Gabriel F Alves
email: gabrielferreiraalves@id.uff.br

Anselmo A Montenegro

Daniela G Trevisan
email: daniela@ic.uff.br

Creating Layouts for Virtual Game Controllers Using Generative Design

Keywords: Generative design, gamepad, virtual controller, genetic algorithm, machine learning

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

With the advancement of technology and thanks to its dynamism, touchscreen devices like smartphones and tablets have also become game controllers: the virtual gamepads. They are dynamic because they have components and devices such as Bluetooth and wireless, accelerometers, gyroscopes and cameras. These components may contribute to the development of different forms of interaction with digital games [START_REF] Baldauf | Your browser is the controller: advanced web-based smartphone remote controls for public screens[END_REF], either in games played on the mobile devices themselves or in games played on another device. Because these controllers can access device components by touching the screen and/or by moving and rotating the device, the buttons on the virtual controller can follow the rules of the game, so it is not necessary to include several buttons that will not be used during the gameplay.

Granted by CAPES, Brazil

Unfortunately, this is not possible in physical controllers, because once the controller has been built, all the buttons will always be available to the user, even if they have no use in certain games. Each game has a finite number of actions and interactions that are not necessarily the same as in other games and can vary between different genres of games, such as action and adventure games; sports and shooting; race and platform; and even within the same genre of games. These actions are usually associated with a single button on the controller. Therefore, there will be a maximum number of buttons required to perform all interactions in the game. On the other hand, generative design is a promising approach to solve various aspects related to User Interface [START_REF] Troiano | Genetic algorithms supporting generative design of user interfaces: Examples[END_REF]. It offers new opportunities for conception and creation problems in areas such as engineering, architecture, and design, making it possible, through the use of meta-heuristics, to determine the layout of UI components, informing only a limited amount of parameters and producing new and sometimes, unexpected results [START_REF] Troiano | Genetic algorithms supporting generative design of user interfaces: Examples[END_REF].

Based on all this knowledge, the goal of this work is to develop a tool that can help game designers to find different and innovative layouts of virtual controllers for their games, since a virtual controller can be implemented especially for a specific game, containing only the actions necessary for it. Therefore, it is not the purpose of this paper to provide layouts for physical video game controllers. Thus, we defined a generative design model [START_REF] Bentley | Evolutionary design by computers[END_REF], where the designer must tell to the system what actions the player should be able to perform within the game, the number of layouts he/she wants the model to deliver as output and how many iterations the algorithm will perform before constructing these layouts. A genetic evolution-based approach has been used to generate a huge variety of layouts and, based on information added to a pre-trained database, the model can generate and suggest a set of valid controller configurations for the desired game. A machine learning approach has been applied to filter the most usable and appropriate set of layouts to be presented to the game designer. This approach was chosen because the generative design provides a wide range of solutions, but it is not feasible for the professional to analyze all of them. Based on the generated and filtered results, the designer will be able to choose which one suits his/her game and, if necessary, make adjustments and refinements to its design. A generative design-based solution can be used as a robust starting point where the results can be refined by software engineers and artists [START_REF] Troiano | Genetic algorithms supporting generative design of user interfaces: Examples[END_REF]. Therefore, it is important to point out that this tool is a software that provides creative support to the designer, the true responsible for the development of the controller of his/her game.

Related Work

In order to help game designers to develop one or more layouts of ideal and innovative virtual controllers for their games, we sought to find approaches, techniques, and methodologies that would aid in the process of creating ideas and supporting the design process. After researching in academic libraries such as Google Scholar and the ACM library, no results were found on the use of Generative Design applied in the area of digital game controllers, so this section will present works and knowledge that have been served as the basis for the development of our solution.

Usually, the virtual gamepads follow some rules and are created based on existing organizational elements in physical controllers, such as the grouping of directional buttons, the possibility of incorporation of d-pads, the existence of buttons action, among others. Looking for solutions to support the design process of game controllers, some researchers have tried to use a set of Design Thinking techniques, such as a day in the life, empathy map, persona, ideation workshop and paper prototyping to design a game controller that pleases its players [START_REF] Alves | Applying design thinking for prototyping a game controller[END_REF]. The developed prototypes have sought to soften and even eliminate the problem of the lack of tactile feedback in touchscreen devices by using personalized skins over smartphones. However, although the work relates to this in the way they look for new solutions that support the design process, no software support for the design process has been provided. Baldauf et al. [START_REF] Baldauf | Investigating on-screen gamepad designs for smartphone-controlled video games[END_REF] has developed four virtual controllers based on existing controllers configurations. A comparative laboratory study was conducted where four gamepad smartphone designs were selected. Each was tested in two popular games: Pac-Man and Super Mario Bros. The construction of different configurations of virtual controllers for games is the main relationship between our work and that of the authors, although our work aims to assist and give creative support to the designer, and not to create the controller itself. Regarding approaches that use artificial intelligence techniques to support the design of game controllers, we point out the work of Torok et al. [START_REF] Torok | Designing game controllers in a mobile device[END_REF] that deals with the design of adaptive game controllers for touchscreen devices. Their solution introduces an adaptation that derives the user's personal preferences from a series of basic events, such as button presses or internal gameplay changes. The main disadvantage of this approach is that the controller layout adaptation takes place during the player interaction time. At times, many changes on the fly to the controller layout may confuse the player, but on the other hand, this can be used by the designer as a way of adding different levels of difficulty to their game. Our work bears similarities to the work of Torok et al. [START_REF] Torok | Designing game controllers in a mobile device[END_REF], mainly in the quest to find solutions for virtual controllers. However, his work builds the solutions in interaction time, while this work produces them in time of design.

In his work, Krish [START_REF] Krish | A practical generative design method[END_REF] proposes a Computer-Aided Design (CAD) Generative Design model suitable for complex multi-criteria design problems in which important performance criteria are incomputable. This method is based on the construction of a genotyping project within a parametric CAD system based on history and then its parameters undergo random variations within predefined limits to generate a set of distinct projects. After the designs are generated, they are filtered through multiple constraint envelopes, representing geometric feasibility, manufacturability, cost, and other performance-related constraints, thus reducing the design space in a smaller, feasible design space, represented by a set of different designs.

By contrast, many of the technologies that masquerade as generative design such as topology optimization, lattice optimization, parametric optimization or similar technologies are focused on improving a preexisting design, not creating new design possibilities as in the generative design. The confusion arises because the inputs to generative design are similar to the inputs to many optimization tools. However, generative design produces many valid designs or solutions instead of one optimized version of a known solution. Genetic Algorithms are metaheuristic methods inspired by natural selection [START_REF] Bjørlykhaug | Mechanical design optimization of a 6dof serial manipulator using genetic algorithm[END_REF] and are unquestionably the most dominant in computational design exploration [START_REF] Krish | A practical generative design method[END_REF].

Methodology and Implementation

The architecture of the proposed model consists of three stages: parameter specification, exploring the design space (by the genetic algorithm) and filtering the most relevant solutions (by machine learning). In the first part, the user -a game designer -informs the model of the essential parameters for the creation of layouts that will be given as results. The model, in its current state, receives the following parameters: the number of buttons, the actions that these buttons execute within the game, the number of layouts that should be generated by the model and the number of iterations it should run. The second part is responsible for creating layouts based on the parameters entered by the user. These layouts will be randomly defined at the beginning of the model execution, pass through genetic operators, will be evaluated by a fitness function and some of the selected individuals will generate the next population. The algorithm will perform these operations n times, where n is the number of iterations defined by the designer. At the end of the iterations, the final set of layouts, also called the final population, will be defined. Many individuals may be produced at this time. The third and last part of the model was precisely defined so that it was possible to present to the designer just those who had some use for him/her. Therefore, the machine learning Support Vector Machine (SVM) algorithm was used to classify individuals between valid and invalid, separating them so the user could analyze a smaller range of results.

The solution development was done using the Java-based language called 'Processing' [START_REF] Reas | Processing. Available in[END_REF], highly recommended in the generative design community because it is a free, easy-to-use and flexible software sketchbook and language. Figure 1 illustrates the pipeline of the proposed generative design model which is detailed in the following subsections.

Input Parameters

In order to start the model's development, it was necessary to carry out a previous study on games, their styles and the ways of controlling them. It is said by Rogers [12] that all games fit into a previously established genre, and a game can fit into one or more genres, where one is dominant. An example is the game The Last of Us [START_REF]PlayStation: The last of us[END_REF] released for PlayStation3 and remastered for PlayStation4, classified as an action game containing features of a puzzle and shooting game.

To classify a game in a given genre, it must be possible to perform some specific actions during gameplay. For example, one of the genres that a game can be classified is the "Platform" genre. In Platform games, the player must be able to perform the actions "walk", "run", "jump" and "act", the latter being a variable determined by the game designer, such as throwing an item, triggering a device or performing a hit.

