Sagar Shenoy Manikar

Joël Jezegou

Pierre De Saqui-Sannes

Emmanuel Benard

Philippe Asseman

Ph D Candidate

Holistic Operability Projection during Early Aircraft Design

Aircraft operational performance is one of the key drivers to airline profitability and punctuality. Along with safety and technical performance, aircraft operational performance needs to be projected from the early stages of development to design an aircraft that can fully meet the expectations of airlines and passengers. The ability of a system to meet its operational requirements in terms of reliability, availability and costs is termed as 'Operability'. This paper proposes a method to model the operability of an aircraft during early design and use it to predict its operational performance. Initially, in-service data is used to create a reference baseline for a system of interest. For a new design, the designers evaluate the changes (deltas) in terms of few high-level metrics from an operations point of view called Consolidated Operability Metrics. An operability model is developed using Bayesian networks that is then used to project the changes in operational performance of the new design in comparison to the baseline. This method will help aircraft architects in conducting trade-off studies during early design from an operational point of view.

I. Acronyms

II. Introduction

A ir transport has proved to be one of the fastest and safest modes of transport today. Despite the temporary reduction in air traffic due to the COVID-19 pandemic, air traffic has grown rapidly over the past decade (2010-2019) and is expected to grow again after the end of the pandemic [1]. Hence, it has become very important for the aviation industry to achieve disruption-less aircraft operations at reduced operating cost. One of the main ways to achieve this is by ensuring very high levels of "operability" of the aircraft through its design. The ability of a system to meet its operational requirements in terms of availability, reliability and operating costs within a given operational environment is called operability. Hence, along with the technical performance and safety, aircraft operability forms an important criterion in trade-off studies during early aircraft design. It dictates the operational performance of the aircraft and has a strong impact on airline operating cost and profitability.

It is important for airlines to have the aircraft available for operations during a large proportion of the time (i.e. high aircraft availability) in order to increase their utilisation for flights which generate revenue. It is also important for airlines to make sure that the flights depart and arrive on time at the scheduled destination (i.e. high operational reliability). Otherwise, if there is a delay in flight departure or arrival, airlines can incur huge expenditures to cover passengers' compensation, additional airport fees, crew replacement, etc. leading to an increase in the overall operating cost for airlines.

There exist different factors which can make an aircraft unavailable for flying [2]. These can be grouped into two categories: maintenance-related and non-maintenance-related. Non-maintenance factors include Air Traffic Control (ATC) decisions, bad weather conditions, labour strikes, etc. which are not related to aircraft technical issues. On the other hand, maintenance-related issues which include both scheduled and unscheduled maintenance can be influenced by the aircraft and airline maintenance policies to a certain extent. Scheduled maintenance involves periodic checks like line checks, base checks and heavy checks while unscheduled maintenance can arise due to unforeseen events like a system failure or structural damage. Any maintenance action on the aircraft has an associated material cost as well as labour cost which contribute to the operating cost of the aircraft.

It is therefore important to take into account aircraft operability early during the development of an aircraft. The usual way of classifying the aircraft major components is using the Air Transportation Association (ATA) 100 specification [START_REF]ATA Specification 100 -Specification for Manufacturers' Technical Data[END_REF] which specifies roughly 100 numbered categories of high-level systems and structural components of aircraft. E.g. ATA 27 refers to the Flight controls system in an aircraft. This paper addresses operability projections of these high-level aircraft components represented by ATA 2-Digits (ATAs 2D).

The objective of this paper is to propose a method that can help project the operability of major aircraft components during early aircraft design. This paper proposes one possible implementation for such a projection method called Holistic Operability Projection (HOP) that uses both in-service data from existing aircraft as well as domain knowledge of operability experts. Some high-level operability metrics called "Consolidated Operability Metrics" (COMs) are defined to address aircraft major components from the operations point of view. There are several factors in aircraft operations that affect aircraft operability which are identified and represented by Operational Environment Parameters (OEPs). A probabilistic model using Bayesian network is built to map the relationships between different input and output parameters.

The rest of the paper is organized as follows: Section III surveys related work in the domain of operability projections. Section IV presents the methodology of the study by defining the different parameters, architecture and bricks of HOP. Section V discusses the preliminary results obtained so far and finally, Section VI concludes the paper.

III. Related work

Different techniques have been used in the aerospace domain to estimate some characteristics of aircraft operability like aircraft reliability and failure rate of aircraft systems [START_REF] Trivedi | Reliability and Availability Engineering: Modeling, Analysis, and Applications[END_REF]. Most of these studies have been focused on evaluating the operational performance at equipment or system level [START_REF] Hugues | Application of Markov processes to predict aircraft operational reliability[END_REF][START_REF] Saintis | Computing in-service aircraft reliability[END_REF] which require detailed definition of the system and knowledge of its failure modes. This kind of information is usually not available during early stages of aircraft development. An attempt to incorporate uncertainty in dependability prediction during early design stages has been made but it is also targeted at individual system-level analysis [START_REF] Limbourg | Dependability Modelling under Uncertainty[END_REF]. There have been some studies which have addressed computing the aircraft reliability during in-service operations in order to assist airline operations and mission planning [START_REF] Tiassou | Aircraft operational reliability-A model-based approach and a case study[END_REF][START_REF] Papakostas | An approach to operational aircraft maintenance planning[END_REF]. Unlike these studies, the focus of this paper is to propose a method to predict the operational performance of major aircraft components during early design stages, which is sparsely addressed in the literature.

Aircraft operations are characterized by randomness due to uncertainty related to weather, air traffic, failures, etc. There is also a huge variability in aircraft operations due to the different ways in which different airlines operate their fleet.

Hence, probabilistic modelling is necessary to efficiently model aircraft operations. Bayesian networks were used to create these models due to their strengths in creating predictive models and performing omnidirectional inferences [START_REF] Fenton | Risk Assessment and Decision Analysis with Bayesian Networks[END_REF]. Another advantage of Bayesian networks is that it allows one to incorporate both expert knowledge and in-service data in the models [START_REF] Jensen | Bayesian Networks and Decision Graphs[END_REF]. Bayesian networks have been used in several studies for reliability and safety applications [START_REF] Cai | Application of Bayesian Networks in Reliability Evaluation[END_REF][START_REF] Kabir | Applications of Bayesian networks and Petri nets in safety, reliability, and risk assessments: A review[END_REF][START_REF] Melani | Mapping SysML Diagrams Into Bayesian Networks: A Systems Engineering Approach for Fault Diagnosis[END_REF]. There has been an attempt to use Bayesian network at functional design stage [START_REF] Yontay | Bayesian network for reliability prediction in functional design stage[END_REF], in which reliability information from both historical products as well as expert elicitation has been utilised for predicting reliability of an automotive system. But, most of these studies focus on safety and reliability predictions of systems whose architectures are already available along with their failure modes. But, this paper addresses major aircraft components during the conceptual design phase when very little information is available regarding the architecture and implementation of the system.

Bayesian networks have also been used in maintenance planning, condition-based monitoring and prognostic applications [START_REF] Nielsen | Computational framework for risk-based planning of inspections, maintenance and condition monitoring using discrete Bayesian networks[END_REF][START_REF] Ferreiro | Application of Bayesian networks in prognostics for a new Integrated Vehicle Health Management concept[END_REF][START_REF] Kipersztok | Evidence-based Bayesian networks approach to airplane maintenance[END_REF]. There have been several works which have addressed learning in Bayesian networks like parameter learning and structure learning [START_REF] Zhao | Analysis of factors that influence hazardous material transportation accidents based on Bayesian networks: A case study in China[END_REF][START_REF] Zhou | When and where to transfer for Bayesian network parameter learning[END_REF]. In this paper, both domain knowledge and in-service data are used to build the Bayesian networks.

IV. Methodology

There are several factors that need to be considered while making operability projections of major aircraft components. They can be divided into two categories, one related to the aircraft design and other related to the operational context of the aircraft. Accordingly, we have two set of input parameters that are taken as inputs in HOP: Consolidated Operability Metrics (COMs) and Operational Environment Parameters (OEPs). COMs are high-level operability metrics that are related to the design of aircraft major components. OEPs represent the different parameters required to characterise the operational context in which an aircraft operates.

The different COMs and OEPs considered in HOP are defined in this section. The implementation architecture of HOP along its major bricks are explained. Finally, the case study used in this paper is also presented in this section.

A. Consolidated Operability Metrics (COMs)

Consolidated Operability Metrics (COMs) are high-level metrics that can be defined for a system from an operational point of view. These COMs drive the operational performance of major aircraft components. The list of COMs were defined through consultation with aircraft operability experts and were refined depending on the feasibility to compute them. Quantification of these COMs during early aircraft design can help designers and system architects to make necessary modifications to achieve better operational performance.

The values for these COMs should be computable directly from the engineering and in-service data available for a reference aircraft. These COMs should also be computable from the different maintainability attributes [START_REF] Wani | Development of maintainability index for mechanical systems[END_REF] of a system like accessibility, testability, etc. that can be evaluated by designers during the development of a system.

A Technical issue (TI) is any event occurring on the aircraft that can degrade its targeted missions and operations. E.g., a system fault, a structural damage. If the scheduled operations of an aircraft are disrupted due to a TI, it is called an Operational Interruption (OI) like delay, cancellation, etc. When TIs occur, they are resolved through appropriate maintenance actions. A TI is said to be rectified when the full functionality of the system is restored. Sometimes, temporary maintenance actions like deactivation of the system are performed to continue flight operations to avoid an OI. All the TIs and the associated maintenance actions performed on an aircraft are recorded in an aircraft logbook by pilots and maintenance crew. The COMs are defined on the different aspects related to the occurrence and treatment of TIs for a given aircraft major component. Definition 1. Consolidated Operability Metric (COM) can be defined as a 3-tuple 𝐶𝑂 𝑀 =< 𝑁, 𝑉, 𝑃𝑣 > where, N is the name of the COM. 𝑉 = 𝑣 1 , 𝑣 2 , ..., 𝑣 𝑛 defines the set of values 𝑣 𝑖 for the COM.

𝑃𝑣 : 𝑉 → [0, 1] specifies the probability distribution of occurrence of each value 𝑣 𝑖 such that

𝑛 ∑ 𝑖=1 𝑃𝑣 𝑖 = 1
The list of COMs to fully characterise and project aircraft operability is shown in Table 1. These COMs represent different dimensions of operability characteristics for an aircraft major component which can be influenced by design. A brief description of each COM, its values and the pre-processing required to specify the probability distribution of values for a reference baseline is presented below.

Reliability

The COM Reliability indicates how many TIs are generated by an aircraft major component per aircraft during one year. Hence, it is represented by a continuous value which will be discretized in the operability model. For a reference baseline, this value is computed by analysing the in-service aircraft logbook data for a given aircraft fleet.

Detectability

The COM Detectability indicates the phase of detection of the TI. It can take three distinct values corresponding to the aircraft phase during which the TI occurs. The three values are Between Pushback and Take-Off (BPTO), Flying and Planned stop. Each of these values can produce different operational impacts. For a reference baseline, this value is deduced from the flight phase entries available in the aircraft logbook data.

Diagnosability

The COM Diagnosability refers to the ability to diagnose a TI after its occurrence. It can take one of the two values: Troubleshooting or No troubleshooting depending on whether troubleshooting procedures are required to identify the TI. For a reference baseline, the value is deduced by processing the defect and resolution descriptions available in the aircraft logbook data.

