + 1 2 dim G.a) = 1 2 dim G.a
and (2) holds.

2.3. Induced and rigid nilpotent orbits. The definitions and results of this subsection are mostly extracted from [START_REF] Dixmier | Algèbres enveloppantes[END_REF], [START_REF] Dixmier | Polarisations dans les Algèbres de Lie II[END_REF], [START_REF] Lusztig | Induced unipotent classes[END_REF] and [START_REF] Borho | Über Bahnen und deren Deformationen bei linearen Aktionen reduktiver Gruppen[END_REF]. We refer to [START_REF] Collingwood | Nilpotent orbits in semisimple Lie algebras[END_REF] and [START_REF] Tauvel | Lie algebras and algebraic groups[END_REF] for recent surveys.

Let p be a proper parabolic subalgebra of g and let l be a reductive factor of p. We denote by p u the nilpotent radical of p. Denote by L the connected closed subgroup of G whose Lie algebra is adl and denote by P the normalizer of p in G. We shall say that e ∈ N(g) is an induced (respectively rigid) nilpotent element of g if the G-orbit of e is an induced (respectively rigid) nilpotent orbit of g. The following results are deeply linked to the properties of the sheets of g and the deformations of its G-orbits. We refer to [START_REF] Borho | Über Bahnen und deren Deformationen bei linearen Aktionen reduktiver Gruppen[END_REF] about these notions.

Theorem 2.9. (i) Let x be a non nilpotent element of g and let O g be the induced nilpotent orbit from the adjoint orbit of x n in g xs . Then O g is the unique nilpotent orbit contained in

C(x) whose dimension is dim G.x. Furthermore, C(x) ∩ N(g) = O g and C(x) ∩ N(g) is the nullvariety in C(x) of f i where i is an element of {1, . . . , ℓ} such that f i (x) = 0. (ii) Conversely, if O g is an induced nilpotent orbit, there exists a non nilpotent element x of g such that C(x) ∩ N(g) = O g .
Proof. (i) Let p be a parabolic subalgebra of g having g xs as a Levi factor. Denote by p u its nilpotent radical and by P the normalizer of p in G. Let O ′ be the adjoint orbit of x n in g xs . Claim 2.10. Let C be the P -invariant closed cone generated by x and let C 0 be the subset of nilpotent elements of C. Then C = kx s + O ′ + p u , C 0 = O ′ + p u and C 0 is an irreducible subset of dimension dim P (x).

Proof. The subset x s + O ′ + p u is an irreducible closed subset of p containing P (x). Moreover, its dimension is equal to

dim O ′ + dim p u = dim g xs -dim g x + dim p u = dim p -dim g x .
Since the closure of P (x) and x s + O ′ + p u are both irreducible subsets of g, they coincide. As a consequence, the set

kx s + O ′ + p u is contained in C. Since the former set is clearly a closed conical subset of g containing x, C = kx s + O ′ + p u . Then we deduce that C 0 = O ′ + p u .
Denote by G × P g the quotient of G × g under the right action of P given by (g, z).p := (gp, p -1 (z)). The map (g, z) → g(z) from G × g to g factorizes through the quotient map from G × g to G × P g. Since G/P is a projective variety, the so obtained map from G × P g to g is closed. Since C and C 0 are closed P -invariant subsets of g, G × P C and G × P C 0 are closed subsets of G × P g. Hence C(x) = G(C) and G(C 0 ) is a closed subset of g. So, by the claim, the subset of nilpotent elements of C(x) is irreducible since C 0 is irreducible. Since there are finitely many nilpotent orbits, the subset of nilpotent elements of C(x) is the closure of one nilpotent orbit. Denote it by Õ and prove Õ = O g . For all k, l in {1, . . . , ℓ}, denote by p k,l the polynomial function

p k,l := f k (x) d l f d k l -f l (x) d k f d l k
Then p k,l is G-invariant and homogeneous of degree d k d l . Moreover p k,l (x) = 0. As a consequence, C(x) is contained in the nullvariety of the functions p k,l , 1 ≤ k, l ≤ ℓ. Hence the nullvariety of f i in C(x) is contained in the nilpotent cone of g since it is the nullvariety in g of the functions f 1 , . . . ,f ℓ . (ii) By hypothesis, O g = Ind g l (O l ), where l is a proper Levi subalgebra of g and O l a nilpotent orbit in l. Let x s be an element of the center of l such that g xs = l, let x n be an element of O l and set x = x s + x n . Since l is a proper subalgebra, the element x is not nilpotent. So by (i), the subset of nilpotent elements of C(x) is the closure of O g .

Then dim Õ = dim C(x) -1 = dim G.x. Since O ′ + p u is contained in C(x),

Proof of Theorem 1.3 for induced nilpotent orbits

Let e be an induced nilpotent element. Let x be a non nilpotent element of g such that C(x) ∩ N(g) = G.e. Such an element does exist by Theorem 2.9(ii).

Theorem 3.1. Assume that inda x = rk a for all reductive subalgebras a strictly contained in g and for all x in a. Then for all induced nilpotent orbits O g in g and for all e in O g , indg e = ℓ.

Proof. Let O g be an induced nilpotent orbit and let e be in O g . Using Theorem 2.9(ii), we let x be a non nilpotent element of g such that C(x) ∩ N(g) = O g . Since x is not nilpotent, g x is the centralizer in the reductive Lie algebra g xs of the nilpotent element x n of g xs . Since g xs is strictly contained in g and has rank ℓ, the index of g x is equal to ℓ by hypothesis. Besides, by Theorem 2.7, (1)⇒(6), applied to x,

dim σ(g × {x}) = 1 2 dim G.x + ℓ. Since σ is G-invariant, σ(g × {x}) = σ(g × G.x).
Hence for all z in a dense subset of σ(g × G.x), the fiber of the restriction of σ to g × G.

x at z has minimal dimension

dim g + dim G.x -( 1 2 dim G.x + ℓ) = dim g + 1 2 dim G.x -ℓ.
Denote by Z the closure of σ(g × C(x)) in k d . We deduce from the above equality that Z has dimension

dim g + dim C(x) -(dim g + 1 2 dim G.x -ℓ) = dim C(x) - 1 2 dim G.x + ℓ = 1 2 dim G.e + ℓ + 1, since dim C(x) = dim G.x + 1 = dim G.e + 1.
Let i be in {1, . . . , ℓ} such that f i (x) = 0. For z ∈ k d , we write z = (z i,j ) 1≤i≤ℓ 0≤j≤d i its coordinates.