With the knowledge about game genres, it was possible to list the actions that can be performed within them. These actions -from moving to pausing a game -are the first input parameters that the designer must inform the model. Other defined parameter is related to the genetic algorithm: population size and the number of iterations.

Exploring the Design Space

In order to explore the design space for the creation of virtual controllers, we used a genetic algorithm. To understand the logic of the system, it is necessary to understand some terms, such as 'population', 'individual', 'fitness' and 'roulette'. A population consists of two or more individuals, while an individual consists of a set of configurations of a controller. By controller configuration, we are talking about a layout and its components, such as the number of buttons, button types, button classifications and their positions (X and Y axes) on the screen. For the development of the system, some data structures were defined. In the data structure associated with each individual, we define three arrays called chromosomes, and each of them stores essential information for the creation of individuals:

-chromosomeX: contains the X-axis positions of all buttons requested by the designer; -chromosomeY: contains the Y-axis positions of the buttons; classification: contains the classification of the button (the role that such button assumes within a control, what action it assumes).

The most basic button classes defined in the model are presented in Table 1. The size of the chromosomes is defined at the beginning of the process, when the user informs the number of buttons that the controller should have, making it possible to perform the actions within the game. In addition to these three vectors, each individual also carries some attributes:

fitness: a numerical value responsible for representing how apt an individual is to follow to a new population; -fitnessPercent: the fitness value represented in percentage; -rouletteTrack: the range that the individual holds over the roulette of individuals' selection. This will be detailed later.

When an individual is created, the position chromosomes containing the X and Y positions of the buttons are filled with random values, causing each button to assume a random position within the controller area and the classification chromosome is filled with labels for the actions listed on Table 1. The value of the fitness attribute is defined by the fitness that an individual has to maintain in the next generations of the algorithm. Let L be a layout candidate, i.e., an individual in the population, with n buttons. Let bx and by be two arrays that store, respectively, the x and y coordinates of each button i, 0 ≤ i < n. Moreover, let us define bt as an array that identifies the type or class of a given button i. The objective function that defines the fitness value of a layout L is given by:

(1)

f (bx, by, bt) = n-1 i=0 n-1 j=i+1 1 -δ(bt i , bt j ) l dist(bx i , by i , bx j , by j ) + αδ(bt i , bt j ) 1 1 + dist(bx i , by i , bx j , by j )
where δ(bt i , bt j ) is 1 if buttons i and j belong to the same class and 0 if they are in different classes. The Euclidean distance between two buttons is given by dist(bx i , by i , bx j , by j ), l = (w + h) is a normalization factor, where w and h are the width and height of the layout, and α is a weight. In our model, we set the α value as six. The first term in the sum is responsible for creating a separation between buttons from different classes, and the second term tries to approximate buttons in the same class. Having defined a fitness value for the individual, the percentage relative to his fitness is calculated and, according to it, the roulet-teTrack is defined. The roulette is a technique of selecting individuals. Through it, all individuals have attached to them a range of a roulette wheel, this range is defined based on their fitness: individuals with higher fitness values will have larger range bands in it, thus having a greater chance of being selected to be part of the new population. Although individuals with lower fitness scores have smaller roulette intervals, they still have a chance of being selected to participate in the new population, causing diversity among individuals. Determining the roulette tracks for each individual is done using the fitnessPercent value to assign the ranges: (1) the first step is to sort the population in ascending order using the fitnessPercent values; (2) after that, each individual is linked to a roulette track -the first receives the initial range, which ranges from zero to his fitnessPercent. The next individual will go from the fitnessPercent value of the previous individual to the sum of this value with their percentage; (3) until 100% of the roulette wheel is distributed. With fitness and fitnessPercent values calculated and having defined the ranges of roulette that each individual possesses, the execution of the genetic operations are triggered. By the number of iterations required, the following steps will be performed: (1) a new population is defined; (2) for each individual in the population, a value between 0 and 100 percent will be drawn. The individual whose track in the roulette contains such value will be selected to undergo the following operations; (3) mutation operators may be triggered to modify the individual; (4) the individual will be added in this new population and old individuals will be removed.

In this work, we did not use any Crossover operators. Instead, we used two mutation operators with 50% probability of a mutation type 1 to occur, while the probability of a mutation type 2 occurring is 80%. The first mutation is responsible for changing the position of any button within the generated layout. This operation calculates an offset vector in any direction and moves the button to a maximum distance of 100 pixels from its original position. The second mutation is responsible for exchanging the position of one or more buttons with other buttons on the same layout -for example, the directional button 'up' exchanges its position with the 'jump' action button.

Finally, it is necessary to check if there is no collision between the buttons, that is if one button does not overlap another within a layout. At the end of n iterations, the final population will be defined and can proceed to the next step: filtering the design space using the Support Vector Machine (SVM) algorithm.

Filtering the Design Space

The search for a solution that applied labels to the results given by the genetic solution was necessary because evaluating all the generated layouts is a tiresome task and the objective function used in the genetic algorithm does not capture all aspects of good design. This was a strategy used to keep the originality and variability of the individuals generated. By embedding too much rules and constraints in the objective function we could drastically reduce the variability of the individuals.

Each button in a controller carries with it a semantics that gives a meaning to its presence in the layout. An 'up' directional button is expected to be positioned above the other directional buttons, which makes the interaction between player and controller more dynamic, easy, and intuitive. Thus, the classification of individuals from different populations was performed considering the position of the buttons in the controller area, the groupings between them and the distances between the buttons of the same class -directional, action or system buttons. This knowledge, orginated in the designer rationale, is learned by the Support Vector Machine (SVM) classifier during the training set.

Based on a previously trained database (training set), the SVM algorithm can classify other databases (test set), automating the process of selecting individuals. For the model, two classifications are made: valid individuals and invalid individuals. The first are those that may be chosen by the designer as candidates for game controller configurations, given the position of the buttons in the controller area, while the second classification is formed by layouts that do not represent configurations that can be used by the final user. After that, the model displays the results for the designer, separating them according to the classification that the SVM has produced. To training the dataset to classify three-button layouts, 301 individuals were labeled manually by a designer, where 150 were labeled 'valid' and 151 as 'invalid'. The SVM training resulted in correct classification of 259 individuals, 86.05% of the entire population. Another population with 306 individuals were manually classified to construct the dataset responsible for classifying controllers configurations with nine buttons. From that, 131 individuals were classified as 'valid' and 175 as 'invalid'. At this case, 273 individuals were correctly classified, corresponding to 89.21% of the settings.

Tests and Results

The experiments were carried out to find out if there is any relation between the number of iterations and the number of valid individuals generated by the model and, in addition, to measure the variability of the results. It is important that the model provides a range of useful results for the designer and that the layouts are not all similar to each other.

First, it was necessary to define which genre of games we will generate the layouts of the controllers. We assume that the game designer wants to build virtual controllers for games of the platform genre, where some specific actions must be made available to be executed in it. One of the controllers should have only three buttons: two directional buttons (left and right) and one action button (jump) and another controller setting should have nine buttons: the four directional buttons (up, down, right and left), four other action buttons (jump, shoot, defend and run) and a system button (start). After chosing the configurations as a function of the number of buttons we define the number of iterations of the evolutionary algorithm and the number of individuals in the final population. Ten different types of iterations were defined: 1000, 2000, 3000, up to 10,000 iterations. The number of individuals was established as 1000, 2000 and 3000 individuals for each controller configuration with three and nine buttons. A total of 60 populations were obtained. After creating all the settings, each population was submitted to the evaluations of the SVM classifier that, according to a previously trained database, classifies each layout between 'valid' or 'invalid'. In this case, specific classification databases were used for three and nine buttons.