Dispatchability

The COM Dispatchability refers to the operational and maintenance procedures that are required to be performed on an aircraft after the occurrence of a TI in order for the aircraft to continue operations. It can take one of the two values: No rectification or Rectification which depends on many factors like severity of TI, system redundancy, etc. For a reference baseline, this value can be deduced by processing the aircraft logbook data as well as referring to the Master Minimum Equipment List (MMEL) and Configuration Deviation List (CDL) of the aircraft.

Deferrability

The COM Deferrability refers to the maximum time duration up to which the rectification of the TI can be postponed after its occurrence. It can take one of the 6 values ranging from Not deferrable to Above 120 days. It takes the value Not deferrable when the rectification of the TI needs to be done immediately before the next flight. For a reference baseline, Deferrability can be obtained from engineering data like MMEL.

Repairability

The COM Repairability refers to the time duration required to carry out the maintenance to rectify the TI. Hence, it is represented by a continuous value which will be discretized in the operability model. For a reference baseline, this value is computed by referring to engineering data like the Aircraft Maintenance Manual (AMM) as well as in-service data like aircraft logbook data.

Resourceability

The COM Resourceability refers to the type of material and human resources required to carry out the maintenance of TI. It can take one of the two values: Standard and Non-standard. Standard resources are those that are not specified by the aircraft manufacturer and are usually available at all airports. Non-standard resources are specified by aircraft manufacturer and are specific to the aircraft type. Hence, airlines might have different levels of provisioning of Non-standard resources at airports. For a reference baseline, this value is computed by referring to the engineering data like the AMM.

Predictability

The COM Predictability refers to the ability to predict the TI before its occurrence using prognostics. It can take two values: Predictable and Not predictable. For a reference baseline, this value needs to be deduced from engineering data.

B. Operational Environment Parameters (OEPs)

The operational context in which an aircraft is operated plays an important role in determining its operational performance [START_REF] Tiassou | Aircraft operational reliability-A model-based approach and a case study[END_REF]. Different airlines can use the same aircraft in different ways depending upon the mission-type, geographical location, airline network, airport type, etc. which influence the occurrences of TIs. The availability of maintenance facilities and maintenance crew also determines the manner in which the generated TIs are handled. Hence, the aircraft manufacturer's experience based on in-service data has shown that an aircraft operating in different operational contexts can generate different profiles of TIs. Therefore, a set of Operational Environment Parameters (OEPs) have to be taken into account in HOP to characterize all these factors related to the operational context which influences aircraft operational performance. Definition 2. Operational Environment Parameter (OEP) can be defined as a 3-tuple

𝑂𝐸 𝑃 =< 𝑁, 𝑉, 𝑃𝑣 >
where, N is the name of the OEP. 𝑉 = 𝑣 1 , 𝑣 2 , ..., 𝑣 𝑛 defines the set of values 𝑣 𝑖 for the OEP.

𝑃𝑣 : 𝑉 → [0, 1] specifies the probability distribution of occurrence of each value 𝑣 𝑖 such that 𝑛 ∑ 𝑖=1 𝑃𝑣 𝑖 = 1
The list of OEPs identified for HOP is shown in Table 2. A brief description of each OEP, its values and the pre-processing required to specify the probability distribution of values for a reference baseline is presented below.

Airport type

The OEP Airport type refers to the type of airport for an airline based upon its ability to carry out maintenance activities at that location with respect to time and cost. An airline usually has different levels of maintenance resources based on the type of airport. The OEP Airport type can take one of the two values: Mainbase or Outstation. At a Mainbase airport, it is assumed that there is high availability of maintenance facilities, support equipment, maintenance crew and spares to carry out the required maintenance activities. On the contrary, Outstation airports are assumed to have low availability of maintenance resources and hence are not preferred by airlines to carry out major maintenance activities. For a reference baseline, the value is deduced by analysing airline operations from in-service data.

Aircraft utilization hours

The annual utilization of aircraft in terms of Flight Hours (FH) (continuous value) 5

Aircraft utilization cycles

The annual utilization of aircraft in terms of Flight Cycles (FC) (continuous value)

Scheduled stop time

The OEP Scheduled stop time refers to the time interval between the scheduled departure time and previous flight's arrival time. Hence, it is represented by a continuous value which will be discretized in the operability model. It is an important factor for airlines to take decisions regarding whether to carry out maintenance activities, especially for ones that can be deferred. For a reference baseline, this value is deduced from the flight schedules available from in-service data.

Mission type

The OEP Mission type refers to the type of mission based upon if it is Extended Range Operations (ETOPS) [START_REF] De Florio | Airworthiness: An Introduction to Aircraft Certification[END_REF] or not. By default, aircraft are only allowed to fly such routes in which the aircraft is within 60 minutes from the closest airport at any point during the flight. This can be particularly challenging for aircraft flying over oceans and uninhabited places where it is hard to find airports on the route. As a result, aircraft were forced to take alternate deviated routes instead of flying the direct routes from departure airports to the destination airports. ETOPS mission bypasses the default rule by allowing aircraft to fly on routes in which it is further than 60 minutes from the closest airport. But, ETOPS mission imposes additional and stricter requirements that need to be met before flying. The OEP Mission type can take one of the two values: ETOPS or non-ETOPS. For a reference baseline, this value is deduced from flight schedules available from in-service data and expert judgement.

Aircraft utilization hours

The OEP Aircraft utilization hours refers to the number of flight hours flown by the aircraft in one year. Hence, it is represented by a continuous value which will be discretized in the operability model. For a reference baseline, this value is computed by analysing the flight schedules and actual flight timings available from in-service data.

Aircraft utilization cycles

The OEP Aircraft utilization cycles refers to the number of flight cycles flown by the aircraft in one year. A flight cycle comprises one take-off and one landing. The OEP Aircraft utilization cycles is represented by a continuous value which will be discretized in the operability model. For a reference baseline, this value is computed by analysing the flight schedules and actual flight timings available from in-service data.

C. Architecture of Holistic Operability Projection (HOP) method

The Holistic Operability Projection (HOP) method is composed of several bricks that aggregate engineering knowledge and in-service data to make the operability projections for an aircraft major component. Initially, the aircraft operations Finite State Machine (FSM) models were created by the authors of this paper in previous work [START_REF] Manikar | A Formal Framework for Modeling and Prediction of Aircraft Operability using SysML[END_REF] to represent the different states an aircraft can occupy in operation (e.g., in flight, in planned stop) and their dynamic behaviour. In these FSM models, all possible transitions between different aircraft states were defined. These models were created using the domain knowledge of operability experts. The aircraft operations FSM model used in this study is explained in subsection IV.F.

A hybrid methodology utilising machine learning and expert knowledge was proposed to project operability in authors' another previous work [START_REF] Manikar | A hybrid approach of machine learning and expert knowledge for projection of aircraft operability[END_REF] which is shown in Fig. 1. This methodology presented high-level steps involved in making operability projections. A reference baseline needs to be initially established by populating aircraft operations FSM using in-service data. Operability models of aircraft major components have to be developed using engineering knowledge and can be trained using machine learning techniques. The operability projections through these hybrid models can then be used to compare the operational performance of future design with respect to baseline. Combining in-service data along with expert knowledge can help achieve more realistic operability projections than pure knowledge-based or pure data-driven methods. The values for the components OEPs (c0), COMs (c1) and OPOs (c2) are calibrated using engineering and in-service data for a reference baseline. Principally, the in-service data regarding flight schedule, actual flight timings, aircraft logbook entries, operational interruptions, etc. are analysed to populate these components. Certain values of components OEPs (c0) and COMs (c1) are populated using engineering data and expert knowledge when in-service data is not available.

For a new design of an aircraft major component, the system designer evaluates the changes (deltas) of the new design in comparison to the reference baseline in terms of different maintainability attributes. These deltas are captured in the form of a questionnaire (c5) which feeds the COMs (c1). The designer also specifies any change in the OEPs (c0) but they are usually kept constant in order to see the individual impact of the new design. The deltas in OEPs (c0) and COMs (c1) are used to project the impact of the new design in terms of deltas of the OPOs (c2). The processes (p2i), (p2ii) and (p2iii) help to propagate the deltas of (c0) and (c1) to (c2). The processes (p2i) and (p2ii) are realized by an Operability model using Bayesian networks which is explained in subsection IV.D. The component (c1i) represents the parameters that are required to simulate the aircraft operations FSM. The process (p2iii) is realized by an Aircraft operations FSM simulator which is explained in subsection IV.F.

D. Operability model using Bayesian networks

The processes (p2i) and (p2ii) of the architecture shown in Fig. 2 are realized by an operability model that maps the deltas of OEPs (c0) and COMs (c1) to the FSM parameters (c1i). Bayesian networks were used to build this operability model by mapping the dependencies between the different elements mentioned above and learning their parameters.

Bayesian networks are probabilistic graphical models that use Bayesian inference for probabilistic computations [START_REF] Pearl | Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference[END_REF]. They consist of two parts, a qualitative one consisting of Directed Acyclic Graphs (DAG) to indicate the dependencies between nodes, and a quantitative one containing the prior conditional probabilities of nodes. Once fully specified, a Bayesian network can represent the joint probability distribution which can be used for computing the posterior probabilities of any set of variables [START_REF] Jensen | Bayesian Networks and Decision Graphs[END_REF].

The Bayesian networks created in this study were modeled using BayesiaLab software [START_REF][END_REF]. The structure of the Bayesian network was built using the domain knowledge of operability experts. The different nodes modelled in the Bayesian network are shown in Fig. 3 which represent the major parameters that are taken into account in the operability model. This operability model allows one to propagate the deltas of COMs or OEPs to the different FSM parameters. First, the parameters for each node have to specified in order to perform inferences from the Bayesian network. The parameters can be either manually specified or learnt directly from data. The latter option was employed in this study which is explained in subsection IV.E

E. Learning Bayesian network parameters

Given a Bayesian network structure, it is possible to estimate the parameters i.e. Conditional Probability Tables (CPTs) of the different nodes of the Bayesian network from a dataset associated with the network. BayesiaLab can use the observations contained in a dataset to populate the CPTs using maximum likelihood estimation [START_REF] Conrady | Bayesian Networks and BayesiaLab: A Practical Introduction for Researchers[END_REF].

For a reference baseline, data for all the nodes in the input and output clusters are gathered by processing the engineering and in-service data available in different datasets. The datasets containing in-service data considered in this study are aircraft logbooks, operational interruptions, flight schedules and actual flight timings. The sources of engineering information considered in this study are the different technical data manuals like MMEL, CDL, AMM, etc. and inputs from operability experts.

For computation of the different input and output nodes of the Bayesian network model, the information from these different datasets are merged into a single dataset based on common identifiers: aircraft manufacturer serial number, flight number and departure time. The aircraft logbooks dataset is chosen as the starting dataset for merging as it contains all the TIs. For each row in aircraft logbooks, all other useful information from other datasets is aggregated. This dataset is then processed based on definitions of different COMs and OEPs to assign a value for each of the input nodes for every row in the dataset. This helps to create one big dataset containing information regarding all the nodes required to learn the parameters of the Bayesian network model.

This dataset is imported in the BayesiaLab software which estimates the parameters (CPTs) for every node using maximum likelihood estimation. The missing values in the dataset are handled by Structural Expectation Maximization algorithm [START_REF] Friedman | The Bayesian structural EM algorithm[END_REF]. Also, the continuous variables in the study (e.g., COM Scheduled stop time) have to be discretized before parameter estimation can take place. The bins of discretization were chosen manually based on manufacturer's experience. But, there exists also options to do it automatically like K-Means [START_REF] Hartigan | Algorithm AS 136: A K-Means Clustering Algorithm[END_REF], equal frequency, etc.