Let V i be the nullvariety in σ(g × C(x)) of the coordinate z i,d i . Then V i is not empty. Since σ(g × C(x)) is an irreducible constructible subset of k d and since z i,d i is not identically zero on σ(g × C(x)), V i has dimension 1 2 dim G.e + ℓ. By Theorem 2.9(i), the nullvariety of

f i in C(x) is equal to G.e. Hence g × G.e = σ -1 (V i ) ∩ (g × C(x))
So σ(g × G.e) is equal to V i and has dimension 1 2 dim G.e + ℓ. Then by Theorem 2.7, (6)⇒(1), the index of g e is equal to ℓ.

From that point, our goal is to prove Theorem 1.3 for rigid nilpotent elements; Theorem 3.1 tells us that this is enough to complete the proof.

The Slodowy slice and the property (P)

In this section, we introduce a property (P) in Definition 4.2 and we prove that e ∈ N(g) has Property (P) if and only if indg e = ℓ (Theorem 4.13). Then, we will show in the next section that all rigid nilpotent orbits of g but seven orbits (one in the type E 7 and six in the type E 8 ) do have Property (P).

4.1. Blowing up of S. Let e be a nilpotent element of g and consider an sl 2 -triple (e, h, f ) containing e as in Subsection 1.3. The Slodowy slice is the affine subspace S := e + g f of g which is a transverse variety to the adjoint orbit G.e. Denote by B e (S) the blowing up of S centered at e and let p : B e (S) → S be the canonical morphism. The variety S is smooth and p -1 (e) is a smooth irreducible hypersurface of B e (S). The use of the blowing-up B e (S) for the computation of the index was initiated by the first author in [START_REF] Charbonnel | Propriétés (Q) et (C). Variété commutante[END_REF] and resumed by the second author in [START_REF] Moreau | Indice du normalisateur du centralisateur d'un élément nilpotent dans une algèbre de Lie semi-simple[END_REF]. Here, we use again this technique to study the index of g e . Describe first the main tools extracted from [START_REF] Charbonnel | Propriétés (Q) et (C). Variété commutante[END_REF] we need.

For Y an open subset of B e (S), we denote by k[Y ] the algebra of regular functions on Y . By [Ch04, Théorème 3.3], we have: such that α(x) = g p(x) if p(x) is regular. Furthermore, the map α is uniquely defined by this condition. In fact, this result is a consequence of [Sh94, Ch. VI, Theorem 1]. From now on, α stands for the so-defined map. Since p -1 (e) is an hypersurface and since Ω is a big open subset of B e (S), note that Ω ∩ p -1 (e) is a nonempty set. In addition, α(x) ⊂ g p(x) for all x ∈ Ω.

Definition 4.2. We say that e has Property (P) if z(g e ) ⊂ α(x) for all x in Ω ∩ p -1 (e).

Remark 4.3. Suppose that e is regular. Then g e is a commutative algebra, i.e. z(g e ) = g e . If

x ∈ Ω ∩ p -1 (e), then α(x) = g e since p(x) = e is regular in this case. On the other hand, ind g e = dim g e = ℓ since e is regular. So e has Property (P) and indg e = ℓ.

4.2. On the property (P). This subsection aims to show: Property (P) holds for e if and only if indg e = ℓ. As a consequence of Remark 4.3, we can (and will) assume that e is a nonregular nilpotent element of g. As a first step, we will state in Corollary 4.12 that, if (P) holds, then so does the assertion (B) of Theorem 4.1. Let L g be the S(g)-submodule of ϕ ∈ S(g)⊗ k g satisfying [ϕ(x), x] = 0 for all x in g. It is known that L g is a free module of basis ϕ 1 , . . . , ϕ ℓ , cf. [START_REF] Dixmier | Champs de vecteurs adjoints sur les groupes et algèbres de Lie semi-simples[END_REF]. We investigate an analogous property for the Slodowy slice S = e + g f . We denote by S reg the intersection of S and g reg . As e is nonregular, the set (S \ S reg ) contains e.

Lemma 4.4. The set S \ S reg has codimension 3 in S and each irreducible component of S \ S reg contains e.

Proof. Let us consider the morphism

G × S -→ g (g, x) -→ g(x)
By a Slodowy's result [START_REF] Slodowy | Simple singularities and simple algebraic groups[END_REF], this morphism is a smooth morphism. So its fibers are equidimensional of dimension dim g f . In addition, by [V72], g \ g reg is a G-invariant equidimensional closed subset of g of codimension 3. Hence S \ S reg is an equidimensional closed subset of S of codimension 3.

Denoting by t → g(t) the one parameter subgroup of G generated by ad h, S and S \ S reg are stable under the action of t -2 g(t) for all t in k * . Furthermore, for all x in S, t -2 g(t)(x) goes to e when t goes to ∞, whence the lemma.

Denote by k[S] the algebra of regular functions on S and denote by L

S the k[S]-submodule of ϕ ∈ k[S] ⊗ k g satisfying [ϕ(x), x] = 0 for all x in S.
Lemma 4.5. The module L S is a free module of basis ϕ 1 | S , . . . , ϕ ℓ | S where ϕ i | S is the restriction to S of ϕ i for i = 1, . . . , ℓ.

Proof. Let ϕ be in L S . There are regular functions a 1 , . . . , a ℓ on S reg satisfying

ϕ(x) = a 1 (x)ϕ 1 | S (x) + • • • + a ℓ (x)ϕ ℓ | S (x)
for all x ∈ S reg , by Lemma 2.1(ii). By Lemma 4.4, S \ S reg has codimension 3 in S. Hence a 1 , . . . , a ℓ have polynomial extensions to S since S is normal. So the maps ϕ 1 | S , . . . , ϕ ℓ | S generate L S . Moreover, by Lemma 2.1(ii) for all x ∈ S reg , ϕ 1 (x), . . . , ϕ ℓ (x) are linearly independent, whence the statement.

The following proposition accounts for an important step to interpret Assertion (B) of Theorem 4.1: Proposition 4.6. Let ϕ be in k[S] ⊗ k g such that ϕ(x) ∈ [g, x] for all x in a nonempty open subset of g. Then there exists a polynomial map ψ ∈ k[S] ⊗ k g such that ϕ(x) = [ψ(x), x] for all x ∈ S.