Variability of Layouts

The variability implies the diversity of results so that it is possible to test several valid configurations and to find those that better suit the designer's need to create an ideal controller for his game. That said, we seek to find a solution that can group similar layouts to make it possible to measure the variability of the valid results generated. The K-means [START_REF] Jain | Data clustering: 50 years beyond k-means[END_REF] clustering algorithm proved to be a possible solution to our problem because it is an algorithm that partitions data into clusters. The K-means clustering algorithm is a commonly used method for automatically partitioning a data set into k groups [START_REF] Wagstaff | Constrained k-means clustering with background knowledge[END_REF]. The purpose of data grouping is to discover the natural grouping of a set of patterns, points or objects so that it is possible to do analyzes and absorb knowledge intrinsic to the data [START_REF] Jain | Data clustering: 50 years beyond k-means[END_REF]. The more traditional definition of this technique is not able to determine the ideal number of clusters, and it is necessary to previously inform the number of clusters that it is desired to separate the data. To define the number of clusters and find the number of clusters that might be ideal for measuring the variability of results, we use a K-means extension that uses the Bayesian Information Criterion (BIC) [START_REF] Fraley | How many clusters? which clustering method? answers via model-based cluster analysis[END_REF]. It is based in part on the likelihood function and can be used to compare models with different parametrizations, with different number of components, or both. Therefore, we use this technique to find the ideal number of clusters and, finally, to separate layouts into similar clusters. As an input to the algorithm, we provide a CSV file containing the positions on the x and y axes of each button, concatenated with the distances between each of the buttons. The algorithm performs clustering and organizes layouts that have been output by the machine learning classifier. It has been observed that both the number of clusters and the number of valid individuals oscillate when the number of iterations varies for each population. This can be explained in part by the fact that the objective function is not able to capture all aspects of a good layout, so this result justifies the use of machine learning to help the designer identify a good design. The stochastic nature of the selection of the individuals can propagate, albeit less likely, individuals with low fitness for subsequent generations, which can then be wrongly classified as valid by the SVM since the technique does not guarantee 100% accuracy in the classification of individuals. One measure that can help us understand the nature of the layout generation process is the ratio between the number of clusters and the number of valid individuals. The largest the ratio, the more efficient is the process is in creating variability inside a group of high quality individuals. As we stated earlier, we are interested in presenting a range of results that are distinct from each other to the end-user. Figure 3 graphically illustrates the calculated ratios in each population created by the model. Analyzing the graphs, we can see that, in most cases, the ratio of clusters by valid individuals is higher in the population of 1000 individuals. This happens in all two configurations containing different amounts of buttons, except in the following iterations: iteration 4000 in layouts with three buttons; iterations 4000, 6000, 7000, and 10,000 in the layouts with nine buttons. As this behavior occurred in a few cases, most likely caused by the randomness of the genetic algorithm when generating and selecting individuals, we realized that a population with 1000 individuals is sufficient to generate sets of valid individuals with greater variability. the model outputs. Five game designers, four male and one female, aged 18-30 years analyzed the four candidate layouts. Those designers have between one and five years of experience with game design. The layouts were presented to them on a smartphone ASUS Zenfone 3 Zoom with a 5.5-inch screen. Their opinion was requested about those layouts, asking them to consider the arrangement and grouping of the buttons in the virtual controller area. In short, the layouts that separated the directional buttons into two groups divided opinions among the designers. The position of the action button in the three-button settings has been reported as too far away, making it difficult to combine with other actions. There were also comments on the user's finger range and button height.

Conclusion and Future Work

In this work, we addressed the problem of assisting a game designer in the process of creating different gamepad layouts using generative design. The main contribution of this work is a generative design pipeline for gamepad design based on evolutionary algorithms and machine learning techniques. The evolution algorithm is responsible for creating a large number of gamepad configurations based on the inputs provided by the user: the number of layouts to be generated, the number of iterations to be executed, quantity, and type of buttons that should be present in the final layout. After execution, the model outputs the exact number of layouts requested by the user. As this is generally a large number, it becomes impractical for the designer to analyze them all. As certainly many of these will be discarded by him because video game controllers must respect some organizational rules, we a Support Vector Machine classifier to divide individuals as 'valid' or 'invalid'. In addition, we propose a way of measuring the variability of the candidate solutions using an extended version of the K-means algorithm, which can automatically determine the number of clusters in the group of individuals. Static database training can be seen as a limitation. Since the data that fed the training base of the SVM classifier have been classified only once, the outputs of the generative design model are, to a certain extent, restricted to those that the classifier knows to be valid or not. In future works we intend to implement a self-feeding of the classifier training base so that it is possible to further increase the variety of results that can be classified as valid. Regarding the analysis of the variability of layouts, the technique used may erroneously group some layouts of controllers in the same cluster, as shown in Figure 5. This occurs because the K-means algorithm does not interpret the semantics present in videogame controllers since it analyzes only the positions of the buttons within the controller's layout space. In this case, the layouts placed in the same cluster are very similar when considering the positioning of the buttons within the gamepad area, but in the context of the player experience they are very different, and that can generate a high cognitive effort on the part of the user when trying to perform actions within a game. Looking to adrress this issue, in future work, we intend to apply techniques such as graph matching approaches [START_REF] Oliveira | An efficient similaritybased approach for comparing xml documents[END_REF] or computational vision-based techniques (e.g image subtraction) [START_REF] Russ | The image processing handbook[END_REF], to improve the way we can measure variability of the solution.

Fig. 1 .

 1 Fig. 1. The proposed generative design model pipeline.

Fig. 2 .

 2 Fig. 2. Samples of layouts with three buttons classified as 'valid' (first row) and invalid (second row).

Fig. 3 .

 3 Fig. 3. Graphical representation of the value of the ratio between the number of clusters and the number of individuals classified as valid in a population of 1000, 2000 and 3000 individuals -layouts containing three (on the left) and nine (on the right) buttons.

Figure 4

 4 Figure 4 shows examples of layouts generated by the model and grouped into certain clusters (smaller figures) and its potential layout proposed by a designer (bigger figure) after analyzing those results. At the end of the tests, we look for other designers' opinions about the 'ideal layouts' constructed after analyzing

Fig. 4 .

 4 Fig. 4. Layouts with different semantic values inserted in the same cluster because of geometric proximity.

Fig. 5 .

 5 Fig. 5. Individuals inserted in the same cluster, however these layouts are not similar because they have buttons with different semantics.

Table 1 .

 1 Basic button classes and actions.

	Classes		Buttons
	Directional buttons up down left right
	Action buttons jump shoot run defend
	Extra buttons	L1	L2 R1 R2
	System buttons start select -	-


Franco Eusébio

Vânia Paula De Almeida Neris

Strategies for Inclusive End-User Co-Creation of Inclusive Storytelling Games

Keywords: End-User Development, Game Development, Game Accessibility, Universal Design, Meta-Design, Human-Centered Computing

As gaming acquires new purposes (for instance, entertainment, education and healthcare), game accessibility becomes increasingly important for multiple domains. For broader inclusion, game accessibility encompass both creation and play. Towards this goal, we have defined a framework to enable more people to create and play digital games In this paper, we present strategies resulting from developing and evaluating Lepi, an inclusive end-user tool for co-creation of inclusive storytelling-based games. Adults with heterogeneous interaction needs, levels of literacy and experience with computers used Lepi to co-create their games over ten creation workshops. Using Lepi and following the framework practices, the participants managed to co-create games accessible for themselves and their peers. From this experience, we have identified some strategies (Creation Commands, Interaction Alternatives for Input, Slots, Creation Alternatives, Assisted and Collaborative Co-Creation, Gentle Slopes, Multimodal Features, Playing Commands and Presenting Game Content) which can contribute towards more inclusive practices of game creation and play of storytelling-based games.

Introduction

Game accessibility aims to enable more people to play digital games. Although the Literature provides techniques, guidelines, and strategies (for instance, in [START_REF] Garcia | Um Motor para Jogos Digitais Universais[END_REF][START_REF] Garcia | A Data-Driven Entity-Component Approach to Develop Universally Accessible Games[END_REF][START_REF] Grammenos | Designing universally accessible games[END_REF][START_REF] Grammenos | Unified Design of Universally Accessible Games[END_REF][START_REF] Yuan | Game Accessibility: A Survey[END_REF]), developers often ignore game accessibility recommendations, for reasons including efforts (for instance, costs and time), impossibility of application and unfamiliarity [START_REF] Aguado-Delgado | Accessibility in video games: A systematic review[END_REF][START_REF] Porter | Understanding and Addressing Real-world Accessibility Issues in Mainstream Video Games[END_REF][START_REF] Westin | Design of a Curriculum Framework for Raising Awareness of Game Accessibility[END_REF]. Developers often makes assumptions of user abilities [START_REF] Pozzi | Individuation and diversity: The need for idiographic HCI[END_REF], which, in practice, do not always hold true. As a result, people whose abilities were not considered during design might become unable to play.

As gaming expands to new applications and domains, accessibility issues may hinder digital and social inclusion. Besides the traditional entertainment value, domain experts can explore digital games as tools to support their activities. Education and healthcare are two examples of serious domains that have been exploring games to enhance learning, training, therapy and rehabilitation practices [START_REF] De Souza | Towards a Semiotic-Based Approach to the Design of Therapeutic Digital Games[END_REF][START_REF] Gee | Video Game Making and Modding[END_REF]. In these domains, experts can use games to aid people with heterogeneous characteristics, including age, socioeconomic status, (dis)abilities, skills, interests and knowledge. With such heterogeneity, assumptions of abilities and skills can limit audiences for games in serious domains. However, the people in these domains often possess the knowledge and skills required for inclusion that the original developers lacked. In education, professors and special educators can remove communication barriers for students. In healthcare, therapists can support use without harming their patients. Moreover, in both cases, students and patients could further provide their skills to enable use.