This process of parameter estimation establishes quantitative dependencies between different nodes of the Bayesian network. The impact of each node on every other node can now be analysed using a metric called mutual information discussed in V.A.

F. Aircraft Operations FSM Simulator

The processes (p2iii) of the architecture shown in Fig. 2 is realized by an aircraft operations Finite State Machine (FSM) simulator. The aircraft operations FSM depicts the different states in which the aircraft can be present during operations, and the possible transitions among them as shown in Fig. 4.

Fig. 4 Aircraft operations Finite State Machine (FSM)

The list of states in the aircraft operations FSM along with their brief descriptions is presented in Table 3. Three kinds of states are identified: nominal (green), interruption (red) and intermediate (orange). Five nominal states represent the usual flight operations, turnarounds and scheduled maintenance activities of an aircraft. Ideally, the aircraft spends most of its time in the nominal states. But, when there are unexpected technical issues which cannot be resolved without disrupting the existing flight schedule, the aircraft can enter one of the five interruption states. These interruption states adversely impact the operational performance of aircraft by inducing delays, additional maintenance costs, etc. There is one intermediate state which is in between the nominal and interruption states. The aircraft can be present in this state when it has diverged from the nominal states but has not yet caused an interruption. When the aircraft is in an intermediate state, it has a high potential to cause an interruption if the technical issue cannot be resolved within the allowed tolerance of 15 minutes. Modeling aircraft operations using FSMs helps to map the time spent by an aircraft or a fleet of aircraft in the different operational states. A distribution of the time spent in different states can be computed when aircraft operations FSM is simulated for a large number of flight cycles. A distribution refers to the average percentage of time spent in different states when the aircraft operations is simulated over a long period of time. The FSM can be initialized with the FSM parameters (c1i) consisting of transition probabilities and time durations. It can then be simulated probabilistically for a given amount of time, which helps to generate a distribution of time spent in different states. These results can be further post-processed to compute the OPOs discussed in subsection IV.G.

There are two categories of FSM parameters that are required to simulate an FSM probabilistically: FSM transition probabilities and FSM time durations. FSM transition probabilities specify the probability distributions among the possible states that an aircraft in a given state can transition into. E.g., From "Planned stop", the aircraft can go to either "BPTO", "Planned maintenance stop" or "Planned stop with delay". There exists 6 FSM transition probabilities as there are 6 states in the aircraft operations FSM having more than one possible transition. 5 of the 11 states have only one possible transition, hence not requiring FSM transition probabilities. FSM time durations specify the probability distribution of the time that the aircraft will spend in a given state. There are 11 FSM time durations, one for each of the states in the aircraft operations FSM.

G. Operability Performance Outputs (OPOs)

Operability Performance Outputs (OPOs) corresponding to the (c2) block of the architecture shown in Fig. 2 captures the different operability KPIs. OPOs computation results are obtained from the simulation of aircraft operations FSM described in section IV.F. Each KPI of OPOs is computed in the context of this simulation. Definition 3. Operability Performance Output (OPO) can be defined as a 6-tuple 𝑂𝑃𝑂 =< 𝐹𝑆𝑀, 𝑂, 𝑇 𝐷, 𝑂𝑅, 𝑀𝑈, 𝐷 𝑀𝐶 > where, FSM is a finite state machine having S set of states and T set of transitions. 𝑂 : 𝑆 → 𝑍 ≥0 defines the number of times the aircraft is present in each state of the FSM. (𝑍 ≥0 = non-negative integers) 𝑇 𝐷 : 𝑆 → 𝑅 ≥0 defines the time spent by the aircraft in each state. (𝑅 ≥0 = non-negative real numbers) OR is the operational reliability of the system. It is computed as the percentage ratio of number of operational interruptions to the total number of take-offs. MU is the maintenance unavailability of the system. It is computed as the total time that the aircraft is on ground due to maintenance expressed in days per year per aircraft. DMC is the direct maintenance cost of the system. It is computed as the sum of the material cost of the replaced part and the labour cost for carrying out the associated maintenance tasks, expressed in $ per Flight Hour (FH).

H. Case study

The Flight Controls system (ATA 27) [START_REF]ATA Specification 100 -Specification for Manufacturers' Technical Data[END_REF], a major aircraft component was selected for the case study to demonstrate the HOP methodology. As a first attempt to realize the full operability model shown in Fig. 3, this case study was defined on a reduced set of parameters. Three COMs and two OEPs were mapped to three FSM transition probabilities and two FSM time duration parameters. Hence, the Bayesian network model comprises 10 nodes, as shown in Fig. 5. Arcs were defined between every pair of input and output nodes. Using Bayesian network learning techniques, it was possible to estimate the parameters of the different nodes in this model using in-service data. Hence, it was possible to automatically fill the Conditional Probability Tables (CPTs) of the nodes.

With parameters estimated from in-service data, the resulting model better represents the real aircraft operations. In the future, this preliminary model will be expanded to cover all the parameters of the full model shown in Fig. 3.

V. Results

A. Preliminary results

For the case study presented in previous section, results have been obtained for some blocks of the architecture presented in Fig. 2. First, in-service data were analysed to create a reference baseline for the component OPOs (c2). One of the main results of OPOs is the Time Distribution (TD) which is the amount of time spent by the aircraft in different states of the FSM. This analysis was performed on a fleet of aircraft comprising of 30 aircraft for a period of 4 years (2015-2018). In this analysis, only 5 major states of the FSM were considered. These results which are at aircraft-level are shown in Table 4. The Bayesian network model comprising 10 nodes was trained using in-service data with a data set of about 2500 entries. Fig. 6 shows the initial probability distribution obtained for the different input nodes: COMs and OEPs. Fig. 7 shows the initial probability distribution obtained for the different output nodes: FSM transition probabilities and FSM time durations. The "Value" shown below each node name in Fig. 6 and Fig. 7 corresponds to the weighted mean value of the different states calculated by assigning weights to each state starting from 0. It gives a quick view of the state closest to the mid-point of the distribution. Mutual Information (MI) is a quantity that is defined between two random variables: it measures the reduction in uncertainty of one variable given the value of the other variable [START_REF] Koller | Probabilistic Graphical Models: Principles and Techniques[END_REF]. In other words, it measures the mutual dependency between the two variables. Similarly, in a Bayesian network, MI values between predictor variables and a target node helps to see which predictor variables have the highest impact on the target node and hence have the highest predictive importance. For the case study considered, MI values between input nodes and output nodes allow designers to see which COMs and OEPs have a major impact on the FSM parameters. Hence, MI can help designers to understand the rationale for the obtained operational performance and guide them to focus on the most important parameters to achieve better operational performance.

The MI values for one of the nodes of the Bayesian network (Planned stop transitions) are shown in Fig. 8 inside white rectangles on the arcs. It can be seen that the most impacting COM is Detectability with a MI of about 3.5%. This can be explained by the fact that the flight phase in which the TI is detected plays a crucial role in determining whether the TI causes an operational interruption or not. For instance, a TI detected during "BPTO" i.e. taxi-out can have a much more severe impact than a TI detected during "Planned Stop" as there is very little time left before departure for carrying out maintenance when the aircraft is in "BPTO". The second most impacting COM is Diagnosability with a MI of 0.2% followed by Dispatchability with a MI of about 0.1%. It can be seen that these two nodes have a relatively lower impact on Planned stop transitions compared to Detectability. This could probably be explained by the reasoning that for very critical aircraft components like ATA 27, the associated maintenance actions have a high tendency to produce interruptions and the requirement of troubleshooting and repair are less important than the phase in which TI is detected. The MI of Scheduled stop time was observed to be 0.2% which is non-negligible. This can be explained by the fact that the amount of time available before next flight is an important factor in determining the next transition from Planned stop. The Airport type node has the lowest MI of 0.002% among all the input nodes signifying that it has a minimal impact on Planned stop transitions. This could probably be explained by the reasoning that for critical aircraft components like ATA 27, associated maintenance actions have a higher tendency of causing a transition to Planned stop with delay irrespective of the type of airport. But, the type of airport may have a significant impact on the time spent in Planned stop with delay. These results were obtained as a proof of concept of this methodology on a limited data set for this specific ATA 27 case study, and they do not represent the overall operability characteristics of the entire aircraft fleet.

Fig. 8 Bayesian network model with Mutual Information (MI) values for one node

In a Bayesian network, when evidence is set on some nodes, it is propagated omni-directionally throughout the network. A sensitivity analysis can be performed by changing the values of input nodes and observing the impact on the output nodes i.e. FSM parameters. There are different ways of setting evidence on the nodes and performing predictions. Hard evidence can be set by fixing just one state for a node i.e. no uncertainty regarding the state of the node. Probabilistic evidence can be set by defining marginal probability distribution on the node. For making projections of a new design, it is usually the probabilistic evidence that will be used. But for the purpose of demonstration, the prediction results for setting hard evidence on COMs Detectability and Diagnosability are shown in Fig. 9 and Fig. 10 respectively.

In Fig. 9, it is assumed that every TI of the case study is detected during BPTO i.e. hard evidence is set on COM Detectability to 100% BPTO. Then, through Bayesian inference, it can be observed that the probability of having a delay increases by 2.5 times from about 6.7% to 16.1%. This reflects the fact that when a failure is detected during BPTO, there is very little time available for the airline to carry out the maintenance before the scheduled departure time and hence, has a high potential to cause delay. An increase in the mean value of "Planned stop with delay time" by about 11 minutes can also be observed from the corresponding distributions. This can be attributed to the fact that when TIs are detected in BPTO, additional maintenance actions and checks have to be performed in the stops leading to an increase in delay time.

Similarly, in Fig. 10, it is assumed that each TI requires troubleshooting procedures i.e. hard evidence is set on the COM Diagnosability to 100% "Troubleshooting". Through Bayesian inference, it is seen that this evidence increases the probability of having a delay by about 1.5 times from 6.7% to 9.2%. This can be explained by the fact that troubleshooting procedures are time consuming and can significantly increase the risk of a delay. The mean value of "Planned stop with delay time" remains almost the same implying that there is no strong relation between it and Diagnosability.

These kind of quantitative results allow designers to clearly see the impact of different COMs and OEPs on the operational performance for a system of interest.

B. Operability projections of new designs

As seen in earlier sections, the Bayesian network model is a powerful tool to propagate the impact of value changes in input nodes to the output nodes of the network. This Bayesian network model can be used to predict the operational performance of new designs for a given aircraft major component. To illustrate this method, a hypothetical new design solution for ATA 27 can be considered (e.g. fully electric flight control system) which might contain variations in COMs values with respect to the reference baseline. These variations have to be captured through a questionnaire answered by the designers by comparing the new design to the existing baseline, which is not addressed in this paper. The COMs values of the new design are taken as inputs in the study and through Bayesian inference, change in values of FSM parameters can be observed as shown in Fig. 11.

As seen in Fig. 11, the values of the three COMs were changed for the hypothetical new design by setting probabilistic evidence in order to illustrate the method. It is assumed that the design features of the new ATA 27 system are improved with respect to failure detection and ability to continue operations when failed. Hence, the TIs can be detected more often during flying phase itself and lesser proportion of TIs will require troubleshooting and rectification. Correspondingly, for the the COM Detectability, value of the state "Flying" was increased by 10%. For the COM Diagnosability, the value of the state "No troubleshooting" was increased by 10%. For the COM Dispatchability, the value of the state "No rectification" was increased by 5%. The effects of these changes can be observed on the FSM parameters. It was seen that the probability of having a delay reduced by about 2% from 6.7% to 4.9%. Similarly, the mean value of the time spent in Planned stop with delay reduced by a minute. This improvement in operational performance can be attributed to all the three changes in COMs. Reduction of TI detection in BPTO minimises the chances of delay as explained earlier in section V.A. Reduction of troubleshooting procedures quicken maintenance actions thereby reducing delay. Finally, the reduction of rectification actions means the aircraft can be dispatched more quickly, again reducing the delay time. These changed FSM parameter values can in turn be used to compute the OPOs of the new design.