Proof. Since g x is the orthogonal complement of [x, g] in g, our hypothesis says that ϕ(x) is orthogonal to g x for all x in a nonempty open subset S ′ of S. The intersection S ′ ∩ S reg is not empty; so by Lemma 2.1(ii), ϕ(x), ϕ i | S (x) = 0 for all i = 1, . . . , ℓ and for all x ∈ S ′ ∩ S reg . Therefore, by continuity, ϕ(x), ϕ i | S (x) = 0 for all i = 1, . . . , ℓ and all x ∈ S. Hence ϕ(x) ∈ [x, g] for all x ∈ S reg by Lemma 2.1(ii) again. Consequently by Lemma 4.4, Lemma 4.5 and the proof of the main theorem of [START_REF] Dixmier | Champs de vecteurs adjoints sur les groupes et algèbres de Lie semi-simples[END_REF], there exists an element ψ ∈ k[S] ⊗ k g which satisfies the condition of the proposition.

Let u 1 , . . . ,u m be a basis of g f and let u * 1 , . . . , u * m be the corresponding coordinate system of S = e + g f . There is an affine open subset Y ⊂ B e (S) with Y ∩ p -1 (e) = ∅ such that k[Y ] is the set of linear combinations of monomials in (u * 1 ) -1 , u * 1 , . . . , u * m whose total degree is nonnegative. In particular, we have a global coordinates system u * 1 , v * 2 , . . . , v * m on Y satisfying the relations:

u * 2 = u * 1 v * 2 , . . . , u * m = u * 1 v * m . (4)
Note that, for x ∈ Y , we so have:

p(x) = e + u * 1 (x)(u 1 + v * 2 (x)u 2 + • • • + v * m (x)u m ).
So, the image of Y by p is the union of {e} and the complementary in S of the nullvariety of u * 1 . Let Y ′ be an affine open subset of Y contained in Ω and having a nonempty intersection with p -1 (e). Denote by L Y ′ the set of regular maps ϕ from Y ′ to g satisfying [ϕ(x), p(x)] = 0 for all x ∈ Y ′ . Lemma 4.7. Suppose that e has Property (P). For each z ∈ z(g e ), there exists

ψ z ∈ k[Y ′ ] ⊗ k g such that z -u * 1 ψ z belongs to L Y ′ .
Proof. Let z be in z(g e ). Since Y ′ ⊂ Ω, for each y ∈ Y ′ , there exists an affine open subset U y of Y ′ containing y and regular maps ν 1 , . . . ,ν ℓ from U y to g such that ν 1 (x), . . . ,ν ℓ (x) is a basis of α(x) for all x ∈ U y . Let y be in Y ′ . We consider two cases:

(1) Suppose p(y) = e. Since e has Property (P), there exist regular functions a 1 , . . . ,a ℓ on U y satisfying

z = a 1 (x)ν 1 (x) + • • • + a ℓ (x)ν ℓ (x),
for all x ∈ U y ∩ p -1 (e). The intersection U y ∩ p -1 (e) is the set of zeroes of u * 1 in U y . So there exists a regular map ψ from U y to g which satisfies the equality:

z -u * 1 ψ = a 1 ν 1 + • • • + a ℓ ν ℓ . Hence [z -u * 1 (x)ψ(x), p(x)] = 0 for all x ∈ U y since α(x) is contained in g p(x) for all x ∈ Ω. (2) Suppose p(y) = e.
Then we can assume that U y ∩ p -1 (e) = ∅ and the map ψ = (u * 1 ) -1 z satisfies the condition: [zu * 1 (x)ψ(x), p(x)] = 0 for all x ∈ U y . In both cases (1) or (2), we have found a regular map ψ y from U y to g satisfying: [z -(u * 1 ψ y )(x), p(x)] = 0 for all x ∈ U y .

Let y 1 , . . . ,y k be in Y ′ such that the open subsets U y 1 , . . . ,U y k cover Y ′ . For i = 1, . . . , k, we denote by ψ i a regular map from U y i to g such that zu * 1 ψ i is in Γ(U y i , L) where L is the localization of L Y ′ on Y ′ . Then for i, j = 1, . . . , k, ψ iψ j is in Γ(U y i ∩ U y j , L). Since Y ′ is affine, H 1 (Y ′ , L) = 0. So, for i = 1, . . . , l, there exists ψ i in Γ(U y i , L) such that ψ iψ j is equal to ψ iψ j on U y i ∩ U y j for all i, j. Then there exists a well-defined map ψ z from Y ′ to g whose restriction to U y i is equal to ψ iψ i for all i, and such that zu * 1 ψ z belongs to L Y ′ . Finally, the map ψ z verifies the required property.

Let z be in z(g e ). We denote by ϕ z the regular map from Y to g defined by:

ϕ z (x) = [z, u 1 ] + v * 2 (x)[z, u 2 ] + • • • + v * m (x)[z, u m ], for all x ∈ Y. (5)
Corollary 4.8. Suppose that e has Property (P) and let z be in z(g e ). There exists

ψ z in k[Y ′ ]⊗ k g such that ϕ z (x) = [ψ z (x), p(x)] for all x ∈ Y ′ .
Proof. By Lemma 4.7, there exists

ψ z in k[Y ′ ] ⊗ k g such that z -u * 1 ψ z is in L Y ′ . Then u * 1 ϕ z (x) = [z, p(x)] = [z -u * 1 ψ z (x), p(x)] + u * 1 [ψ z (x), p(x)], for all x ∈ Y ′ . So the map ψ z is convenient, since u *
1 is not identically zero on Y ′ . The following lemma is easy but helpful for Proposition 4.10: Lemma 4.9. Let v be in g e . Then, v belongs to z(g e ) if and only if

[v, g f ] ⊂ [e, g].
Proof. Since [x, g] is the orthogonal complement of g x in g for all x ∈ g, we have:

[v, g f ] ⊂ [e, g] ⇐⇒ [v, g f ], g e = 0 ⇐⇒ [v, g e ], g f = 0 ⇐⇒ [v, g e ] ⊂ [f, g].
But g is the direct sum of g e and [f, g] and [v, g e ] is contained in g e since v ∈ g e . Hence [v, g f ] is contained in [e, g] if and only if v is in z(g e ).