In this scenario, co-creation could provide a different strategy to promote inclusion. To achieve a scenario on which end-users could co-create inclusion, they would have to act as non-professional software developers to improve the original game. Game modification (popularized as "modding") [START_REF] Gee | Video Game Making and Modding[END_REF] is a popular term to describe End-User Development (EUD) [START_REF] Lieberman | End-User Development: An Emerging Paradigm[END_REF] practices in games. EUD provides methods, techniques, and tools to allow end-users to create, modify, or extend software [START_REF] Lieberman | End-User Development: An Emerging Paradigm[END_REF]. Game creation and modding expand these practices to game making.

Although, traditionally, EUD is concerned with functional software features, it may be employed to improving non-functional features. In particular, EUD could contribute to co-creation of accessibility and usability, which we have been exploring by the means of a framework to promote inclusive co-creation of inclusive games [START_REF] Garcia | An Inclusive End-User Development Framework for Tailorable Games[END_REF][START_REF] Garcia | Able to Create, Able to (Self-)Improve: How an Inclusive Game Framework Fostered Self-Improvement Through Creation and Play in Alcohol and Drugs Rehabilitation[END_REF]. With the framework, we can implement software without assuming particular interactions, enabling communities of end-users to co-create alternatives to enable use and promote inclusion. This way, the community may provide broader accessibility even if the original developers failed to do so. The community, thus, practices "accessibility modding" towards broader inclusion.

Lepi (Figure 1) was the first game creation tool developed for the framework, serving both as a proof of concept and a study of a viability. As a proof of concept, Lepi targeted a single genre (storytelling-based games) with a subset of features considering interaction needs of potential creators and players (mainstream audiences, hearing disabilities and low literacy) 1 . Participants with heterogeneous interaction needs (adults with different levels of literacy, computer skills, and emotional characteristics) used Lepi in a public healthcare service during ten collaborative workshops performed at a period of four months at an alcohol and drugs rehabilitation program2 [START_REF] Garcia | Able to Create, Able to (Self-)Improve: How an Inclusive Game Framework Fostered Self-Improvement Through Creation and Play in Alcohol and Drugs Rehabilitation[END_REF]. To enable every participant to play every game, participants collaborated to provide accessible content to the game according to their abilities and knowledge. As they were not programmers, we explored an iterative and incremental process to adapt Lepi to suit and support the participants' interaction needs for use and co-creation. In this paper, we have generalized adaptations and features that we have explored into strategies that we had followed to enable the creation and workflow, promoting inclusive end-user co-creating of inclusive storytelling games.

2 Related Work

Game Accessibility Strategies

Although enabling end-users to create inclusive game is an incipient practice (for instance, there are solutions to support the development of memory games by blind users under supervision [START_REF] Giannakopoulos | Accessible electronic games for blind children and young people[END_REF] and to form communities of audio game designers [START_REF] Urbanek | Building a Community of Audio Game Designers -Towards an Online Audio Game Editor[END_REF]), existing studies regarding game accessibility seem to be aimed at professionals developers. For instance, Universally Accessible Games (UA-Games) have techniques to support professionals designing [START_REF] Grammenos | Designing universally accessible games[END_REF][START_REF] Grammenos | Unified Design of Universally Accessible Games[END_REF], and implementing them [START_REF] Garcia | Um Motor para Jogos Digitais Universais[END_REF][START_REF] Garcia | A Data-Driven Entity-Component Approach to Develop Universally Accessible Games[END_REF]. Another useful artifact for developers is the game interaction model from [START_REF] Yuan | Game Accessibility: A Survey[END_REF]. The model defines a finite state machine based on how players interact with a game during play. A player cycles among three states ( "receive stimuli", "determine response", and "provide input") to play. Based on the model, highlevel strategies for game accessibility include: enhancing stimuli and replacing stimuli (for the state receive stimuli); reduce stimuli, reduce time constraints and reduce input (for the state determine response); reduce input and replace input (for the state provide input). We have explored these strategies to define the framework's software architecture, as well as to provide interaction alternatives for creation tools and generated games.

Collaboration and Accessibility

Communities are an important element for game modding [START_REF] Burke | Decade of Game Making for Learning: From Tools to Communities[END_REF]. In this paper, in particular, we explore communities as receivers and providers of inclusion. This is similar, for instance, to "communistic interactions" ("from each according to their abilities, to each according to their needs") [START_REF] Liu | Helping Others Makes Me Happy": Social Interaction and Integration of People with Disabilities[END_REF] and to accessibility as interdependence [START_REF] Bennett | Interdependence As a Frame for Assistive Technology Research and Design[END_REF]. For the former, inclusion should aim for groups of people instead of individuals, and on abilities instead of disabilities. For the latter, interdependence can highlight relations and dependencies between people and things.

We argue that, from a combination of both (individuals and groups, independence and interdependence), communities could start providing and receive support to enable game creation and play by more audiences. To achieve this result, communities would need computational support to co-create.

Storytelling in Game Creation and Modding

Storytelling has been previously explored to promote game creation by end-users. For instance, ScriptEase [START_REF] Carbonaro | Interactive story authoring: A viable form of creative expression for the classroom[END_REF] is a tool to convert text into game scripts for Neverwinter Nights. <e-Game Project> [START_REF] De Leeuw | A documental approach to adventure game development[END_REF] (aimed at writers) describes a documental approach employing a Domain Specific Language (DSL) to create graphical adventure games. Writing Environment for Education Video Games [START_REF] Marchiori | A narrative metaphor to facilitate educational game authoring[END_REF] (aimed at educators) provides a methodology for the creation of stories in point-and-click games. As with <e-Game Project>, it also employs a DSL to promote content creation.

However, to the best of our knowledge, intended audiences for existing solutions are literate people. In this paper, we aimed at broader inclusion to support people with low literacy as well other interaction needs (for instance, providing signs for hearing disabilities and audio-description for visual disabilities). We explored for in-game adaptation (created by the end-user acting as non-professional designers/developers) supported by the creation tool (Lepi) itself. This contrasts to current approaches in two ways, which may enable more inclusive creation and result into better quality and experience of use. First, the tool provides features for adaptation (later discussed in Section 4 and Section 5) to tailor generated games to players during use, potentially enabling new audiences to play. Accessibility features are typically are not available in creation tools [START_REF] Aguado-Delgado | Accessibility in video games: A systematic review[END_REF]), forcing players to resort to external tools (for instance, assistive technologies) which were not considered during design, resulting into low quality of interaction and use [START_REF] Aguado-Delgado | Accessibility in video games: A systematic review[END_REF][START_REF] Yuan | Game Accessibility: A Survey[END_REF]). Second, creating the tool with the features them allows for defining different ways to promote game creation, potentially enabling new audiences to create. To support this workflow, we fostered community collaborations.

Research Approach

This paper results from lessons learned from past studies regarding game design and implementation for game accessibility (visual, hearing, motor and cognitive disabilities), and serious games education and healthcare (depression, and alcohol and drugs rehabilitation) [START_REF] De Souza | Towards a Semiotic-Based Approach to the Design of Therapeutic Digital Games[END_REF][START_REF] Garcia | Um Motor para Jogos Digitais Universais[END_REF][START_REF] Garcia | Able to Create, Able to (Self-)Improve: How an Inclusive Game Framework Fostered Self-Improvement Through Creation and Play in Alcohol and Drugs Rehabilitation[END_REF][START_REF] Garcia | A Data-Driven Entity-Component Approach to Develop Universally Accessible Games[END_REF][START_REF] Rodrigues | Personas-Driven Design for Mental Health Therapeutic Applications[END_REF]. We adopted Participatory Design [START_REF] Muller | Participatory practices in the software lifecycle[END_REF] and Organizational Semiotics [START_REF]Encyclopedia of Semiotics[END_REF] as research frameworks for design activities, gathering requirements from participatory workshops performed at public schools, hospitals and healthcare services. For use (design and play), we started the research gathering requirements for designing and implementing serious games for these institutions [START_REF] De Souza | Towards a Semiotic-Based Approach to the Design of Therapeutic Digital Games[END_REF][START_REF] Rodrigues | Personas-Driven Design for Mental Health Therapeutic Applications[END_REF]. For implementation, we had defined approaches to improve game accessibility [START_REF] Garcia | A Data-Driven Entity-Component Approach to Develop Universally Accessible Games[END_REF] based on recommendations from the Literature, which resulted into a game engine [START_REF] Garcia | Um Motor para Jogos Digitais Universais[END_REF]. The game engine was simplified to its core architecture [START_REF] Garcia | An Inclusive End-User Development Framework for Tailorable Games[END_REF], which we briefly outline in Section 4.