The Bayesian network model also allows the designer to scrutinize the impact of each parameter. The designer can perform what-if scenario analysis by setting different values to input parameters and computing the resulting operational performance. Reverse engineering can also be performed thanks to the bi-directional inferencing in Bayesian networks.

The designer can set evidence on FSM parameter nodes and through Bayesian inference can find the target values of COMs required to achieve the desired values of FSM parameters. These capabilities make the Bayesian network model very useful to perform trade-off analyses during early design. It also guides the designers to focus on improving the COMs that are the most pertinent to achieve required operational performance targets.

C. Discussion

The operability model realized for the case study of ATA 27 using Bayesian networks shows promising results regarding its ability to project the FSM parameters of a major aircraft component using a few COMs and OEPs. The current model deals with just one aircraft major component. To observe the operational performance of a new design at aircraft-level, the effects of multiple aircraft major components have to be handled in HOP.

The simulation of aircraft operations FSM (Step 4 in Fig. 2) was not addressed in this paper as it requires as inputs all the FSM parameter values to run the probabilistic simulation. Currently, the Bayesian network model for the case study handles only a subset of the FSM parameters. The final Bayesian network model is expected to contain all the FSM parameters (6 FSM transition probabilities and 11 FSM time durations) which will make it possible to run the aircraft operations FSM simulator. This will help in computing the different OPOs presented in section IV.G like operational reliability, maintenance unavailability, etc.

VI. Conclusions

The projection of aircraft operational performance during early development stages is essential to deliver an aircraft that meets the expectations of the airlines and passengers right from the entry-into-service of the aircraft. Hence, it becomes important to make operability projections for aircraft major components during early design. This paper proposes a method to estimate the operational performance of an aircraft major component using a combination of in-service data and expert knowledge.

The implementation architecture for the Holistic Operability Projection (HOP) method is presented in this paper along with the major components and processes involved. Bayesian networks are used to create the operability model that map the different Consolidated Operability Metrics (COMs) and Operational Environment Parameters (OEPs) to the aircraft operations FSM parameters. The structure of the Bayesian network model is built using the domain knowledge of operability experts. A case study (Flight controls system) is defined and modelled using a subset of COMs, OEPs and FSM parameters. Bayesian network parameters (CPTs) are estimated directly from engineering and in-service data. Three COMs, two OEPs and five FSM nodes are modeled and their Conditional Probability Tables (CPTs) are estimated.

The preliminary results from the Bayesian network model prove the feasibility of the method to estimate COMs and OEPs from engineering and in-service data for an aircraft major component. With the use of Mutual Information (MI) metric, the main contributing COMs and OEPs towards the FSM parameters could be identified. For instance, Detectability was observed to be the most impacting COM for the FSM parameter Planned stop transitions. For a new design, this trained Bayesian network model can help to project the changes in input nodes to the FSM parameters. It also allows designers to make a sensitivity analysis and see how the changes in COMs and OEPs affect the operational performance, thereby helping them to concentrate on improving the most important aspects.

The next step is to expand the Bayesian network model by incorporating more COMs, OEPs and FSM parameters. This expanded model will make it possible to simulate the aircraft operations FSM simulator that will help compute the different operability KPIs. By computing the operability KPIs of different candidate solutions, aircraft architects can make operability projections and perform trade-off studies at aircraft-level from an operability point of view. Another improvement of the Bayesian network model could be to use machine learning techniques to learn the structure of the Bayesian network directly from a dataset of different parameters. This technique could help to create a Bayesian network model that is more representative of in-service data.

Fig. 1

 1 Fig. 1 High-level overview of the hybrid methodology to project operability [24].

Fig. 2

 2 Fig. 2 Proposed implementation of Holistic Operability Projection (HOP) method showing the involved components and processes.

Fig. 3

 3 Fig. 3 Bayesian network nodes for the operability model of HOP architecture

Fig. 5

 5 Fig. 5 Bayesian network model of the case study.

Fig. 6 Fig. 7

 67 Fig. 6 Probability distribution of input nodes in the Bayesian network model

Fig. 9

 9 Fig. 9 Evidence set on Detectability to 100% 'BPTO'

Fig. 10

 10 Fig. 10 Evidence set on Diagnosability to 100% "Troubleshooting"

Table 1 List of Consolidated Operability Metrics (COMs) defined for aircraft major components in HOP. No. Name of COM (N) Definition Values (V)

 1

	1	Reliability	The number of Technical Issues (TI) per aircraft per year	(continuous value)
	2	Detectability	The flight phase of detection of TI	Between Pushback and Take-Off (BPTO), Flying,
				Planned stop
	3	Diagnosability	The need for troubleshooting to diagnose the TI	Troubleshooting, No troubleshooting
	4	Dispatchability	The operational and maintenance procedures required to	Rectification, No rectification
			dispatch the aircraft after occurrence of TI
	5	Deferrability	The maximum number of days upto which the TI can be	Not deferrable, Upto 1 day, From 1 to 3 days, From
			deferred for rectification	3 to 10 days, From 10 days to 120 days, Above 120
				days
	6	Repairability	The maintenance duration required to rectify the TI	(continuous value)
	7	Resourceability	The type of facility and human resources required to rectify	Standard, Non-standard
			the TI
	8	Predictability	The ability to predict the occurrence and type of TI	Predictable, Not predictable

Table 2 List of Operational Environment Parameters (OEPs) identified in HOP.

 2

	No. Name of OEP (N)	Definition	Values (V)
	1	Airport type	The type of airport based upon the airline's capacity to perform	Mainbase, Outstation
			maintenance activities at this location
	2	Scheduled stop time The time duration between the scheduled departure time of next	(continuous value)
			flight and previous flight's arrival time
	3	Mission type	The type of aircraft mission based upon if it is Extended Range	ETOPS, non-ETOPS
			Operations (ETOPS) or not
	4

Table 3 Different states of the Aircraft operations FSM

 3

	State	Type	Description
	Between Pushback and Take-off (BPTO)	nominal	The phase between aircraft leaving the gate and aircraft taking-off
	Flying	nominal	The phase between aircraft taking-off and aircraft landing
	Taxiing-In	nominal	The phase between aircraft landing and aircraft reaching the gate
	Planned stop	nominal	The phase between aircraft reaching the gate and aircraft leaving the gate for
			next flight (turnaround, night stop, etc.)
	Planned maintenance stop	nominal	The phase in which scheduled maintenance activities are carried out on the
			aircraft
	Planned stop with delay	intermediate	The phase when the aircraft is in a planned stop and the departure is delayed
	Return to Gate	interruption	The phase in which the aircraft was previously in BPTO and returns back to
			the gate instead of taking-off
	Return to departure airport	interruption	The phase in which the aircraft was previously flying and returns back immedi-
			ately to the departure airport
	Diversion to another airport	interruption	The phase in which the aircraft was previously flying and diverts immediately
			to another airport
	Unplanned stop	interruption	The phase in which aircraft is in a temporary stop that was not in the scheduled
			plan
	Out of Service	interruption	The phase in which aircraft is out of the scheduled flight operations plan

Table 4 Percentage of elapsed time in different aircraft states

 4

	Aircraft state	% of elapsed time
	Between Push-back and Take-off (BPTO)	1.88%
	Flying	40.84 %
	Taxi-in	0.74 %
	Planned stop	53.89 %
	Planned stop with delay	2.65 %

Acknowledgments

This work has been funded by Airbus and Association Nationale de Recherche et Technologie (ANRT), France in the frame of a CIFRE project (n°2019/0993).


Time-domain Wave Propagation in Rigid Porous Media using Equivalent Fluid Model with a Quadratic Nonlinearity Ilyes Moufid * , Denis Matignon † , Rémi Roncen * , and Estelle Piot * * ONERA, The French Aerospace Laboratory -Université de Toulouse, 31055 Toulouse, France † ISAE-SUPAERO -Université de Toulouse, 31055 Toulouse, France Summary. The acoustic properties of rigid porous media can be described by the equivalent fluid model (EFM) in the frequency domain, involving complex-valued functions. These physical quantities can be irrational, which leads to fractional derivatives in the time domain. Besides, this model is built with a constant flow resistivity, which is known to grow linearly with the flow velocity in the Forchheimer regime. Hence, a correction on the EFM is made according to the Darcy-Forchheimer law, leading to a more general model with an additional nonlinear term. Here, an approach is presented to formulate the EFM equations with the Forchheimer's correction in the time domain, where the fractional derivatives described by causal convolution are approximated by additional differential equations.

It results in a nonlinear system on which an energy-based analysis is performed to ensure its stability under suitable conditions.

Nonlinear equivalent fluid model equations

When the material can be assumed rigid, acoustic wave propagation in porous media can be well represented by:

ρ 0 α ∞ ∂ t u + ∇p + M u + N (g ∂ t u) = -F ξ |u|u , χ 0 ∂ t p + ∇ • u + (γ -1)N (h ∂ t p) = 0 , (1a) (1b)
where ρ 0 is the ambient fluid density, α ∞ the high frequency limit of the material dynamic tortuosity, χ 0 the ambient fluid adiabatic compressibility, and γ the heat capacity ratio; u and p are the particle velocity and pressure, respectively. Factor M is defined through Darcy's law. The two terms involving the kernel functions g and h and the convolution operator account for viscous and thermal losses in the porous medium. Definitions of M , N , N , g and h are based on the dynamic tortuosity α and the dynamic compressibility β, for which complex-valued analytical models exist, such as the JCAPL model [START_REF] Allard | Propagation of Sound in Porous Media: Modelling Sound Absorbing Materials[END_REF], presented below with the same parameters:

α(s) = α ∞     1 + M s + N 1 + s L -1 s     , (2)
β(s) = γ -(γ -1)     1 + M s + N 1 + s L -1 s     -1 , (3)
where f denotes the Laplace transform of f , s is the Laplace variable and all the coefficients are defined from physical quantities, detailed in [START_REF] Allard | Propagation of Sound in Porous Media: Modelling Sound Absorbing Materials[END_REF]. Lastly, an additional nonlinear term to the initial EFM is taken into account in the right-hand side of (1a), representing the inertial effects induced by high amplitude waves travelling in the material. It comes from the nonlinear Fochheimer equation [START_REF] Cimolin | Navier-Stokes/Forchheimer models for filtration through porous media[END_REF] where the multiplying factor F ξ depends on the Forchheimer nonlinearity parameter ξ, computed in experimental studies [START_REF] Umnova | Behavior of rigid-porous layers at high levels of continuous acoustic excitation: Theory and experiment[END_REF][START_REF] Turno | Influence of Forchheimer's nonlinearity and transient effects on pulse propagation in air saturated rigid granular materials[END_REF] by considering a linear relation between the total flow resistivity σ and the velocity amplitude:

σ = σ 0 (1 + ξφ|u|) , (4)
where φ is the porosity and σ 0 is the static flow resistivity. This relation is known as Forchheimer's correction.