Proposition 4.10. Suppose that e has Property (P) and let ϕ be in k[Y ]⊗ k g such that ϕ(x) ∈ [g, p(x)] for all x ∈ Y . Then there exists

ψ in k[Y ′ ] ⊗ k g such that ϕ(x) = [ψ(x), p(x)] for all x ∈ Y ′ .
Proof. Since ϕ is a regular map from Y to g, there is a nonnegative integer d and ϕ

∈ k[S] ⊗ k g such that (u * 1 ) d (x)ϕ(x) = ( ϕ•p)(x), ∀x ∈ Y (6)
and ϕ is a linear combination of monomials in u * 1 , . . . , u * m whose total degree is at least d. By hypothesis on ϕ, we deduce that for all x ∈ S such that u

* 1 (x) = 0, ϕ(x) is in [g, x]
. Hence by Proposition 4.6, there exists

ψ in k[S] ⊗ k g such that ϕ(x) = [ ψ(x), x] for all x ∈ S.
Denote by ψ ′ the sum of monomials of degree at least d in ψ and denote by

ψ ′ the element of k[Y ] ⊗ k g satisfying (u * 1 ) d (x)ψ ′ (x) = ( ψ ′ •p)(x), ∀x ∈ Y. (7) Then we set, for x ∈ Y , ϕ ′ (x) := ϕ(x) -[ψ ′ (x), p(x)]. We have to prove the existence of an element ψ ′′ in k[Y ′ ] ⊗ k g such that ϕ ′ (x) = [ψ ′′ (x), p(x)] for all x ∈ Y ′ .
• If d = 0, then ϕ = ϕ•p, ψ ′ = ψ and ϕ ′ = 0; so ψ ′ is convenient in that case.

• If d = 1, we can write

u * 1 (x)ϕ(x) = ϕ(p(x)) = [ ψ(p(x)), e + u * 1 (x)(u 1 + v * 2 (x)u 2 + • • • + v * m (x)u m )], for all x ∈ Y , whence we deduce u * 1 (x)(ϕ(x) -[ψ ′ (x), p(x)]) = [ ψ(e), e + u * 1 (x)(u 1 + v * 2 (x)u 2 + • • • + v * m (x)u m )
] for all x ∈ Y . Hence ψ(e) belongs to g e and [ ψ(e), u i ] ∈ [e, g] for all i = 1, . . . , m, since ϕ(x) ∈ [e, g] for all x ∈ Y ∩ p -1 (e). Then ψ(e) is in z(g e ) by Lemma 4.9. So by Corollary 4.8, ϕ ′ has the desired property.

• Suppose d > 1. For i = (i 1 , . . . , i m ) ∈ N m , we set |i|

:= i 1 + • • • + i m and we denote by ψ i the coefficient of (u * 1 ) i 1 • • • (u * m ) im in ψ.
By Corollary 4.8, it suffices to prove:

ψ i = 0 if |i| < d -1; ψ i ∈ z(g e ) if |i| = d -1 .
For i ∈ N m and j ∈ {1, . . . , m}, we define the element i(j) of N m by: i(j) := (i 1 , . . . , i j-1 , i j + 1, i j+1 , . . . , i m ).

It suffices to prove:

Claim 4.11. For |i| ≤ d-1, ψ i is an element of g e such that [ψ i , u j ]+[ψ i(j) , e] = 0 for j = 1, . . . , m.
Indeed, by Lemma 4.9, if

[ψ i , u j ] + [ψ i(j)
, e] = 0 and ψ i ∈ g e for all j = 1, . . . , m, then ψ i ∈ z(g e ). Furthermore, if

[ψ i , u j ] + [ψ i(j)
, e] = 0 and ψ i ∈ g e and ψ i(j) ∈ g e for all j = 1, . . . , m, then ψ i = 0 since z(g e ) ∩ g f = 0. So only remains to prove Claim 4.11. We prove the claim by induction on |i|. Arguing as in the case d = 1, we prove the claim for |i| = 0. We suppose the claim true for all |i| ≤ l -1 for some 0 < l ≤ d -2. We have to prove the statement for all |i| ≤ l. By what foregoes and by induction hypothesis,

ψ i = 0 for |i| ≤ l -2. For k = l + 1, l + 2, we consider the ring k[τ k ] where τ k k = 0. Since (u * 1 ) d vanishes on the set of k[τ l+1 ]-points x = x 0 + x 1 τ l+1 + • • • + x l τ l l+1 of Y whose source x 0 is a zero of u * 1 , 0 = [ ψ(e + τ l+1 v), e + τ l+1 v] = |i|=l τ l l+1 [ψ i , e](u * 1 ) i 1 • • • (u * m ) im (v),
for all v ∈ g f . So ψ i ∈ g e for |i| = l.

For |i| equal to l, the term in

τ l+1 l+2 (u * 1 ) i 1 • • • (u * i j-1 ) i j-1 (u * i j +1 ) i j +1 (u * i j+1 ) i j+1 • • • (u * m ) im (v) of [ ψ(e + τ l+2 v), e + τ l+2 v] is equal to [ψ i(j) , e] + [ψ i , u j ]. Since (u * 1 ) d vanishes on the set of k[τ l+2 ]-points of Y whose source is a zero of u *
1 , this term is equal to 0, whence the claim.

Recall that Y ′ is an affine open subset of Y contained in Ω and having a nonempty intersection with p -1 (e).

Corollary 4.12. Suppose that e has Property (P). Let ϕ be in k

[Y ′ ]⊗ k g such that ϕ(x) ∈ [g, p(x)] for all x ∈ Y ′ . Then there exists ψ in k[Y ′ ] ⊗ k g such that ϕ(x) = [ψ(x), p(x)] for all x ∈ Y ′ . Proof. For a ∈ k[Y ],
, . . . , a k in k[Y ]. Since ϕ is a regular map from Y ′ to g, there is m i ≥ 0 such that a m i i ϕ is the restriction to Y ′ of some regular map ϕ i from Y to g. For m i big enough, ϕ i vanishes on Y \ D(a i ); hence ϕ i (x) ∈ [g, p(x)] for all x ∈ Y . So, by Proposition 4.6, there is a regular map ψ i from Y ′ to g such that ϕ i (x) = [ψ i (x), p(x)] for all x ∈ Y ′ . Then for all x ∈ D(a i ), we have ϕ(x) = [a i (x) -m i ψ i (x), p(x)]. Since Y ′ is
an affine open subset of Y , there exists a regular map ψ from Y ′ to g which satisfies the condition of the corollary.

We are now in position to prove the main result of this section: Theorem 4.13. The equality indg e = ℓ holds if and only if e has Property (P).