Over time, we have progressed towards end-user co-creation of digital games [START_REF] Garcia | An Inclusive End-User Development Framework for Tailorable Games[END_REF][START_REF] Garcia | Able to Create, Able to (Self-)Improve: How an Inclusive Game Framework Fostered Self-Improvement Through Creation and Play in Alcohol and Drugs Rehabilitation[END_REF]. In our considered domains (students, patients, and educational and healthcare professionals), we found storytelling as a suitable introduction to end-user game creation via visual novels 3 . Lepi was developed to assist inclusive end-user co-creation of inclusive storytelling games. It was designed and initially implemented based on our previous experience, and improved in an incremental and iterative process in participatory activities as research participants created games with it.

Design considerations and evaluation are detailed in [START_REF] Garcia | Able to Create, Able to (Self-)Improve: How an Inclusive Game Framework Fostered Self-Improvement Through Creation and Play in Alcohol and Drugs Rehabilitation[END_REF]. As anticipated in Section 1, we have performed co-creation workshops in ten activities over four months. In the workshops, participants co-created their games using Lepi according to their own abilities and preferences. For instance, participants able to write and who had previous experience with computers opted to type their stories. Participants unable to write either recorded their story in audio (then inserted the resulting file into the audio slots of Lepi) or dictated their stories to Collaborators, who either typed the content for them or spelled the characters to the Creators. Likewise, some participants opted to draw or sketch their stories before inserting them into Lepi. Others opted to use Lepi directly (especially on their following creations).

To enable every participant to play every game, Collaborators enhanced the projects with complementary content to fill the other slots in Lepi (performing the proposed "accessibility modding"). This included, for instance, voice content for written stories and text content for voiced stories. Therefore, they co-created inclusion together, as a community, based on the skills and abilities of each member. Similarly to the creation, every participant could play the resulting games based on their interaction needs and/or preferences. Without the collaborative enhancements, this would have been impossible: for instance, people unable to read would not be able to play text-only stories. In the remainder of this paper, we described the strategies that we have explored to support broader inclusion for creation and play.

An Inclusive End-User Development Framework for Tailorable Games

To enable more people to create and play games, we have designed a framework to foster community co-creation of inclusion games [START_REF] Garcia | An Inclusive End-User Development Framework for Tailorable Games[END_REF]. The framework defines (i) a software architecture for game implementation, (ii) a collaborative working model to support co-creation and (iii) game creation approaches to enable endusers to develop their games.

Based on the concept of tailoring [START_REF] Mørch | Three levels of end-user tailoring: Customization, integration, and extension[END_REF] (adaptations of software to suit the practices of a user), games implementing the architecture were named "tailorable games". The central idea of the architecture was allowing the implementation of interaction-abstract games, defining commands and semantics of use instead of pre-defined interactions to enable play. This way, developers can define interaction alternatives to suit different abilities and skills from players iteratively, one audience at a time, via "interaction add-ons". These add-ons can be attached to (or removed from) a game, defining how a player sensory perceives the game (for the "receive stimuli" phase of game interaction model from [START_REF] Yuan | Game Accessibility: A Survey[END_REF]) and controls it (for the "provide input" phase of the game interaction model).

To foster community co-creation, we have defined a collaborative working model around the flexibility of the architecture. The central idea of the model was enabling members of a community to provide accessibility features to a game project. In the same way modding enables a community to improve aesthetics, content and gameplay for games, the collaborative working model promotes "accessibility modding" towards broader inclusion. As with "communistic interactions", members of a community can provide their own abilities and skills to create (or improve) accessibility features (interaction alternatives) to enable new audiences to play. In particular, if these people had support to co-create the game, they could further contribute to improve it. Their own abilities and skills could, potentially, include other new audiences, defining "cycles of inclusion". This way, inclusion could become an iterative and dynamic process, because people who become able to play may further contribute to improve the game.

As end-users are (usually) not programmers, game creation approaches enable and support the co-creation. In special, systems for co-creation can also implement the architecture (for the same benefits) to become tailorable systems and implement tailorable games as their projects. From this, it results inclusive end-user creation tools able to define inclusive games -that is, game making tools as well as games that members of a community can improve with content, accessibility and usability features to promote play and inclusion. Lepi was the first tool that we have defined to explore this idea.

Strategies of the Framework to Foster Inclusion

Decoupled Interaction from Logic. The architecture simplifies the engine from [START_REF] Garcia | Um Motor para Jogos Digitais Universais[END_REF][START_REF] Garcia | A Data-Driven Entity-Component Approach to Develop Universally Accessible Games[END_REF] to a few elements -components, entities and subsystems from Entity-Component-Systems (ECS) events and event handlers from Event-Driven Architectures. These elements, combined, allow for modifying game interaction at use-time (run-time). These elements allow developers to change input and output (IO) features of an entity at run-time [START_REF] Garcia | Um Motor para Jogos Digitais Universais[END_REF][START_REF] Garcia | A Data-Driven Entity-Component Approach to Develop Universally Accessible Games[END_REF], as they can make arbitrary interaction toggle-able. For instance, entities with graphical components are graphically represented into a screen. Once the component is removed, the entity stops being represented (although it still exists). This idea allows swapping ways to control (input to command) and represent (output to convey) an entity, even when the game is already running. For accessibility, particularly useful applications of EDA include providing immediate feedback to players (for instance, with sound, haptic, or graphical effects) and implement game agents able to provide input to the game.

Interaction-Abstract Implementation. To achieve interaction-abstract digital games using the architecture, the implementation should not impose any fixed physical-level IO-related interactions for use. Rather, the architecture proposes that developers define semantics of use (that is, define what the play can do) first, then provide interaction alternatives (that is, define how the player will do it) to allow users to apply the semantic to play. This allows developers to implement interactive systems without assumptions of how a player will use it to, and to interaction alternatives to define physical-level interactions. This way, developers can re-define the "receive stimuli" and "provide input" states from the game interaction model at use-time.

Commands. In the architecture, commands abstract the proposed semantics of use. They are implemented as events. Commands define possible actions to express intents of what a user wants to accomplish when using the system. Commands are usually verbs expressing actions available to modify the current state of the system. For instance, in a storytelling game, commands can include "choose an option", "confirm / cancel a choice", "forward / rewind a piece of dialogue". The implementation processes these commands; as they are events, an interaction-abstract implementation reacts to them whenever they are issued. In turn, the system performs the required processing to change its internal state. This makes it easier to implement interaction alternatives for input, input (re-)mapping [START_REF] Gregory | Game Engine Architecture, Second Edition[END_REF] and provide automation features. For the latter, the system can dispatch a command to provide input on behalf of a player, assisting players who may need help at the "determine response" and "provide input" states of the game interaction model.

Interaction Alternatives for Input. For human-interaction, developers can offer multiple implementations of physical-level interactions to map users' abilities into commands to allow for user input. Thus, instead of single, fixed and predefined mappings, developers can provide users with choice. If a pair consisting of an input device (for instance, controllers, mice, keyboard, microphones, cameras) and mechanism (for instance, button presses, stick movements, spoken words, gestures) can be translated into a command, then the pair can be used to control the game. These pairings can be defined via input (re-)mapping (for instance, "press ENTER" or "say CONFIRM" perform the command "confirm a choice").

Interaction Alternatives for Output. As the design should not impose any physical-level IO-related interactions, the concepts of sign and signifier from Semiotics [START_REF]Encyclopedia of Semiotics[END_REF] are useful for the architecture. These concepts enable separating a message (abstract meanings, conceptions and functions for things in signs) from its representation (denotation using symbols through signifiers). If game output is considered as the means to continuously providing message to translate the current state of the abstract simulation (a sign derived from data) into concrete representation (signifiers representing information) to a player, then providing accessible stimuli becomes the task of matching signs to accessible signifiers. As a result, the final user interface can be a composition of output alternatives to convey the internal state of the game to the player. This allows modifying game presentation to support the "receive stimuli" state of the game interaction model.

Strategies for Inclusive End-User Co-Creation of Inclusive Storytelling Games Defined in Lepi

The strategies from Section 4.1 were applied to the iterative development of Lepi over the workshops to enable our initial audiences (including people with hearing disabilities and low literacy) to create and play storytelling-based games. In this section, we describe how we abstracted from the end-users acting as creators and players.

Strategies to Enable Inclusive Creation of Storytelling Games

Creation Commands and Interaction Alternatives for Input. Lepi, as an editor, provides commands underlying user interfaces and widgets to perform the processing (for instance, "add / edit / remove an entity", "add / edit / remove a piece of dialogue" and "add / edit / remove a new scene"). Creation commands can be bound to available input devices to enable use. The architecture idea of interaction add-ons enable combinations of commands to define macros, which Creators can attach to their software to perform compound commands. Slots. In storytelling games, entities can be characters, places and objects as well as pieces of dialogues and narrations. In Lepi, every game entity can be multimodal. To support the current audiences, entities have text, image, audio and video slots. To abstract the architecture from the Creators, Lepi provides slots to represent different signifiers of a same sign. Creators can attach data and media files into these slots to convey a very same sign in different ways. For instance, entities support text, audio, image and sign language (video) descriptions. Likewise, text, voice audio or sign language video can convey dialogues (Figure 1d). Once a Creator fills a slot providing its media, the content can be reproduced to players to sensory convey the desired information.