Nonlinear equivalent fluid model with additional differential equations

In order to build an efficient numerical scheme for (1), a reformulation of the equations is performed using the additional differential equations (ADE) method [START_REF] Dragna | A generalized recursive convolution method for time-domain propagation in porous media[END_REF]. The irrational functions ĝ and ĥ are first approximated by rational functions Ĝ and Ĥ, written as multipole models and parameterized by a set of real or complex conjugate weights and poles:

Ĝ(s) = K k=1 r k s -s k , (5)
Ĥ(s) = K k=1 r k s -s k . ( 6
)
All the parameters r k , r k , s k and s k are here assumed to be real and can be computed using different available methods [START_REF] Horoshenkov | Padé approximants for the acoustical properties of rigid frame porous media with pore size distributions[END_REF][START_REF] Gustavsen | Rational approximation of frequency domain responses by vector fitting[END_REF][START_REF] Monteghetti | Time-local discretization of fractional and related diffusive operators using Gaussian quadrature with applications[END_REF]. Then, using ( 5) and (6) in system (1) written in the Laplace domain, leads to the following system:

             ρ 0 α ∞ s û + ∇ p + M û + N K k=1 r k + r k s k s -s k û = -F ξ | u|u , χ 0 s p + ∇ • û + (γ -1)N K k=1 r k + r k s k s -s k p = 0 . (7)
In order to express the discrete equivalent in the time domain, the inverse Laplace transform is applied, leading to a new set of equations with causal convolutions which are computed by introducing additional variables as follows:

ϕ k ϕ k ϕ k (t) = (e s k u)(t) , (8)
ψ k (t) = (e s k p)(t) , (9
) where e x : t → e xt H(t) and H is the Heaviside function.

Each of these additional variables is solution to a first-order ordinary differential equation and can be computed with a standard time-integration scheme. Finally, the global system of equations in the time domain reads:

                       ρ 0 α ∞ ∂ t u + ∇p + M + N K k=1 r k u + N K k=1 r k s k ϕ k ϕ k ϕ k = -F ξ |u|u , χ 0 ∂ t p + ∇ • u + (γ -1)   N K k=1 r k   p + (γ -1)N K k=1 r k s k ψ k = 0 , ∂ t ϕ k ϕ k ϕ k = s k ϕ k ϕ k ϕ k + u (∀k ∈ [1, K]) , ∂ t ψ k = s k ψ k + p (∀k ∈ [1, K ]) . (10)
Note that in (10), there are no spatial derivatives in the ADE. Hence, when the system is discretized with a numerical scheme based on fluxes, these fluxes depend on the velocity and pressure variables, but not on the additional variables.

Consequently, the problem to solve at each mesh interface does not grow with the number of additional variables.

Stability analysis

Hereafter, a stability analysis of system (10) is performed thanks to the energy functional defined below:

E(t) := 1 2 ρ 0 α ∞ Ω u 2 dx -N K k=1 r k s k Ω ϕ k ϕ k ϕ k 2 dx + 1 2   χ 0 Ω p 2 dx -(γ -1) N K k=1 r k s k Ω ψ 2 k dx   , (11) the derivative of which is dE dt (t) = -M Ω u 2 dx -F ξ Ω |u| u 2 dx -N K k=1 r k Ω ∂ t ϕ k ϕ k ϕ k 2 dx -(γ -1) N K k=1 r k Ω (∂ t ψ k ) 2 dx - ∂Ω (p u) • n dσ . (12)
From ( 11) and ( 12), combined with the fact that ρ 0 , α ∞ , χ 0 and (γ -1) are necessarily positive for porous media, as well as the parameters M , N , N , F ξ , we impose the following sufficient conditions to ensure stability:

(C 1 ) the weights (r k , r k ) k are positive, (C 2 ) the poles (s k , s k ) k are negative.

When both these conditions are met, E is a positive-definite functional, which is decreasing in time without external contributions at the boundary ∂Ω of the domain Ω.

Conclusion

This work formulates the nonlinear EFM equations for porous media in the time domain, with the diffusive part of the complex-valued functions α and β described by a set of weights and poles. In addition, a proof of stability is given under conditions on the sign of these, conditions satisfied by the JCAPL model for a realistic range of physical parameters. A numerical study will be carried out in order to check the theoretical results and to investigate the effects of nonlinearity.

Acknowledgment This research has been financially supported by ONERA and by ISAE-SUPAERO, through the EUR TSAE under grant ANR-17-EURE-0005.


Jean-Louis Milhorat

A REMARK ON THE FIRST EIGENVALUE OF THE LAPLACE OPERATOR ON 1-FORMS FOR COMPACT INNER SYMMETRIC SPACES

We remark that on a compact inner symmetric space G/K, indowed with the Riemmannian metric given by the Killing form of G signedchanged, the first (non-zero) eigenvalue of the Laplace operator on 1-forms is the Casimir eigenvalue of the highest either long or short root of G, according as the highest weight of the isotropy representation is long or short. Some results for the first (non-zero) eigenvalue on functions are derived.

Introduction

It is well-known that symmetric spaces provide examples where the spectrum of Laplace or Dirac operators can be (theoretically) explicitly computed. However this explicit computation is far from being simple in general and only a few examples are known. On the other hand, several classical results in geometry involve the first (non-zero) eigenvalue of those spectra, so it seems interesting to get this eigenvalue without computing all the spectrum. The present paper is a proof of the following remark:

Proposition 1.1. Let G/K be a compact inner symmetric space of "type I", indowed with the Riemmannian metric given by the Killing form of G signed-changed. The first eigenvalue 1 of the Laplace operator acting on 1-forms is given by the Casimir eigenvalue of the highest either long or short root of G (relative to the choice of a common maximal torus T in G and K), according as the highest weight of the isotropy representation is long of short.

Note that, although the result involves the choice of a basis of roots, it does not depend on this particular choice, by the transitivity of the Weyl group W G of G on root bases. Indeed, by the Freudenthal formula, the Casimir eigenvalue of the highest (long of short) root β is given by

β + 2δ G , β ,
where δ G is the half-sum of the positive roots of G, and , the scalar product on the set of weights induced by the Killing form of G signed-changed. Hence, by the W G -invariance of the scalar product, two choices of a basis of roots (relative to the choice of a common maximal torus T in G and K) will lead to the same Casimir eigenvalue. On the other hand, recall that for the symmetric spaces considered Acknowledgements: The author thanks Francis Burstall for having pointed out an incorrect statement in a first version of the present paper, and for the references about spherical representations.

1 by Bochner's vanishing theorem, there are no harmonic 1-forms on the symmetric spaces considered here, since their Ricci curvature is positive.

here, the group G is simple, hence at most two lenghts occur in the sets of roots (cf. for instance [START_REF] Humphreys | Introduction to Lie Algebras and Representation Theory[END_REF]). So, if only one lenght occurs, the Casimir eigenvalue of the highest root has only to be considered.

The study of subgroups of maximal rank in a compact Lie group was initiated by A. Borel and J. De Siebenthal in [START_REF] Borel | Les sous-groupes fermés de rang maximum des groupes de Lie clos[END_REF], with an explicit description for compact simple groups, resulting in the following complete list of irreducible compact simplyconnected Riemannian inner symmetric spaces G/K of type I (cf. J. A. Wolf's book [START_REF] Wolf | Spaces of constant curvature[END_REF]), where the first eigenvalue is given.

G/K

SU(p+q) S(U(p)×U(q)) , 1 ≤ p ≤ q 1 long 1 SO(2p+2q+1) SO(2p)×SO(2q+1) , p ≥ 1, q ≥ 0 2 short if q = 0 p 2p-1 (G/K = S 2p ) long if q ≥ 1 1 Sp(p+q) Sp(p)×Sp(q) , 1 ≤ p ≤ q 2 short p+q p+q+1 Sp(n) U(n) 2 long 1 SO(2p+2q) SO(2p)×SO(2q) , 1 ≤ p ≤ q 1 long 1 SO(2n) U(n) , n > 2 1 long 1 G2 SO(4) 2 long 1 F4 Sp(3)•Sp(1) 2 long 1 F4 Spin(9) 2 short 2/3 E6 SO(10)•SO(2) 1 long 1 E6 SU(6)•SU(2) 1 long 1 E7 E6•SO(2) 1 long 1 E7 SU(8)/{±I} 1 long 1 E7 SO ′ (12)•SU(2) 1 long 1 E8 SO ′ (16) 1 long 1 E8 E7•SU(2) 1 long 1
(By convention, if all roots have same lenght, they are called long. The notation SO ′ (n) is used to mention that SO(n) acts by means of a spin representation). The (rather puzzling) fact that the eigenvalue is equal to 1 in most of the cases is explained below. Some results for the spectrum on functions may be derived. Indeed, if a function f verifies ∆f = λ f , where λ is the first (nonzero) eigenvalue, then ∆df = λ df , hence λ ≥ µ, where µ is the first eigenvalue on 1-forms. We classify in the following the symmetric spaces for which this inequality is an equality in all the cases considered here.

It can be checked that the values given in the above table agree with already konwn results (mainly on the spectrum of functions): compare with the table given in [START_REF] Nagano | On the Minimum Eigenvalues of the Laplacians in Riemannian Manifolds[END_REF], with the explicit computations of the whole spectrum given in [START_REF] Cahn | Zeta Functions and Their Asymptotic Expansions for Compact Symmetric Spaces of Rank One[END_REF] and [START_REF] Besse | Manifolds all of Those Geodesics are Closed[END_REF] for Compact Rank One Symmetric Spaces: RP n , CP n , HP n , CaP 2 = F 4 /Spin(9), with the partial results (not always very explicit) obtained for the spectrum of Grassmannians ([IT78], [START_REF] Strese | Spektren symmetrischer Räume[END_REF], [START_REF] Tsukamoto | Spectra of Laplace-Beltrami operators on SO(n + 2)/SO(2) × SO(n) and Sp(n + 1)/Sp(1) × Sp(n)[END_REF], [START_REF] Gr | The Spectrum of the Laplace Operator for the Manifold SO(2p + 2q + 1)/SO(2p) × SO(2q + 1)[END_REF], [START_REF] Chami | Spectra of the Laplace Operator on Grassmann Manifolds[END_REF], [START_REF] Halima | Branching rules for unitary groups and spectra of invariant differential operators on complex Grassmannians[END_REF], [START_REF]The Spectrum of the Laplace Operator on the Manifold Sp(n)/Sp(q)×Sp(nq)[END_REF]) and the spectrum of Sp(n)/U(n), ([TK03b], [HC11]) 2 .

Preliminaries for the proof

We consider a compact simply connected irreducible symmetric space G/K of "type I", where G is a simple compact and simply-connected Lie group and K is the connected subgroup formed by the fixed elements of an involution σ of G. This involution induces the Cartan decomposition of the Lie algebra g of G into

g = k ⊕ p ,
where k is the Lie algebra of K and p is the vector space {X ∈ g ; σ * • X = -X}. This space p is canonically identified with the tangent space to G/K at the point o, o being the class of the neutral element of G. We consider here irreducible symmetric spaces, that is, the isotropy representation

ρ : K -→ GL(p) k -→ Ad(k) |p
is irreducible. Hence all G-invariant scalar products on p, and so all G-invariant Riemannian metrics on G/K are proportional. We consider the metric induced by the Killing form of G sign-changed. With this metric, G/K is an Einstein space with scalar curvature Scal = n/2, (cf. for instance Theorem 7.73 in [START_REF]Einstein Manifolds[END_REF]).

As G/K is an homogeneous space, the bundle of p-forms on G/K may be identified with the bundle G × ∧ p ρ ∧ p p. Any p-form ω is then identified with a K-invariant function G → ∧ p p, that is a function verifying ∀g ∈ G , ∀k ∈ K , ω(gk) = ∧ p ρ(k -1 ) ω(g) .