Proof. By Corollary 4.12, if e has Property (P), then Assertion (B) of Theorem 4.1 is satisfied. Conversely, suppose that indg e = ℓ and show that e has Property (P). By Theorem 4.1, (A)⇒(B), Assertion (B) is satisfied. We choose an affine open subset Y ′ of Y , contained in Ω, such that Y ′ ∩ p -1 (e) = and verifying the condition of the assertion (B). Let z ∈ z(g e ). Recall that the map ϕ z is defined by (5). Let x be in Y ′ . If u * 1 (x) = 0, then ϕ z (x) belongs to [g, p(x)] by (5). If u * 1 (x) = 0 , then by Lemma 4.9, ϕ z (x) belongs to [e, g]. So there exists a regular map ψ from Y ′ to g such that ϕ z (x) = [ψ(x), p(x)] for all x ∈ Y ′ by Assertion (B). Hence we have [zu * 1 ψ(x), p(x)] = 0, for all x ∈ Y ′ since (u * 1 ϕ z )(x) = [z, p(x)] for all x ∈ Y . So α(x) contains z for all x in Ω ∩ Y ′ ∩ p -1 (e). Since p -1 (e) is irreducible, we deduce that e has Property (P).

A new formulation of the property (P).

Recall that Property (P) is introduced in Definition 4.2. As has been noticed in the proof of Lemma 4.4, the morphism G × S → g, (g, x) → g(x) is smooth. As a consequence, the set S reg of v ∈ S such that v is regular is a nonempty open subset of S. For x in S reg , g e+t(x-e) has dimension ℓ for all t in a nonempty open subset of k since x = e + (xe) is regular. Furthermore, since k has dimension 1, [Sh94, Ch. VI, Theorem 1] asserts that there is a unique regular map

β x : k → Gr ℓ (g)
satisfying β x (t) = g e+t(x-e) for all t in a nonempty open subset of k.

Recall that Y is an affine open subset of B e (S) with Y ∩ p -1 (e) = ∅ and that u * 1 , v * 2 , . . . , v * m is a global coordinates system of Y , cf. (4). Let S ′ reg be the subset of x in S reg such that u * 1 (x) = 0. For x in S ′ reg , we denote by x the element of Y whose coordinates are 0, v * 2 (x), . . . , v * m (x).

Lemma 4.14. Let x be in S ′ reg . (i) The subspace β x (0) is contained in g e . (ii) If x ∈ Ω, then α( x) = β x (0).

Proof. (i) The map β x is a regular map and [β x (t), e + t(xe)] = 0 for all t in a nonempty open subset of k. So, β x (0) is contained in g e .

(ii) Since S ′ reg has an empty intersection with the nullvariety of u * 1 in S, the restriction of p to p -1 (S ′ reg ) is an isomorphism from p -1 (S ′ reg ) to S ′ reg . Furthermore, β x (t) = α(p -1 (e + txte)) for any t in k such that e + t(xe) belongs to S ′ reg and p -1 (e + txte) goes to x when t goes to 0. Hence β x (0) is equal to α( x) since α and β are regular maps. reg by the map x → x is dense in Y ′′ . Since α is regular, e has property (P) if and only if α(x) contains z(g e ) for all x in a dense subset of Y ′′ . By Lemma 4.14(ii), the latter property is equivalent to the fact that β x (0) contains z(g e ) for all x in a dense open subset of S ′′ reg . Corollary 4.16. (i) If z(g e ) is generated by ϕ 1 (e), . . . , ϕ ℓ (e), then e has Property (P).

(ii) If z(g e ) has dimension 1, then e has Property (P).

Proof. Recall that ϕ i (e) belongs to z(g e ), for all i = 1, . . . , ℓ, by Lemma 2.1(i). Moreover, for all x in S reg and all i = 1, . . . , ℓ, ϕ i (e + t(xe)) belongs to g e+t(x-e) for any t in k. So by continuity, ϕ i (e) belongs to β x (0). As a consequence, whenever z(g e ) is generated by ϕ 1 (e), . . . ,ϕ ℓ (e), e has Property (P) by Corollary 4.15.

(ii) is an immediate consequence of (i) since ϕ 1 (e) = e by our choice of d 1 .

Proof of Theorem 1.3 for rigid nilpotent orbits

We intend to prove in this section the following theorem:

Theorem 5.1. Suppose that g is reductive and let e be a rigid nilpotent element of g. Then the index of g e is equal to ℓ.

Theorem 5.1 will complete the proof of Theorem 1.3 by Theorem 3.1. As explained in introduction, we can assume that g is simple. We consider two cases, according to g has classical type or exceptional type.

5.1. The classical case. Assume that g is simple of classical type. More precisely, assume that g is one of the Lie algebras sl ℓ+1 (k), so 2ℓ+1 (k), sp 2ℓ (k), so 2ℓ (k).

Lemma 5.2. Let m be a positive integer such that x mtr x m belongs to g for all x in g. Then e m belongs to the subspace generated by ϕ 1 (e), . . . ,ϕ ℓ (e).

Proof. Recall that L g is the submodule of elements ϕ of S(g) ⊗ k g such that [x, ϕ(x)] = 0 for all x in g. According to [START_REF] Dixmier | Champs de vecteurs adjoints sur les groupes et algèbres de Lie semi-simples[END_REF], L g is a free module generated by the ϕ ′ i s. For all x in g, [x, x m ] = 0. Hence there exist polynomial functions a 1 , . . . ,a ℓ on g such that

x m -tr x m = a 1 (x)ϕ 1 (x) + • • • + a ℓ (x)ϕ ℓ (x)
for all x in g, whence the lemma.

Theorem 5.3. Let e be a rigid nilpotent element. Then z(g e ) is generated by powers of e. In particular, the index of g e is equal to ℓ.

Proof. Let us prove the first assertion. If g has type A or C, then z(g e ) is generated by powers of e by [Mo06c, Théorème 1.1.8] or [Y06b]. So we can assume that g has type B or D.

Set n := 2ℓ + 1 if g has type B ℓ and n := 2ℓ if g has type D ℓ . Denote by (n 1 , . . . ,n k ), with n 1 ≥ • • • ≥n k , the partition of n corresponding to the nilpotent element e. By [Mo06c, Théorème 1.1.8] or [Y06b], z(g e ) is not generated by powers of e if and only if n 1 and n 2 are both odd integers and n 3 < n 2 . On the other hand, since e is rigid, n k is equal to 1, n i ≤ n i+1 ≤ n i + 1 and all odd integers of the partition (n 1 , . . . ,n k ) have a multiplicity different from 2 [Ke83, Sp82, ch. II] or [CMa93, Corollary 7.3.5]. Hence, the preceding criterion is not satisfied for e. Then, the second assertion results from Lemma 5.2, Corollary 4.16(i) and Theorem 4.13.

Remark 5.4. Yakimova's proof of Elashvili's conjecture in the classical case is shorter and more elementary [Y06a]. The results of Section 4 will serve the exceptional case in a more relevant way.