Creation Alternatives. Slots featuring multimodal alternatives mean that people with different knowledge, skills, abilities and background can contribute with the development of the project. Thus, people have opportunities to co-create based on their preferences and skills, instead of being unable to contribute based on what they cannot do. For instance, the current slots enable people to collaborate based on writing, speaking, drawing and acting skills. Thus, besides the traditional text composition in storytelling games, Creators can provide content to Lepi by adding voice narrations, drawings, and sign language videos into their slots. Initial prototypes can start with low-level technology (for instance, paper) to explore ideas (such as a sketch of the story in a comic book format). Once the idea is refined, creators can insert their content into Lepi.

Assisted and Collaborative Co-Creation. Although independence is often strove in accessibility [START_REF] Bennett | Interdependence As a Frame for Assistive Technology Research and Design[END_REF], people can assist one another to co-create. Their abilities can complement each other to overcome barriers that each one could not address individually ("cross-ability cooperation" [START_REF] Bennett | Interdependence As a Frame for Assistive Technology Research and Design[END_REF]). In the collaborative work model, Collaborators provide these external assistance whenever Creators cannot satisfy their own abilities -or those from their Players -alone. Collaborators can create and provide accessible media, add support for assistive technologies or even support use (for instance, a therapist can help a patient to perform tasks that he/she would not be able to do on her/his own). Every collaboration contributes to broader inclusion. For creation activities in serious contexts, there will often be people who have skills and knowledge to contribute towards inclusion (for instance, with audio-description or sign language content in inclusive education).

Gentle Slopes. Making syntactic errors hard (preferably impossible) and supporting incremental development are desirable features of EUD tools [START_REF] Repenning | What Makes End-User Development Tick? 13 Design Guidelines[END_REF]. Practices can also scaffold creation. Creation practices should aim for gentle slopes [START_REF] Basawapatna | The Consume -Create Spectrum: Balancing Convenience and Computational Thinking in Stem Learning[END_REF], gently introducing complexity over time. Lepi explores gentle slopes within storytelling activities: Creators can start with small, linear stories and explore more complex features (for instance, branches and decisions for branching storytelling) over time. With slots, this same reasoning applies to media.

Multimodal Features. Inclusive creation requires multimodal features. Although visual programming languages and approaches are commonly explored in EUD tools (because they accessible for sighted users), they are not accessible for everyone. Rather, constructs should be multimodal. For instance, instance, in Lepi, Creators can define story branches and decisions with visual (linking connectors in a graph; Figure 1b) and text constructs (providing the next scene number; Figure 1c). For content creation, slots allow Creators to co-create according to their needs and skills (written, spoken and acted content, for instance).

Strategies to Enable Inclusive Play of Storytelling Games

Playing Commands. As Players' interaction needs can differ from Creator's, the resulting games should be IO flexible as well. The first strategy is similar to "Creation Commands and Interaction Alternatives for Input" in Lepi. Players should be able to use their preferred devices to play the game. In this first prototype, Lepi supports traditional game input devices (keyboard, mice and controllers).

Presenting Game Content. The second strategy results from the Slots. Players can choose what slots they want to build the user interface of their games (text, image, audio and video slots). If a combination of available content is slots fulfill the interaction needs of a Player, she/he can play the game.

As, at this time, the interaction alternatives do not conflict with each other, a resulting game can have any combination of slots. For instance, one Player may interact with a traditional storytelling game, composed of text and graphics. Another play with graphics and sign language videos. Finally, a third Player may enable all features to read, listen and watch the game content.

Concluding Remarks and Current Work

As digital games are becoming increasingly important for different domains and purposes (including entertainment, education and healthcare), it becomes fundamental to enable more people to create and play them. Otherwise, instead of reducing social and digital inclusion, technology would contribute to increase them and to create more barriers.

In this paper, we described some strategies that we have employed towards promoting inclusive co-creation of inclusive storytelling-based games. These strategies resulted from the development of inclusive end-user tool (Lepi) for the creation of inclusive storytelling games. We have briefly described a game framework aimed at game accessibility and its three pillars, with special focus on its software architecture and Lepi, its first co-creation tool. For these pillars, we have described the strategies that we have defined and have been exploring to enable new audiences to co-create and play digital games based on their abilities and skills. The strategies resulted from our experiences in participatory workshops involving participants with heterogeneous interaction needs making games for themselves and for their peers. They focus on the concept of tailoring to adapt software to better suit interaction needs of users -in this case, to promote more inclusive game creation and use practices.

We aim to keep improving the framework. We are currently introducing creation and play features (for instance, text-to-speech and simple voice commands) to Lepi aimed at making it more accessible for visual disabilities. Some of these features should also provide the first resources towards inclusion for motor disabilities. Once we address those, the goal is introducing new mechanics for creation and playing practices, introducing new practices and tools over time.

  (a) Creation interface. (b) Linking scenes (graphics). (c) Linking scenes (text). (d) Slots for text, audio, and sign language video.

Fig. 1 :

 1 Fig. 1: Lepi provides interfaces to suit different interactions needs.

Although some of these features can assist people with vision disabilities to play, we have not yet addressed creation for these audiences.

We complied with research ethics protocols throughout the entire evaluation. Certificado de Apresentação de Apreciação Ética from Plataforma Brasil: CAAE: 89477018.5.0000.5504.

Reasons included: focus on story over programming; closeness to traditional media (for instance, books and films); potential for introducing programming practices over time[START_REF] Garcia | Able to Create, Able to (Self-)Improve: How an Inclusive Game Framework Fostered Self-Improvement Through Creation and Play in Alcohol and Drugs Rehabilitation[END_REF].

This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior -Brasil (CAPES) -Finance Code 001.


Louise Nyssen
email: lnyssen@math.univ-montp2.fr

Test vectors for trilinear forms, given two unramified representations

Introduction

Let F be a finite extension of Q p , with ring of integers O F , and uniformizing parameter ̟, whose residual field has q elements. For G = GL 2 (F ), let (π 1 , V 1 ), (π 2 , V 2 ) and (π 3 , V 3 ) be three irreducible, admissible, infinite dimensional representations of G. Using the theory of Gelfand pairs, Dipendra Prasad proves in [P] that the space of G-invariant linear forms on V 1 ⊗ V 2 ⊗ V 3 has dimension at most one. He gives a precise criterion for this dimension to be one, that we will explain now.

Let D * F be the group of invertible elements of the quaternion division algebra D F over F . When (π i , V i ) is a discrete series representation of G, denote by (π ′ i , V ′ i ) the irreducible representation of D * F associated to (π i , V i ) by the Jacquet-Langlands correspondence. Again, by the theory of Gelfand pairs, the space of D * F -invariant linear forms on V ′ 1 ⊗ V ′ 2 ⊗ V ′ 3 has dimension at most one.

Let σ i be the two dimensional representations of the Weil-Deligne group of F associated to the irreducible representations π i . The triple tensor product σ 1 ⊗ σ 2 ⊗ σ 3 is an eight dimensional symplectic representation of the Weil-Deligne group, and has local root number ε(σ 1 ⊗ σ 2 ⊗ σ 3 ) = ±1. When ε(σ 1 ⊗ σ 2 ⊗ σ 3 ) = -1, one can prove that the representations π i 's are all discrete series representations of G.

Theorem 1. (Prasad,Theorem 1.4 of [P] ) Let (π 1 , V 1 ), (π 2 , V 2 ), (π 3 , V 3 ) be three irreducible, admissible, infinite dimensional representations of G such that the product of their central characters is trivial. If all the representations V i 's are cuspidal, assume that the residue characteristic of F is not 2. Then ε(σ 1 ⊗ σ 2 ⊗ σ 3 ) = 1 if and only if there exists a non-zero G-invariant linear form on

V 1 ⊗ V 2 ⊗ V 3 ε(σ 1 ⊗ σ 2 ⊗ σ 3 ) = -1 if and only if there exists a non-zero D * k invariant linear form on V ′ 1 ⊗ V ′ 2 ⊗ V ′ 3 . Given a non zero G-invariant linear form ℓ on V 1 ⊗ V 2 ⊗ V 3 , or a non-zero D * k -invariant linear form ℓ ′ on V ′ 1 ⊗ V ′ 2 ⊗ V ′ 3 , the goal is to find a vector in V 1 ⊗ V 2 ⊗ V 3 which is not in the kernel of ℓ, or a vector in V ′ 1 ⊗ V ′ 2 ⊗ V ′
3 which is not in the kernel of ℓ ′ . Such a vector is called a test vector. At first sight, it appears to have strong connections with the new vectors v 1 , v 2 and v 3 of the representations π 1 , π 2 and π 3 .