Let L 2 K (G, ∧ p p) be the Hilbert space of L 2 K-equivariant functions G → ∧ p p. The Laplacian operator ∆ p extends to a self-adjoint operator on L 2 K (G, ∧ p p). Since it is an elliptic operator, it has a (real) discrete spectrum. By the Peter-Weyl theorem, the natural unitary representation of G on the Hilbert space L 2 K (G, ∧ p p) decomposes into the Hilbert sum

γ∈ G V γ ⊗ Hom K (V γ , ∧ p p) ,
where G is the set of equivalence classes of irreducible unitary complex representations of G, (ρ γ , V γ ) represents an element γ ∈ G and Hom K (V γ , ∧ p p) is the vector space of K-equivariant homomorphisms V γ → ∧ p p, i.e.

Hom K (V γ , ∧ p p) = {A ∈ Hom(V γ , ∧ p p) s.t. ∀k ∈ K , A • ρ γ (k) = ∧ p ρ(k) • A} . The injection V γ ⊗ Hom K (V γ , ∧ p p) ֒→ L 2 K (G, ∧ p p) is given by v ⊗ A → g → (A • ρ γ (g -1 ) ) • v .
The Laplacian ∆ p respects the above decomposition, and its restriction to the space V γ ⊗ Hom K (V γ , ∧ p p) is nothing else but the Casimir operator C γ of the representation (ρ γ , V γ ), (see [START_REF] Ikeda | Spectra and eigenforms of the Laplacian on S n and P n (C)[END_REF]):

∆(v ⊗ A) = v ⊗ (A • C γ ) .
2 Many references for the explicit computations of spectra may be found in https://mathoverflow.net/questions/219109/explicit-eigenvalues-of-the-laplacian.

But since the representation is irreducible, the Casimir operator is a scalar multiple of identity, C γ = c γ id, where the eigenvalue c γ only depends of γ ∈ G. Hence the spectrum of ∆ p is the set of the c γ for which Hom K (V γ , ∧ p p) is non trivial. Denote by ∧ p ρ = ⊕ N j=1 ρ p j , the decomposition of the representation K → ∧ p p into irreducible components. Note that for p = 0 or 1, the decomposition has only one component, since the representations are respectively the trivial one, and the isotropy representation, which are both irreducible. Now, by the Frobenius reciprocity theorem, one has

dim(Hom K (V γ , ∧ p p) = N j=1 mult(ρ p j , Res G K (ρ γ )) ,
where Res G K (ρ γ ) is the restriction to K of the representation ρ γ . So, finally,

Spec(∆ p ) = {c γ ; γ ∈ G s.t. ∃j s.t. mult(ρ p j , Res G K (ρ γ )) = 0} . In particular Spec(∆ 0 ) = {c γ ; γ ∈ G s.t. mult(triv.repr., Res G K (ρ γ )) = 0} , (1)
and

Spec(∆ 1 ) = {c γ ; γ ∈ G s.t. mult(ρ, Res G K (ρ γ )) = 0} . (2)
3. Proof of the result 1.1.

We furthermore assume that G and K have same rank and consider a fixed common maximal torus T . Let Φ be the set of non-zero roots of the group G with respect to T . According to a classical terminology, a root θ is called compact if the corresponding root space is contained in k C (that is, θ is a root of K with respect to T ) and noncompact if the root space is contained in p C . Let Φ + G be the set of positive roots of G, with respect to a choice of a basis of simple roots. The half-sum of the positive roots of G is denoted by δ G . The space of weights is endowed with the W G -invariant scalar product < , > induced by the Killing form of G sign-changed. The symmetric spaces considerer here being irreducible, the space p C is irreducible. Let α be the highest weight of this representation. As the group G is simple, there are at most two root lenghts, and all roots of a given lenght are conjugate under the Weyl group W G of G (see [START_REF] Humphreys | Introduction to Lie Algebras and Representation Theory[END_REF], $.10.4, lemma C). So α is conjugate to either the maximal root of G or the highest short root. Denote by β this root, and let w be any element in W G such that w • α = β. We claim that Lemma 3.1. The multiplicity mult(ρ, Res G K (ρ β )) is = 0. As the proof differs if either α is long or short, we first have a glance to symmetric spaces for which α is short.

3.1. Symmetric spaces for which the highest weight of the isotropy representation of K is a short root. First note that if G has only one root-lenght, then the highest weight α of the isotropy representation is necessarily a long root. So we only have to consider symmetric spaces G/K for which G has two rootlenghts. Using for instance the table 2, p. 66 in [START_REF] Humphreys | Introduction to Lie Algebras and Representation Theory[END_REF], we have to look to the following symmetric spaces.

(1) SO(2p+2q+1)/SO(2p)×SO(2q+1). We consider here G = Spin(2p+2q+1).

Identifying R 2p and R 2q+1 with the subspaces spanned respectively by e 1 , . . . , e 2p and e 2p+1 , . . . , e 2p+2q+1 , where (e 1 , . . . , e 2p+2q+1 ) is the canonical basis of R 2p+2q+1 , K is the subgroup of G defined by

Spin(2p) • Spin(2q + 1) = ψ ∈ Spin(2p + 2q + 1) ; ψ = ϕ φ , ϕ ∈ Spin(2p) , φ ∈ Spin(2q + 1) ,
(Note that K = Spin(2p), when q = 0). We consider the common torus of G and K defined by

T = p+q k=1 cos(β k ) + sin(β k ) e 2k-1 • e 2k ; β 1 , . . . , β p+q ∈ R .
The Lie algebra of T is

t = p+q k=1 β k e 2k-1 • e 2k ; (1)β 1 , . . . , β p+q ∈ R .
We denote by (x 1 , . . . , x p+q ) the basis of t * given by

x k • p+q j=1 β j e 2j-1 • e 2j = β k .
We introduce the basis ( x 1 , . . . , x p+q ) of i t * defined by

x k := 2i x k , k = 1, . . . , p + q . A vector µ ∈ i t * such that µ = p+q k=1 µ k x k , is denoted by µ = (µ 1 , µ 2 , . . . , µ p+q ) .
The restriction to t of the Killing form B of G is given by B(e 2k-1 • e 2k , e 2l-1 • e 2l ) = -8(2p + 2q -1) δ kl .

It is easy to verify that the scalar product on i t * induced by the Killing form sign changed is given by

∀µ = (µ 1 , . . . , µ p+q ) ∈ i t * , ∀µ ′ = (µ ′ 1 , . . . , µ ′ p+q ) ∈ i t * , < µ, µ ′ > = 1 2(2p + 2q -1) p+q k=1 µ k µ ′ k . (3)
Considering the decomposition of the complexified Lie algebra of G under the action of T , it is easy to verify that T is a common maximal torus of G and K, and that the respective roots are given by (see for instance chapter 12.4 in [START_REF] Bourguignon | A spinorial approach to Riemannian and conformal geometry[END_REF] for details),

±( x i + x j ) , ±( x i -x j ) , 1 ≤ i < j ≤ p + q , ± x i , 1 ≤ i ≤ p + q , for G , ±( x i + x j ) , ±( x i -x j ) , 1 ≤ i < j ≤ p , p + 1 ≤ i < j ≤ p + q , ± x i , p + 1 ≤ i ≤ p + q , for K .
We consider as sets of positive roots

Φ + G = { x i + x j , x i -x j , 1 ≤ i < j ≤ p + q , x i , 1 ≤ i ≤ p + q} , and
Φ + K = x i + x j , x i -x j , 1 ≤ i < j ≤ p , p + 1 ≤ i < j ≤ p + q , x i , p + 1 ≤ i ≤ p + q .
so the set of positive non-compact roots is

Φ + p = x i + x j , x i -x j , 1 ≤ i ≤ p , p + 1 ≤ j ≤ p + q , x i , 1 ≤ i ≤ p .
Note that the sets

∆ G = { x i -x i+1 , 1 ≤ i ≤ p + q -1 , x p+q } and ∆ K = x i -x i+1 , 1 ≤ i ≤ p -1 , x p-1 + x p , x i -x i+1 , p + 1 ≤ i ≤ p + q -1 , x p+q , ,
are basis of G-roots and K-roots respectively. So,

Any µ = (µ 1 , . . . , µ p+q ) ∈ i t * • is a dominant G-weight if and only if µ 1 ≥ µ 2 ≥ • • • ≥ µ p+q ≥ 0 ,
and the µ i are all simultaneously integers or half-integers, • is a dominant K-weight if and only if

µ 1 ≥ µ 2 ≥ • • • ≥ µ p-1 ≥ |µ p | , µ p+1 ≥ µ p+2 ≥ • • • ≥ µ p+q ≥ 0 ,
and the µ i , for 1 ≤ i ≤ p or p + 1 ≤ i ≤ p + q, are all simultaneously integers or half-integers. Hence

• If q = 0, the highest weight of p C is the short root α = x 1 , (which is also the highest shortest root β of G). • If q > 0, the highest weight of p C is the long root α = x 1 + x p+1 .

(2) Sp(p + q)/Sp(p) × Sp(q). The space H p+q is viewed as a right vector space on H in such a way that G may be identified with the group

A ∈ M p+q (H) ; t AA = I p+q ,
acting on the left on H p+q in the usual way. The group K is identified with the subgroup of G defined by

A ∈ M p+q (H) ; A = B 0 0 C , t BB = I p , t CC = I q .
Let T be the common torus of G and K (4)

T :=         e iβ1 . . . e iβp+q    , β 1 , . . . , β p+q ∈ R      ,
where ∀β ∈ R , e iβ := cos(β) + sin(β) i , (1, i, j, k) being the standard basis of H.

The Lie algebra of T is

t =         iβ 1 . . . iβ p+q    ; β 1 , β 2 , , . . . , β p+q ∈ R      .
We denote by(1) (x 1 , . . . , x p+q ) the basis of t * given by

x k •    iβ 1 . . . iβ p+q    = β k . A vector µ ∈ i t * such that µ = p+q k=1 µ k x k , in the basis ( x k ≡ i x k ) k=1,...,p+q , is denoted by µ = (µ 1 , µ 2 , . . . , µ p+q ) .
The restriction to t of the Killing form B of G is given by

∀X ∈ t , ∀Y ∈ t , B(X, Y ) = 4 (p + q + 1) ℜ tr(X Y ) .
It is easy to verify that the scalar product on i t * induced by the Killing form sign changed is given by

∀µ = (µ 1 , . . . , µ p+q ) ∈ i t * , ∀µ ′ = (µ ′ 1 , . . . , µ ′ p+q ) ∈ i t * , < µ, µ ′ > = 1 4(p + q + 1) p+q k=1 µ k µ ′ k . (5)
Now, considering the decomposition of the complexified Lie algebra of G under the action of T , it is easy to verify that T is a common maximal torus of G and K, and that the respective roots are given by

     ±( x i + x j ) , ±( x i -x i ) , 1 ≤ i < j ≤ p + q , ± 2 x i , 1 ≤ i ≤ p + q for G ,          ±( x i + x j ) , ±( x i -x j ) , 1 ≤ i < j ≤ p , p + 1 ≤ i < j ≤ p + q , ± 2 x i , 1 ≤ i ≤ p + q for K ,
We consider as sets of positive roots

Φ + G = { x i + x j , x i -x i , 1 ≤ i < j ≤ p + q , 2 x i , 1 ≤ i ≤ p + q} and Φ + K = x i + x j x i -x j , 1 ≤ i < j ≤ p p + 1 ≤ i < j ≤ p + q , 2 x i , 1 ≤ i ≤ p + q ,
so the set of positive non-compact roots is

Φ + p = { x i + x j , x i -x i , 1 ≤ i ≤ p , p + 1 ≤ j ≤ p + q} . Note that the sets ∆ G = { x i -x i+1 , 1 ≤ i ≤ p + q -1 , 2 x p+q } , and ∆ K = x i -x i+1 , 1 ≤ i ≤ p -1 , 2 x p , x i -x i+1 , p + 1 ≤ i ≤ p + q -1 , 2 x p+q ,
are basis of G-roots and K-roots respectively. So, Any µ = (µ 1 , . . . , µ p+q ) ∈ i t *

• is a dominant G-weight if and only if

µ 1 ≥ µ 2 ≥ • • • ≥ µ p+q ≥ 0 ,
and the µ i are all integers, • is a dominant K-weight if and only if

µ 1 ≥ µ 2 ≥ • • • ≥ µ p ≥ 0 , µ p+1 ≥ µ p+2 ≥ • • • ≥ µ p+q ≥ 0
and the µ i are all integers. Hence the highest weight of p C is the short root α = x 1 + x p+1 .