5.2. The exceptional case. We let in this subsection g be simple of exceptional type and we assume that e is a nonzero rigid nilpotent element of g. The dimension of the center of centralizers of nilpotent elements has been recently described in [LT08, Theorem 4]. On the other hand, we have explicit computations for the rigid nilpotent orbits in the exceptional types due to A.G. Elashvili. These computations are collected in [Sp82, Appendix of Chap. II] and a complete version was published later in [E85b]. From all this, we observe that the center of g e has dimension 1 in most cases. In more details, we have: Proposition 5.5. Let e be nonzero rigid nilpotent element of g.

(i) Suppose that g has type G 2 , F 4 or E 6 . Then dim z(g e ) = 1.

(ii) Suppose that g has type E 7 . If g e has dimension 41, then dim z(g e ) = 2; otherwise dim z(g e ) = 1.

(iii) Suppose that g has type E 8 . If g e has dimension 112, 84, 76, or 46, then dim z(g e ) = 2, if g e has dimension 72, then dim z(g e ) = 3; otherwise dim z(g e ) = 1. By Corollary 4.16(ii), ind g e = ℓ whenever dim z(g e ) = 1. So, as an immediate consequence of Proposition 5.5, we obtain: Corollary 5.6. Suppose that either g has type G 2 , F 4 , E 6 , or g has type E 7 and dim g e = 41, or g has type E 8 and dim g e ∈ {112, 84, 76, 72, 46}. Then dim z(g e ) = 1 and the index of g e is equal to ℓ.

According to Corollary 5.6, it remains 7 cases; there are indeed two rigid nilpotent orbits of codimension 46 in E 8 . We handle now these remaining cases. We process here in a different way; we study technical conditions on g e under which ind g e = ℓ. For the moment, we state general results about the index.

Let a be an algebraic Lie algebra. Recall that the stabilizer of ξ ∈ a * for the coadjoint representation is denoted by a ξ and that ξ is regular if dim a ξ = inda. Choose a commutative subalgebra t of a consisted of semisimple elements of a and denote by z a (t) the centralizer of t in a. Then a = z a (t) ⊕ [t, a]. The dual z a (t) * of z a (t) identifies to the orthogonal complement of [t, a] in a * . Thus, ξ ∈ z a (t) * if and only if t is contained in a ξ . Lemma 5.7. Suppose that there exists ξ in z a (t) * such that dim (a ξ ∩ [t, a]) ≤ 2. Then ind a ≤ indz a (t) + 1.

Proof. Let T be the closure in z a (t) * × Gr 3 ([t, a]) of the subset of elements (η, E) such that η is a regular element of z a (t) * and E is contained in a η . The image T 1 of T by the projection from z a (t) * × Gr 3 ([t, a]) to z a (t) * is closed in z a (t) * . By hypothesis, T 1 is not equal to z a (t) * since for all η in T 1 , dim (a η ∩ [t, a]) ≥ 3. Hence there exists a regular element ξ 0 in z a (t) * such that dim (a

ξ 0 ∩ [t, a]) ≤ 2. Since t is contained in a ξ 0 , a ξ 0 = z a (t) ξ 0 ⊕ a ξ 0 ∩ [t, a].
If [t, a] ∩ a ξ 0 = {0} then inda is at most indz a (t). Otherwise, a ξ 0 is not a commutative subalgebra since t is contained in a ξ 0 . Hence ξ 0 is not a regular element of a * , so ind a < dim a ξ 0 . Since dim a ξ 0 ≤ indz a (t) + 2, the lemma follows.

From now on, we assume that a = g e . As a rigid nilpotent element of g, e is a nondistinguished nilpotent element. So we can choose a nonzero commutative subalgebra t of g e consisted of semisimple elements. Denote by l the centralizer of t in g. As a Levi subalgebra of g, l is a reductive Lie algebra whose rank is ℓ. Moreover its dimension is strictly smaller than dim g. In the preceding notations, we have z g e (t) = z g (t) e = l e . Let t 1 be a commutative subalgebra of l e containing t and consisting of semisimple elements of l. Then [t, g e ] is stable under the adjoint action of t 1 . For λ in t * 1 , denote by g e λ the λ-weight space of the adjoint action of t 1 in g e . Lemma 5.8. Let λ ∈ t * 1 be a nonzero weight of the adjoint action of t 1 in g e . Then -λ is also a weight for this action and λ and -λ have the same multiplicity. Moreover, g e λ is contained in [t, g e ] if and only if the restriction of λ to t is not identically zero.

Proof. By definition, g e λ ∩ l e = {0} if and only if the restriction of λ to t is not identically zero. So g e λ is contained in [t, g e ] if and only if the restriction of λ to t is not equal to 0 since g e λ = (g e λ ∩ l e ) ⊕ (g e λ ∩ [t, g e ]). The subalgebra t 1 is contained in a reductive factor of g e . So we can choose h and f such that t 1 is contained in g e ∩ g f . As a consequence, any weight of the adjoint action of t 1 in g f is a weight of the adjoint action of t 1 in g e with the same multiplicity. Furthermore, the t 1 -module g f for the ajoint action is isomorphic to the t 1 -module (g e ) * for the coadjoint action. So -λ is a fi;; end;; return List([1..4],i->M(k)[n(i,k,p)]);; end;; r := function(t) if t=1 then return 1; elif t=-1 then return 1;; elif t=0 then return 2;; fi;; end;; Q := function(k,s,t) local q;; q := function In conclusion, Condition (1) of Proposition 5.9 is satisfied for t := kt and t 1 :=span(Bt1). if k=1 and s=1 and t=-1 then return 2;; elif k=-1 and s=-1 and t=1 then return 2;; elif k=1 and s=1 and t=0 then return 2;; elif k=-1 and s=-1 and t=0 then return 2;; elif k=1 and s=-1 and t=1 then return 2;; elif k=-1 and s=1 and t=-1 then return 2;; elif k=1 and s=-1 and t=0 then return 2;; elif k=-1 and s=1 and t=0 then return 2;; elif k=1 and s=0 and t=1 then return 2;; elif k=-1 and s=0 and t=-1 then return 2;; elif k=1 and s=0 and t=-1 then return 2;; elif k=-1 and s=0 and t=1 then return 2;; elif k=1 and s=0 and t=0 then return 5;; elif k=-1 and s=0 and t=0 then return 5;; fi;; end;; Q := function(k,s,t) local q;; q := function In conclusion, Condition (2) of Proposition 5.9 is satisfied for t := kt and t 1 :=span(Bt1).