Theorem 2. (Prasad,Theorem 1.3 of [P]) When all the π i 's are unramified principal series representations of G, v 1 ⊗ v 2 ⊗ v 3 is a test vector for ℓ.

Theorem 3. (Gross and Prasad,) When all the π i 's are unramified twists of the special representation of

G : if ε(σ 1 ⊗ σ 2 ⊗ σ 3 ) = 1, then v 1 ⊗ v 2 ⊗ v 3 is a test vector for ℓ, if ε(σ 1 ⊗ σ 2 ⊗ σ 3 ) = -1, let R ′ be the unique maximal order in D F . Then the open compact subgroup R ′ * × R ′ * × R ′ * fixes a unique line in V ′ 1 ⊗ V ′ 2 ⊗ V ′ 3 .
Any vector on this line is a test vector for ℓ ′ .

The proof by Gross and Prasad of the first statement of this theorem actually contains another result: Theorem 4. When two of the π i 's are unramified twists of the special representation of G and the third one belongs to the unramified principal series of G, v 1 ⊗ v 2 ⊗ v 3 is a test vector for ℓ.

But the paper [G-P] gives evidence that v 1 ⊗ v 2 ⊗ v 3 is not always a test vector for ℓ. Let K = GL(O F ) be the maximal compact subgroup of G. If π 1 and π 2 are unramified and if

π 3 has conductor n ≥ 1, ℓ being G-invariant, v 1 and v 2 being K-invariant, one gets a K-invariant linear form V 3 -→ C v -→ ℓ(v 1 ⊗ v 2 ⊗ v) which must be 0 since π 3 is ramified. Then ℓ(v 1 ⊗ v 2 ⊗ v 3 ) = 0.
Now Gross and Prasad make the following suggestion. Let I n be the congruence subgroup

I n = a b c d ∈ K | c ≡ 0 mod ̟ n
and R be a maximal order M 2 (F ) such that

R * ∩ K = I n . If v * 1 is a R * -invariant vector in V 1 , the linear form V 3 -→ C v -→ ℓ(v * 1 ⊗ v 2 ⊗ v) is invariant under the action of R * ∩ K = I n , and one can still hope that v * 1 ⊗ v 2 ⊗ v 3 is a test vector for ℓ.
The purpose of this paper is to prove that v * 1 ⊗ v 2 ⊗ v 3 actually is a test vector for ℓ. This is the object of Theorem 5. The case n = 1, together with Theorems 2, 3 and 4, complete the study of test vectors when the π i 's have ramification 0 or 1.

In the long term, the search for test vectors is motivated by the subconvexity problem for L-functions. Roughly speaking, one wants to bound some L-functions along the critical line ℜ(z) = 1 2 . A recent and successful idea in this direction has been to relate triple products of automorphic forms to special values of L-functions on the critical line. In [B-R 1] and [B-R 2] Joseph Bernstein and Andre Reznikov established a so called subconvexity bound the for the L-function of a triple of representations : each representation is attached to the eigenvalue of a certain operator, and the eigenvalue of one representation varies. Philippe Michel and Akshay Venkatesh considered the case when the level of one representation varies. More details about subconvexity and those related techniques can be found in [V] or [M-V]. Test vectors are key ingredients. Bernstein and Reznikov use an explicit test vector. Venkatesh uses a theoretical one, but explains that the bounds would be better with an explicit one (see paragraph 5 of [V]). Unfortunately, the difficulty of finding them increases with the ramification of the representations involved.

There is an extension of Prasad's result in [H-S], where Harris and Scholl prove that the dimension of the space of G-invariant linear forms on V 1 ⊗V 2 ⊗V 3 is one when π 1 , π 2 and π 3 are principal series representations, either irreducible or reducible with their unique irreducible subspace, infinite dimensional. They apply the global setting of this to the construction of elements in the motivic cohomology of the product of two modular curves constructed by Beilinson.

I would like to thank Philippe Michel for suggesting this problem, Wen-Ching Winnie Li who invited me to spend one semester at PennState University where I wrote the first draft of this paper, and of course Benedict Gross and Dipendra Prasad for the inspiration. I would also like to thank Paul Broussous and Nicolas Templier for many interesting discussions, and Eric Bahuaud for his help with English.

In a previous version of this paper, I obtained Theorem 5 under an unpleasant technical condition. I am extremely grateful to Malden Dimitrov, because, thanks to our discussions on the subject, I found the way to remove the condition. In [D-N], we are working on a more general version of Theorem 5.

Statement of the result 2.1 About induced and contragredient representations

Let (ρ, W ) be a smooth representation of a closed subgroup H of G. Let ∆ H be the modular function on H. The induction of ρ from H to G is a representation π whose space is the space Ind G H ρ of functions f from G to W satisfying the two following conditions :

(1

) ∀h ∈ H, ∀g ∈ G, f (hg) = ∆ H -1 2 (h)ρ(h)f (g), (2) there exists an open compact subgroup K f of G such that ∀k ∈ K f , ∀g ∈ G, f (gk) = f (g)
where G acts by right translation. The resulting function will be denoted π(g), f that is

∀g, g 0 ∈ G, π(g), f (g 0 ) = f (g 0 g).
With the additional condition that f must be compactly supported modulo H, one gets the compact induction denoted by ind G H . When G/H is compact, there is no difference between Ind G H and ind G H . Let B the Borel subgroup of upper triangular matrices in G, and let T be the diagonal torus. The character ∆ T is trivial and we will use

δ = ∆ B -1 with δ a b 0 d = | a d | where | • | is the normalised valuation of F .
The quotient B\G is compact and can be identified with P 1 (F ).

For a smooth representation V of G, V * is the space of linear forms on V . The contragredient representation π is given by the action of G on V , the subspace of smooth vectors in

V * . If H is a subgroup of G, V ⊂ V |H ⊂ V * .
We refer the reader to [B-Z] for more details about induced and contragredient representations.

New vectors and ramification

Let (π, V ) be an irreducible, admissible, infinite dimensional representation of G with central character ω. To the descending chain of compact subgroups of

G K ⊃ I 1 ⊃ • • • ⊃ I n ⊃ I n+1 • • •
one can associate an ascending chain of vector spaces

V 0 = V K and ∀n ≥ 1, V n = v ∈ V | ∀ a b c d ∈ I n , π a b c d v = ω(a)v .
There exists a minimal n such that the vector space V n is not {0}. It is necessarily one dimensional and any generator of V n is called a new vector of (π, V ). The integer n is the conductor of (π, V ). The representation (π, V ) is said to be unramified when n = 0. Else, it is ramified. More information about new vectors can be found in [C].

The main result

Let (π 1 , V 1 ), (π 2 , V 2 ) and (π 3 , V 3 ) be three irreducible, admissible, infinite dimensional representations of G such that the product of their central characters is trivial. Assume that π 1 and π 2 are unramified principal series, and that π 3 has conductor n ≥ 1. According to Theorem 1, since π 1 and π 2 are not discrete series, there exists a non-zero, G-invariant linear form ℓ on V 1 ⊗ V 2 ⊗ V 3 . We are looking for a vector v in V 1 ⊗ V 2 ⊗ V 3 which is not in the kernel of ℓ. In order to follow the suggestion of Gross and Prasad we consider

γ = ̟ 0 0 1 and R n = γ -n M 2 (O F )γ n .
One can easily check that

R * n = γ -n Kγ n and R * n ∩ K = I n . If v 1 , v 2
and v 3 denote the new vectors of π 1 , π 2 and π 3 , the vector

v * 1 = π 1 (γ -n ) • v 1 is invariant under the action of R * n . Hence we can write v * 1 ∈ V 1 R * n , v 2 ∈ V 2 K and v 3 ∈ V 3 R * n ∩K . Theorem 5. Under those conditions, v * 1 ⊗ v 2 ⊗ v 3 is a test vector for ℓ.
The proof will follow the same pattern as Prasad's proof of Theorem 2 in [P], with the necessary changes.

3 Going down Prasad's exact sequence

Central characters

Let ω 1 , ω 2 and ω 3 be the central characters of π 1 , π 2 and π 3 . Notice that the condition ω 1 ω 2 ω 3 = 1 derives from the G-invariance of ℓ. Since π 1 and π 2 are unramified, ω 1 and ω 2 are unramified too, and so is ω 3 because ω 1 ω 2 ω 3 = 1. Let η i , for i ∈ {1, 2, 3} be unramified quasi-characters of F * with η 2 i = ω i and η 1 η 2 η 3 = 1. Then

V 1 ⊗ V 2 ⊗ V 3 ≃ V 1 ⊗ η -1 1 ⊗ V 2 ⊗ η -1 2 ⊗ V 3 ⊗ η -1 3
as a representation of G. Hence it is enough to prove Theorem 5 when the central characters of the representations are trivial.