(3) Sp(n)/U(n). With the same notations as above, we consider the subgroup K of G = Sp(n):

K = {A = (a ij ) ∈ Sp(n) ; a ij ∈ R + i R} ≃ U(n) .
Note that K is the set of fixed points of the inner involution:

σ : Sp(n) → Sp(n) , A → IAI -1 ,
where

I = i I n . The subspace p = {A ∈ sp(n) ; σ * (A) = -A} is then the set p = {A = (a ij ) ∈ sp(n) ; a ij = j R + k R} .
The torus T introduced above (4) is a common torus of G and K. Considering the decomposition of the complexified Lie algebra of G under the action of T , it is easy to verify that T is a common maximal torus of G and K, and that the respective roots are given by

{±( x i -x j ) , 1 ≤ i < j ≤ n , ±( x i + x j ) , 1 ≤ i ≤ j ≤ n} for G , {±( x i -x j ) , 1 ≤ i < j ≤ n} for K ,
We consider as sets of positive roots

Φ + G = { x i -x j , 1 ≤ i < j ≤ n , x i + x j , 1 ≤ i ≤ j ≤ n} and Φ + K = { x i -x j ) , 1 ≤ i < j ≤ n} , so the set of positive non-compact roots is Φ + p = { x i + x j , 1 ≤ i ≤ j ≤ n} . Note that the sets ∆ G = { x i -x i+1 , 1 ≤ i ≤ n -1 , 2 x n } , and ∆ K = { x i -x i+1 , 1 ≤ i ≤ n -1} ,
are basis of G-roots and K-roots respectively (there are only n -1 simple K-roots as K is not semi-simple). So, Any µ = (µ 1 , . . . , µ n ) ∈ i t * is(1)

• a dominant G-weight if and only if

µ 1 ≥ µ 2 ≥ • • • ≥ µ n ≥ 0 ,
and the µ i are all integers, • a dominant K-weight if and only if

µ i -µ i+1 ∈ N , 1 ≤ i ≤ n -1 .
Hence there are two dominant K-weights in the representation p C : 2 x 1 and x 1 + x 2 , but the highest weight is 2 x 1 since x 1 + x 2 ≺ 2 x 1 . Hence the highest weight of p C is the long root α = 2 x 1 . (4) G 2 /SO(4). We use here the results given in [START_REF] Cahen | Spin Structures on Compact Simply Connected Riemannian Symmetric Spaces[END_REF] (see page 226), which follow from a general result of Borel-de Siebenthal, [START_REF] Borel | Les sous-groupes fermés de rang maximum des groupes de Lie clos[END_REF]. A set of Groots Φ G is given by the elements x ∈ R 3 , whose coordinates are integers verifying 3 i=1

x i = 0 and x 2 = 2 or 6 , hence Φ G = {±(e 1 -e 2 ) , ±(e 2 -e 3 ) , ±(e 3 -e 1 ) , ±(2 e 1 -e 2 -e 3 ) , ±(2 e 2 -e 1 -e 3 ), ±(2 e 3 -e 1 -e 2 )} .

The following system of positive G-roots is choosen:

Φ + G = {e 1 -e 2 , e
3 -e 2 , e 3 -e 1 , -2 e 1 + e 2 + e 3 , -2 e 2 + e 3 + e 1 , 2 e 3 -e 1 -e 2 } . It can be checked that a basis of G-roots is given by ∆ G = {e 1 -e 2 , -2 e 1 + e 2 + e 3 } .

A system of positive K-roots (which appears to be also a basis of K-roots) is then given by Φ + K = {e 1 -e 2 , -e 1 -e 2 + 2 e 3 } . The set of positive non-compact roots is Φ + p = {e 3 -e 2 , e 3 -e 1 , -2 e 1 + e 2 + e 3 , -2 e 2 + e 3 + e 1 } . There are two dominant weights in p C : -2 e 2 + e 3 + e 1 and e 3 -e 2 , but the highest weight is -2 e 2 + e 3 + e 1 since e 3 -e 2 ≺ -2 e 2 + e 3 + e 1 . Hence the highest weight of p C is the long root α = -2 e 2 + e 3 + e 1 .

(5) F 4 /Sp(3) • Sp(1). We use here also results given in [START_REF] Cahen | Spin Structures on Compact Simply Connected Riemannian Symmetric Spaces[END_REF] (see page 227).

A set Φ G of G-roots is given by the elements x ∈ R 4 whose coordinates are integers or half-integers satisfying x 2 = 1 or 2, [START_REF] Humphreys | Introduction to Lie Algebras and Representation Theory[END_REF], hence

Φ G = ±e i , 1 ≤ i ≤ 4 , ±e i ± e j , 1 ≤ i < j ≤ 4 , 1 2 (±e 1 ± e 2 ± e 3 ± e 4 ) .
We consider the system of positive roots

Φ + G = e i , 1 ≤ i ≤ 4 , e i ± e j , 1 ≤ i < j ≤ 4 , 1 2 (e 1 ± e 2 ± e 3 ± e 4 ) .
It can be check that a basis of G-roots is given by ∆ G = {α 1 := e 2 -e 3 , α 2 := e 3 -e 4 , α 3 := e 4 , α 4 := 1 2 (e 1 -e 2 -e 3 -e 4 )} .

A system of positive K-roots is then given by There are two dominant weights in p C : e 1 and e 1 + e 3 , but the highest weight is e 1 + e 3 since e 1 ≺ e 1 + e 3 . Hence the highest weight of p C is the long root α = e 1 + e 3 . (6) F 4 /Spin(9). In that case, a system of positive K-roots is

Φ + K = e 3 ,
Φ + K = {e i , 1 ≤ i ≤ 4 , e i ± e j , 1 ≤ i < j ≤ 4} .
A basis of simple K-roots is given by ∆ K = {e 1 -e 2 , e 2 -e 3 , e 3 -e 4 , e 4 } .

(1) The set of positive non-compact roots is

Φ + p = 1 2 (e 1 ± e 2 ± e 3 ± e 4 ) ,
hence the highest weight of the representation p C is the only dominant weight: the short root α = 1 2 (e 1 + e 2 + e 3 + e 4 ).

Proof of the lemma 3.1.

Proof. The proof is very simple if there is only one root lenght, or if α is a long root. In that case, β is necessarily the maximal root, hence the highest weight of the adjoint representation of the simple group G in its complexified Lie algebra g C . But the decomposition of g C = k C ⊕ p C into K-invariant subspaces implies at once that ρ is contained in the restriction of ρ β to K.

The proof is a little more involved when two roots lenghts occur and α is a short root. As we saw it just before, there are only three cases to be considered here: SO(2p + 1)/SO(2p), Sp(p + q)/Sp(p) × Sp(q), and F 4 /Spin(9).

Let v β be the maximal vector (unique up to a scalar multiple) of the representation V β and let g ∈ G be some representant of w -1 ∈ W G . First ρ β (g) • v β is a weightvector for the weight α since forall X ∈ t,

ρ β * (X) • (ρ β (g) • v β ) = d dt |t=0 (ρ β (exp(t X)g) • v β ) = d dt |t=0 ρ β (gg -1 exp(t X)g) • v β = ρ β (g) • ρ β * (Ad(g -1 ) • X) • v β = ρ β (g) • β(w • X) v β = w -1 (β)(X) ρ β (g) • v β = α(X) ρ β (g) • v β .
Now, we claim that there exists w ∈ W G such that β = w • α, and the weight-vector ρ β (g) • v β , where g is a representative of w -1 , is a maximal vector for the action of K. We may consider only the action of a basis of simple K-roots {θ 1 , . . . , θ r }. First note that if θ is a positive K-root, and E θ a root-vector associated to it, then Ad(g -1 ) • E θ is a root-vector for the root w • θ, since forall X ∈ t,

[X, Ad(g -1 ) • E θ ] = Ad(g -1 ) • [Ad(g) • X, E θ ] = Ad(g -1 ) • [w -1 • X, E θ ] = θ(w -1 • X) Ad(g -1 ) • E θ = (w • θ)(X) Ad(g -1 ) • E θ .
Now, if w • θ is a positive root, then, since v β is a maximal vector killed by any root-vector associated to a positive root,

ρ β * (E θ ) • (ρ β (g) • v β ) = d dt |t=0 ρ β (gg -1 exp(t E θ )g) • v β = ρ β (g) • ρ β * (Ad(g -1 ) • E θ ) • v β = ρ β (g) • 0 = 0 .
So ρ β (g) • v β is killed by the action of any positive K-root θ such that w • θ is a positive root. Hence we may conclude by proving that, for each symmetric space under consideration, there exists w ∈ W G such that w •α = β, and w •θ i is a positive G-root, for any simple K-root θ i .

(1) SO(2p + 1)/SO(2p). We saw above that the highest weight of the representation p C is the short root α = x 1 , which is also the highest shortest root β of G. We may choose w = id, and as all the K-simple roots in ∆ K are G-positive roots, the claim is verified in that case. (2) Sp(p + q)/Sp(p) × Sp(q). We saw above that the highest weight of the representation p C is the short root α = x 1 + x p+1 . Now the highest short G-root is β = x 1 + x 2 . Let w be the element of the Weyl group W G given by the p-cycle permutation (2 3 • • • p + 1). One has w • α = β, and it is easily verified that w • θ i is a positive G-root for any simple K-root θ i , hence the claim is also proved in that case. (3) F 4 /Spin(9). We saw above that the highest weight of the representation p C is the short root α = 1 2 (e 1 + e 2 + e 3 + e 4 ). Now, the highest short G-root is β = e 1 , and the reflection σ α4 across the hyperplane α ⊥ 4 verifies σ α4 • α = β.

It is easily verified that σ α4 • θ i is a positive G-root for any simple K-root θ i since σ α4 : e 1 -e 2 → e 3 + e 4 , e 2 -e 3 → e 2 -e 3 , e 3 -e 4 → e 3 -e 4 , e 4 → 1 2 (e 1 -e 2 -e 3 + e 4 ) .

Hence the claim is also proved in that case.

3.3. First eigenvalue of the Laplace operator acting on 1-forms. In order to conclude, we have to verify that the Casimir eigenvalue c β is the lowest eigenvalue of the Laplacian.