  Theorem 2.8 ([CMa93],Theorem 7.1.1). Let O l be a nilpotent orbit of l. There exists a unique nilpotent orbit O g in g whose intersection with O l +p u is a dense open subset of O l +p u . Moreover, the intersection of O g and O l + p u consists of a single P -orbit and codim g (O g ) = codim l (O l ).The orbit O g only depends on l and not on the choice of a parabolic subalgebra p containing it [CMa93, Theorem 7.1.3]. By definition, the orbit O g is called the induced orbit from O l ; it is denoted by Ind g l (O l ). If O l = 0, then we call O g a Richardson orbit. For example all even nilpotent orbits are Richardson[START_REF] Collingwood | Nilpotent orbits in semisimple Lie algebras[END_REF] Corollary 7.1.7]. In turn, not all nilpotent orbits are induced from another one. A nilpotent orbit which is not induced in a proper way from another one is called rigid.

Theorem 4. 1 .

 1 The following two assertions are equivalent: (A) the equality ind g e = ℓ holds, (B) there exists an affine open subset Y ⊂ B e (S) such that Y ∩ p -1 (e) = ∅ and satisfying the following property:for any regular map ϕ ∈ k[Y ] ⊗ k g such that ϕ(x) ∈ [g, p(x)] for all x ∈ Y , there exists ψ ∈ k[Y ] ⊗ k g such that ϕ(x) = [ψ(x), p(x)] for all x ∈ Y . An open subset Ω ⊂ B e (S)is called a big open subset if B e (S) \ Ω has codimension at least 2 in B e (S). As explained in [Ch04, Section 2], there exists a big open subset Ω of B e (S) and a regular map α : Ω → Gr ℓ (g)

  denote by D(a) the principal open subset defined by a. Let D(a 1 ), . . . , D(a m ) be an open covering of Y ′ by principal open subsets of Y , with a 1

  (j,k,s,t) if j=1 then return Position(List([1..4], i->Bt1[3]*Bc[N(k,s)[i]]-t*Bc[N(k,s)[i]]),0*x[1]);; else return q(j-1,k,s,t)+Position(List([q(j-1,k,s,t)+1..4], i->Bt1[3]*Bc[N(k,s)[i]]-t*Bc[N(k,s)[i]]),0*x[1]);; fi;; end;; return List([1..r(t)],i->N(k,s)[q(i,k,s,t)]);; end;; W := [[1,1,1],[1,-1,1],[1,1,-1],[1,-1,-1],[1,1,0],[1,-1,0]];; S := 2*(1+Sum(List([1..Length(W)],i->Length(Q(W[i][1],W[i][2],W[i][3]))))) +Dimension(le)-Dimension(c); > 0 A := function(i) return List([1..r(W[i][3])],t->List([1..r(W[i][3])], s->Bc[Q(W[i][1],W[i][2],W[i][3])[s]]*Bc[Q(-W[i][1],-W[i][2],-W[i][3])[t]]));; end;; A(1);A(2);A(3);A(4);A(5);A(6); > [ [ (-1)*v.63 ] ] > [ [ v.63 ] ] > [ [ v.63 ] ] > [ [ (-1)*v.63 ] ] > [ [ (-1)*v.57+(-1)*v.60, (-1)*v.63 ], [ (-1)*v.63, 0*v.1 ] ] > [ [ (-1)*v.57+(-1)*v.60, (-1)*v.63 ], [ (-1)*v.63, 0*v.1 ] ] a := Bc[M(2)[1]]*Bc[M(-2)[1]]; > v.133

  p := function(j,k,t) if j=1 then return Position(List([1..n(k)], i->Bt1[2]*Bc[M(k)[i]]-t*Bc[M(k)[i]]),0*x[1]);; else return p(j-1,k,t)+Position(List([p(j-1,k,t)+1..n(k)], i->Bt1[2]*Bc[M(k)[i]]-t*Bc[M(k)[i]]),0*x[1]);; fi;; end;; return List([1..r(k,t)],i->M(k)[p(i,k,t)]);; end;; m := function(k,s,t)

  (j,k,s,t) if j=1 then return Position(List([1..r(k,s)], i->Bt1[3]*Bc[N(k,s)[i]]-t*Bc[N(k,s)[i]]),0*x[1]);; else return q(j-1,k,s,t)+Position(List([q(j-1,k,s,t)+1..r(k,s)], i->Bt1[3]*Bc[N(k,s)[i]]-t*Bc[N(k,s)[i]]),0*x[1]);; fi;; end;; return List([1..m(k,s,t)],i->N(k,s)[q(i,k,s,t)]);; end;; W := [[1,1,-1],[1,1,0],[1,-1,1],[1,-1,0],[1,0,1],[1,0,-1],[1,0,0]];; S := 2 + 2*Sum(List([1..Length(W)],i->Length(Q(W[i][1],W[i][2],W[i][3])))) + Dimension(le)-Dimension(c);; A := function(i) return List([1..m(W[i][1],W[i][2],W[i][3])], t->List([1..m(W[i][1],W[i][2],W[i][3])], s->Bc[Q(W[i][1],W[i][2],W[i][3])[s]]*Bc[Q(-W[i][1],-W[i][2],-W[i][3])[t]]));; end;; # A(1), A(2), A(3), A(5), A(6) are nonsingular. # A(7) is singular of order 5 of rank 4; its minor List([1..4],s->List([1..4], t->Bc[Q(W[7][1],W[7][2],W[7][3])[s]]*Bc[Q(-W[7][1],-W[7][2],-W[7][3])[t]]));; # is different from 0. a := Bc[M(2)[1]]*Bc[M(-2)[1]];;

  Theorem 2.8 tells us that O g is contained in C(x). Moreover by Theorem 2.8, O g has dimension dim G.x, whence Õ = O g . All statements of (i) are now clear.

  Corollary 4.15. The element e has Property (P) if and only if z(g e ) ⊂ β x (0) for all x in a nonempty open subset of S reg . Proof. The map x → x from S ′ reg to Y is well-defined and its image is an open subset of Y ∩p -1 (e). Let S ′′ reg be the set of x ∈ S ′ reg such that x ∈ Ω and let Y ′′ be the image of S ′′ reg by the map x → x. Then S ′′ reg is open in S reg and Y ′′ is dense in Ω ∩ p -1 (e) since p -1 (e) is irreducible. Furthermore, the image of a dense open subset of S ′′

weight of the adjoint action of t 1 in g f with the same multiplicity as λ. Hence -λ is a weight of the adjoint action of t 1 in g e with the same multiplicity as λ, whence the lemma.