Lemma 3.2. Let (ρ γ , V γ ) be an irreducible G representation such that the multi- plicity mult(ρ, Res G K (ρ γ )) = 0. Then c β ≤ c γ .
Proof. Recall that the highest weight of ρ is α, hence if mult(ρ, Res G K (ρ γ )) = 0, then α and β = w • α are actually weights of the representation ρ γ . But then, as γ is the highest weight,

β + δ G , β + δ G ≤ γ + δ G , γ + δ G , cf. Lemma C, p.71 in [Hum72], hence c β = β + δ G 2 -δ G 2 ≤ γ + δ G 2 -δ G 2 = c γ .
Remark 3.3. The fact that c β = 1 when β is the highest long root may seem to be rather puzzling. This is indeed a consequence of Freudenthal's formula (cf. 48.2 in [START_REF] Freudenthal | Linear Lie Groups[END_REF] or p. 123 in [START_REF] Humphreys | Introduction to Lie Algebras and Representation Theory[END_REF]): for any G-weight µ of the representation (ρ β , V β ) = (Ad, g C ) , the multiplicity mult(µ) of µ is given recursively by the formula

( β + δ G , β + δ G -µ + δ G , µ + δ G ) mult(µ) = 2 θ≻0 ∞ i=1 mult(µ + iθ) µ + iθ, θ .
Applying this formula to the weight µ = 0, one obtains since • mult(0) = dim(t), as T is a maximal common torus,

• for any integer i ≥ 1, mult(iθ) = 0 ⇔ i = 1, and mult(θ) = 1, by properties of roots, since the only multiple of a root θ which is itself a root is ±θ,

( β + δ G 2 -δ G 2 ) dim(t) = 2 θ≻0 θ 2 .
But Gordon Brown's formula [START_REF] Brown | A remark on semi-simple Lie algebras[END_REF] or 21.5 in [START_REF] Freudenthal | Linear Lie Groups[END_REF]) states that

2 θ≻0 θ 2 = dim(t) , hence c β = 1 .

The first non-zero eigenvalue of the Laplace operator on functions

As it was noticed in the introduction, any (non-zero) eigenvalue λ of the Laplace operator on functions is greater or equal to the first eigenvalue of the Laplace operator on 1-forms. We now examine in which case this inequality is an equality.

Proposition 4.1. Let λ be the first non-zero eigenvalue of the Laplace operator on functions, and let µ be the first eigenvalue of the Laplace operator on 1-forms.

If G/K is a Hermitian symmetric space:

SU(p + q)/S(U(p) × U(q)) , SO(p + 2)/SO(n) × SO(2) , Sp(n)/U(n) , SO(2n)/U(n) , E 6 /SO(10) • SO(2) , E 7 /E 6 • SO(2) ,
or if G/K is one of the symmetric spaces for which the highest weight of the isotropy representation is a short root:

SO(2p + 1)/SO(2p) , Sp(p + q)/Sp(p) × Sp(q) or F 4 /Sp(3) • Sp(1) , then λ = µ.
In all the other cases, λ > µ.

Proof. By (1), the spectrum of the Laplace operator on functions is given by considering the irreducible representations γ ∈ G such that mult(triv.repr., Res G K (ρ γ )) = 0, or equivalently, such that there exists a non-zero weight-vector trivially acted by the group K. Such representations are called spherical representations [START_REF] Sugiura | Representations of Compact Groups Realized by Spherical Functions on Symmetric Spaces[END_REF], [START_REF] Takeuchi | Modern Spherical Functions, Translations of Mathematical Monographs[END_REF]. Hence, to prove the equality λ = µ, we have to verify that mult(triv.repr., Res G K (ρ β )) = 0, where β is the highest long or short root. Indeed, if that is verified, then c β ≥ λ, as c β belongs to the spectrum. But, as it was remarked in the introduction, λ ≥ c β , since λ has to be greater or equal to the first eigenvalue of Laplace operator on 1-forms. Let us first examine the case when β is the highest long root. In that case, ρ β is the adjoint representation of the simple group G on its complexified algebra g C , hence a non-zero root-vector is trivially acted by the subgroup K if and only if it belongs to the center of K. But this is only possible if and only if G/K is hermitian, see for instance [START_REF] Wolf | Spaces of constant curvature[END_REF]. Let us consider now the three symmetric spaces where two root lengths occur and β is the highest short root: SO(2p + 1)/SO(2p), Sp(p + q)/Sp(p) × Sp(q), and and F 4 /Spin(9). Going back to the case-by-case study of those three symmetric spaces above, one gets:

(1) SO(2p + 1)/SO(2p). The highest short root is β = x 1 , which is the highest weight of the fundamental standard representation of the group Spin(2p+1) (or SO(2p+1)) in the space C 2p+1 , see for instance chapter 12 in [START_REF] Bourguignon | A spinorial approach to Riemannian and conformal geometry[END_REF]. Now, the group K acts trivially on the last vector e 2p+1 of the canonical basis (e 1 , . . . , e 2p+1 ) of C 2p+1 since the inclusion K ⊂ G is induced by the natural inclusion

A ∈ SO(2p) -→ A 0 0 1 ∈ SO(2p + 1) .
Hence mult(triv.repr., Res G K (ρ β )) = 0.

(2) Sp(p + q)/Sp(p) × Sp(q). The highest short root is β = x 1 + x 2 , which is the highest weight of the fundamental representation of the group Sp(p + q) in the space ∧ 2 0 (C 2(p+q) ). To be more explicit, H p+q is identified with C 2(p+q) here, such that Sp(p + q) ≃ SU(2(p + q)) ∩ Sp(2(p + q), C), and the representation of Sp(p + q) in the space ∧ 2 (C 2(p+q) ) decomposes into two irreducible pieces:

(6) ∧ 2 (C 2(p+q) ) = ∧ 2 0 (C 2(p+q) ) ⊕ C • ω p+q ,
where the first piece is the fundamental representation with highest weight x 1 + x 2 , and the second one the trivial representation on the space generated by the symplectic 2-form:

ω p+q := e 1 ∧ e -1 + e 2 ∧ e -2 + • • • + e p+q ∧ e -(p+q) ,
where (e 1 , e 2 , . . . , e p+q , e -1 , e -2 , . . . , e -(p+q) ) is the canonical basis of C 2(p+q) , see for instance chapter 12 in [START_REF] Bourguignon | A spinorial approach to Riemannian and conformal geometry[END_REF]. Note then that the 2-form

e 1 ∧ e -1 + e 2 ∧ e -2 + • • • + e p ∧ e -p , is K = Sp(p) × Sp(q)-invariant, so its (non-zero) component in ∧ 2 0 (C 2(p+q)
) under the decomposition (6) is also an invariant of the group K, hence mult(triv.repr., Res G K (ρ β )) = 0. (3) F 4 /Spin(9). The highest short root is β = e 1 . The function "branch" in the LiE Program, [START_REF] Van Leeuwen | A Package for Lie Group Computations[END_REF],

branch([0, 0, 0, 1], B4, res mat(B4, (F 4)), (F 4)) , returns that the dominant weights in the decomposition of Res G K (ρ β ) (expressed in the basis of fundamental weights) are [0, 0, 0, 0] , [0, 0, 0, 1] and [1, 0, 0, 0] , hence mult(triv.repr., Res G K (ρ β )) = 1 = 0. The result may be also verified with the help of the Satake diagram of F 4 /Spin(9) (see [START_REF] Sugiura | Representations of Compact Groups Realized by Spherical Functions on Symmetric Spaces[END_REF]). Consider now all the remaining symmetric spaces. The highest weight of their isotropy representation is long. Let γ ∈ G be such that c γ = λ and mult(triv.repr., Res G K (ρ γ ) = 0. The irreducible representation γ is not the adjoint representation of group G on its complexified algebra g C , since otherwise G/K should be hermitian. Let v be a weight-vector trivially acted by the group K. This vector v is not killed by the action of at least one root-vector associated to a simple (non-compact) root, since otherwise γ should not be irreducible. The corresponding positive non-compact root belongs then to the set of weights. We claim that such a root is long. This is obviously true if there is only one root-lenght. But this is also true if there are two root-lenghts: going back to the symmetric spaces considered in section 3.1, it can be checked that in all the cases under consideration, any short simple root of G is always also a simple root of K. Now, since all roots of a given lenght are conjugate under the Weyl group (see for instance Lemma C, p. 71 in [START_REF] Humphreys | Introduction to Lie Algebras and Representation Theory[END_REF]), one can deduce that the highest long root β belongs to the set of weights of γ. As β is not the highest weight of γ, since otherwise G/K should be hermitian, one can conclude using Lemma C, p.71 in [START_REF] Humphreys | Introduction to Lie Algebras and Representation Theory[END_REF] that c γ > c β = 1, hence λ > µ.

Francis Burstall has pointed out to us that the three cases : λ < 1, λ = 1 and λ > 1 exactly amount the three possibilities for the second homotopy group of G/K : λ < 1 if π 2 (G/K) is trivial, λ = 1 if π 2 (G/K) = Z, and λ > 1 if π 2 (G/K) = Z/Z 2 , see chapter 3 in [START_REF] Burnstall | Twistor theory for Riemannian symmetric spaces[END_REF].

To conclude, we have to compute the Casimir eigenvalue c β for the three symmetric spaces for which the highest weight of the isotropy representation is a short root.

(1) SO(2p + 1)/SO(2p). Here β = x 1 . The half-sum of the positive G-roots is

δ G = p - 1 2 , p - 3 2 , . . . , 3 2 , 1 2 .
Using (3), one gets

c β = β + 2 δ G , β = 2 p 2 (2p -1) = p 2p -1 .
(2) Sp(p + q)/Sp(p) × Sp(q). Here β = x 1 + x 2 . The half-sum of the positive roots is δ G = (p + q, p + q -1, . . . , 2, 1) . Using (5), one gets c β = β + 2 δ G , β = 4 (p + q) 4 (p + q + 1) = p + q p + q + 1 .

(3) F 4 /Spin(9). Here β = e 1 . The half-sum of the positive roots is

δ G = 1 2
(11 e 1 + 5 e 2 + 3 e 3 + e 4 ) .

We considered above the scalar product on weights induced by the usual scalar product on R 4 . In order to compare it with the scalar product induced by the Killing form sign-changed, we use the "strange formula" of Freudenthal and de Vries (see 47-11 in [START_REF] Freudenthal | Linear Lie Groups[END_REF]). For the scalar product ( , ) induced by the usual scalar product on R 4 :

(δ G , δ G ) = 39 , whereas for the scalar product , induced by Killing form sign-changed:

δ G , δ G = dim(G) 24 = 13 6 .

Hence, as the two Ad G -invariant scalar products have to be proportional since G is simple,

  e 4 , e 1 + e 2 , e 1 -e 2 , e 3 + e 4 , e 3 -e 4 , -e 2 + e 3 + e 4 ) . can be check that a basis of K-roots is given by∆ K = e 1 + e 2 ,e 4 , e 3 -e 4 , 1 2 (e 1 -e 2 -e 3 -e 4 ) . , e 2 , e 1 + e 3 , e 1 + e 4 , e 2 + e 3 , e 2 + e 4 , e 1 -e 3 , e 1 -e 4 , e 2 -e 3 , e 2 -e 4 ,

					1 2	(e 1 -e 2 -e 3 -e 4 ) ,
	1 2 (e 1 It The set of positive non-compact roots is (e 1 -e 2 -e 3 + e 4 ) , 1 2 (e 1 -e 2 + e 3 -e 4 ) , 1 2
	Φ + p = e 1 1 2	(e 1 + e 2 + e 3 + e 4 ) ,	1 2	(e 1 + e 2 + e 3 -e 4 ) ,
	1 2	(e 1 + e 2 -e 3 + e 4 ) ,	1 2	(e

1 + e 2 -e 3 -e 4 ) .