Choose pairwise different elements λ 1 , . . . ,λ r of t * 1 so that the weights of the adjoint action of t 1 in g e which are not identically zero on t are precisely the elements ±λ i . For i = 1, . . . , r, let v i,1 , . . . ,v i,m i and w i,1 , . . . ,w i,m i be basis of g e λ i and g e -λ i respectively. Then we set: q i := det ([v i,k , w i,l ]) 1≤k,l≤m i ∈ S(l e ).

Proposition 5.9. Suppose that indl e = ℓ and suppose that one of the following two conditions is satisfied:

(1) for i = 1, . . . , r, q i = 0, (2) there exists j in {1, . . . , r} such that q i = 0 for all i = j and such that the basis v j,1 , . . . ,v j,m j and w j,1 , . . . ,w j,m j can be chosen so that

Then, ind g e = ℓ.

Proof. First, observe that indg eind g is an even integer. Indeed, we have:

But the integers indg edim g e , dim g edim g and dim gind g are all even integers. Thereby, if ind g e ≤ indg + 1, then indg e ≤ indg. In turn, by Vinberg's inequality (cf. Introduction), we have ind g e ≥ ind g. Hence, it suffices to prove ind g e ≤ ind l e + 1 since our hypothesis says that ind l e = ℓ = indg. Now, by Lemma 5.7, if there exists ξ in (l e ) * such that (g e ) ξ ∩ [t, g e ] has dimension at most 2, then we are done. Denote by l 1 the centralizer of t 1 in g. Then l 1 is contained in l and l e = l e 1 ⊕ [t 1 , l e ] and (l e 1 ) * identifies to the orthogonal of [t 1 , l e ] in the dual of l e . Moreover, for i = 1, . . . , r, q i belongs to S(l e 1 ). For ξ in (l e 1 ) * , denote by B ξ the bilinear form [t,

and denote by ker B ξ its kernel. For i = 1, . . . , r, -q i (ξ) 2 is the determinant of the restriction of B ξ to the subspace (g e λ i ⊕ g e -λ i ) × (g e λ i ⊕ g e -λ i ) in the basis v i,1 , . . . ,v i,m i , w i,1 , . . . ,w i,m i .

If (1) holds, we can find ξ in (l e 1 ) * such that ker B ξ is zero. If (2) holds, we can find ξ in (l e 1 ) * such that ker B ξ has dimension 2 since B ξ is invariant under the adjoint action of t 1 . But ker B ξ is equal to (g e ) ξ ∩ [t, g e ]. Hence such a ξ satisfies the required inequality and the proposition follows.

The proof of the following proposition is given in Appendix A since it relies on explicit computations:

Proposition 5.10. (i) Suppose that either g has type E 7 and dim g e = 41 or, g has type E 8 and dim g e ∈ {112, 72}. Then, for suitable choices of t and t 1 , Condition (1) of Proposition 5.9 is satisfied.

(ii) Suppose that g has type E 8 and that g e has dimension 84, 76, or 46. Then, for suitable choices of t and t 1 , Condition (2) of Proposition 5.5 is satisfied. 5.3. Proof of Theorem 1.3. We are now in position to complete the proof of Theorem 1.3:

Proof of Theorem 1.3. We argue by induction on the dimension of g. If g has dimension 3, the statement is known. Assume now that indl e ′ = rk l for any reductive Lie algebras l of dimension at most dim g -1 and any e ′ ∈ N(l). Let e ∈ N(g) be a nilpotent element of g. By Theorem 3.1 and Theorem 5.3, we can assume that e is rigid and that g is simple of exceptional type. Furthermore by Corollary 5.6, we can assume that dim z(g e ) > 1. Then we consider the different cases given by Proposition 5.10.

If, either g has type E 7 and dim g e = 41, or g has type E 8 and dim g e equals 112, 72, or 46, then Condition (1) of Proposition 5.9 applies for suitable choices of t and t 1 by Proposition 5.10. Moreover, if l = z g (t), then l is a reductive Lie algebra of rank ℓ and strictly contained in g. So, from our induction hypothesis, we deduce that indg e = ℓ by Proposition 5.9.

If g has type E 8 and dim g e equals 84, 76, or 46, then Condition (2) of Proposition 5.9 applies for suitable choices of t and t 1 by Proposition 5.10. Arguing as above, we deduce that indg e = ℓ.

Appendix A. Proof of Proposition 5.10: explicit computations.

This appendix aims to prove Proposition 5.10. We prove Proposition 5.10 for each case by using explicit computations made with the help of GAP; our programmes are presented below (two cases are detailed; the other ones are similar). Explain the general approach. In our programmes, x[1], . . . are root vectors generating the nilradical of the Borel subalgebra b of g and the representative e (denoted by e in our programmes) of the rigid orbit is chosen so that e and h belong to b and h respectively. The element e is given by the tables of [START_REF] Quitté | Bases de Chevalley et sl 2 -triplets des algèbres de Lie simples exceptionnelles[END_REF]. In fact, in [START_REF] Quitté | Bases de Chevalley et sl 2 -triplets des algèbres de Lie simples exceptionnelles[END_REF], they use the programme Lie which induces minor changes in the numbering. Then, we exhibit suitable tori t and t 1 of g contained in g e which satisfies conditions (1) or (2) of Proposition 5.9. In each case, our torus t is one dimensional; we define it by a generator, called t in our programmes. Its centralizer in g e is denoted by le. The torus t 1 has dimension at most 4. It is defined by a basis denoted by Bt1. The weights of t 1 for the adjoint action of t 1 on g e are given by their values on the basis Bt1 of t 1 . We list in a matrix W almost all weights which have a positive value at Bt1. The other weights have multiplicity 1. In our programmes, by the term S we check that no weight is forgotten; this term has to be zero. Then, the matrices corresponding to the weights given by W are given by a function A. Their determinants correspond to the q i 's in the notations of Proposition 5.9. If there is only one other weight, the corresponding matrix is denoted by a. At last, we verify that these matrices have the desired property depending on the situations (i) or (ii) of Proposition 5.10.

As examples, we detail below two cases:

(1) the case of E 7 with dim g e = 41 where we intend to check that Condition (1) of Proposition 5.9 is satisfied;

(2) the case of E 8 with dim g e = 84 where we intend to check that Condition (2) of Proposition 5.9 is satisfied.

(1) E 7 , dim g e = 41: In this case, with our choices, dim t = 1, dim l e = 23 and dim t 1 = 3. The order of matrices to be considered is at most 2.

(2) E 8 , dim g e = 84: In this case, with our choices, dim t = 1, dim l e = 48 and dim t 1 = 3. The matrix A(7) has order 5 and it is singular of rank 4. The order of the other matrices is at most 2.