When n = 1, it is also enough to prove Theorem 5 when V 3 is the special representation Sp of G : take η 3 to be the unramified character such that V 3 = η 3 ⊗ Sp.

Prasad's exact sequence

Let us now explain how Prasad finds ℓ. It is equivalent to search for ℓ or to search for a non-zero element in Hom G V 1 ⊗ V 2 , V 3 . Since the central characters of π 1 and π 2 are trivial, there are unramified characters µ 1 and µ 2 such that for i = 1 and i = 2

π i = Ind G B χ i with χ i a b 0 d = µ i a d . Hence V 1 ⊗ V 2 = Res G Ind G×G B×B χ 1 × χ 2
where G is diagonally embedded in G×G for the restriction. The action of G on B×B\G×G = P 1 (F )× P 1 (F ) has precisely two orbits. The first is {(u, v) ∈ P 1 (F )× P 1 (F ) | u = v} which is open and can be identified with T \G. The second orbit is the diagonal embedding of P 1 (F ) in P 1 (F ) × P 1 (F ), which is closed and can be identified with B\G. Then, we have a short exact sequence of G-modules

0 → ind G T χ 1 χ 2 ext --→ V 1 ⊗ V 2 res --→ Ind G B χ 1 χ 2 δ 1 2 → 0. (1)
The surjection res is the restriction of functions from G × G to the diagonal part of B\G × B\G, that is

∆ B\G = (g, bg) | b ∈ B, g ∈ G . The injection ext takes a function f ∈ ind G T χ 1 χ 2
to a function F ∈ Ind G×G B×B χ 1 × χ 2 vanishing on ∆ B\G , and is given by the relation

F g, 0 1 1 0 g = f (g),
on the other orbit. Applying the functor Hom G • , V 3 , one gets a long exact sequence

0 → Hom G Ind G B χ 1 χ 2 δ 1 2 , V 3 → Hom G V 1 ⊗ V 2 , V 3 → Hom G ind G T χ 1 χ 2 , V 3 ↓ • • • ← Ext 1 G Ind G B χ 1 χ 2 δ 1 2 , V 3 (2)

The simple case

The situation is easier when n = 1 and

µ 1 µ 2 | • | 1 2 = | • | -1 2 , as π 3 is special and there is a natural surjection Ind G B χ 1 χ 2 δ 1 2 -→ V 3
whose kernel is the one dimensional subspace of constant functions. Thanks to the exact sequence (1) one gets a surjection Ψ

V 1 ⊗ V 2 res --→ Ind G B χ 1 χ 2 δ 1 2 Ψ ց ւ V 3 which corresponds to ℓ V 1 ⊗ V 2 ⊗ V 3 -→ C v ⊗ v ′ ⊗ v ′′ -→ Ψ(v ⊗ v ′ ).v ′′
The surjection Ψ vanishes on v * 1 ⊗ v 2 if and only if res(v * 1 ⊗ v 2 ) has constant value on P 1 (F ) ≃ B\G. An easy computation proves that res(v * 1 ⊗ v 2 ) is not constant : the new vectors v 1 and v 2 are functions from G to C such that

∀i ∈ {1, 2}, ∀b ∈ B, ∀k ∈ K, v i (bk) = χ i (b) • δ(b) 1 2 and ∀g ∈ G, v * 1 (g) = v 1 (gγ -1 ). Then (v * 1 ⊗ v 2 ) 1 0 0 1 = v 1 γ -1 v 2 1 0 0 1 = v 1 ̟ -1 0 0 1 = µ 1 (̟) -1 |̟| -1 2 = √ q µ 1 (̟) and (v * 1 ⊗ v 2 ) 0 1 1 0 = v 1 0 1 1 0 ̟ -1 0 0 1 = v 1 1 0 0 ̟ -1 0 1 1 0 = µ 1 (̟) √ q .
The representation π 1 is principal so

√ q µ 1 (̟) = µ 1 (̟) √ q and (v * 1 ⊗ v 2 ) 1 0 0 1 = (v * 1 ⊗ v 2 ) 0 1 1 0 . Hence Ψ does not vanish on v * 1 ⊗ v 2 . Now, v * 1 being R * 1 -invariant and v 2 being K-invariant, Ψ(v * 1 ⊗ v 2 ) is a non-zero I 1 -invariant element of V 3
, that is, a new vector for π 3 . Consequently it does not vanish on v 3 :

ℓ(v * 1 ⊗ v 2 ⊗ v 3 ) = Ψ(v * 1 ⊗ v 2 ).v 3 = 0 and v * 1 ⊗ v 2 ⊗ v 3 is a test vector for ℓ.

The other case

If n ≥ 2 or µ 1 µ 2 | • | 1 2 = | • | -1 2 then Hom G Ind G B χ 1 χ 2 δ
1 2 , V 3 = 0 and by Corollary 5.9 of [P] Ext

1 G Ind G B χ 1 χ 2 δ 1 2 , V 3 = 0.
Through the long exact sequence (2) we get an isomorphism

Hom G V 1 ⊗ V 2 , V 3 ≃ Hom G ind G T χ 1 χ 2 , V 3 ,
and by Frobenius reciprocity

Hom G ind G T χ 1 χ 2 , V 3 ≃ Hom T χ 1 χ 2 , V 3|T ,
where V 3|T is the space of the contragredient representation of π 3|T . By Lemmas 8 and 9 of [W], the latter space is one dimensional. Thus, we have a chain of isomorphic one dimensional vector spaces

HomG " V1 ⊗ V2 ⊗ V3, C " → HomG " V1 ⊗ V2, f V3 " → HomG " ind G T " χ 1 χ 2 " , f V3 " → HomT "" χ 1 χ 2 " , g V3 |T " ℓ → Ψ → Φ → ϕ
with generators ℓ, Ψ, Φ and ϕ corresponding via the isomorphisms. Notice that ϕ is a linear form on V 3 such that

∀t ∈ T, ∀v ∈ V 3 , ϕ π 3 (t)v = χ 2 (t) χ 1 (t) ϕ(v) (3)
which is identified to the following element of Hom

T χ 1 χ 2 , V 3|T C -→ V 3|T z -→ zϕ Lemma 1. ϕ(v 3 ) = 0.
Proof : this is Proposition 2.6 of [G-P] with the following translation :

-the local field F is the same, -the quadratic extension K/F of Gross and Prasad is F × F and their group K * is our torus T , -the infinite dimensional representation V 1 of Gross and Prasad is our π 3 , -the one dimensional, unramified representation V 2 of Gross and Prasad is χ 1 χ 2 . Then the representation that Gross and Prasad call V is χ 1 χ 2 ⊗ π 3 and their condition (1.3) is exactly our condition (3). In order to apply Gross and Prasad's Proposition, we need to check the equality ε(σ ⊗ σ 3 ) = α K/F (-1)ω(-1).

Basically, it is true because K is not a field. Let us give some details.

-In [G-P], ω is the central character of the representation V 1 which is trivial for us.

-The character α K/F is the quadratic character of F * associated to the extension K/F by local class-field theory. Here, it is trivial because K is F × F .

-To compute ε(σ ⊗ σ 3 ) we will use the first pages of [T].

∀ x 0 0 z ∈ T χ 1 χ 2 x 0 0 z = µ 1 µ 2 (x) µ 2 µ 1 (z) ⇒ ε(σ ⊗ σ 3 ) = ε( µ 1 µ 2 ⊗ σ 3 )ε( µ 2 µ 1 ⊗ σ 3 )
Since the determinant of σ 3 is the central character of π 3 which is trivial, σ 3 is isomorphic to its own contragredient and the contragredient representation of

µ 1 µ 2 ⊗ σ 3 is µ 2 µ 1 ⊗ σ 3 . Formula (1.1.6) of [T] leads to ε(σ ⊗ σ 3 ) = det σ 3 (-1) = 1 = α K/F (-1)ω(-1).
According to [G-P], the restriction of χ 1 χ 2 ⊗ π 3 to the group

M = x 0 0 z | x, y ∈ O * F × I n
fixes a unique line in V 3 : it is the line generated by the new vector v 3 . Still according to Gross and Prasad, a non-zero linear form on V 3 which satisfies (3) cannot vanish on v 3 .

We will deduce from lemma 1 that ℓ

(v * 1 ⊗ v 2 ⊗ v 3 ) = 0.
4 Going up Prasad's exact sequence

4.1 From ϕ(v 3 ) to f Let f be the element of ind G T χ 1
χ 2 which is the characteristic function of the orbit of the unit in the decomposition of T \G under the action of I n . This means :

f (g) = χ 1 (t) χ 2 (t) if g = tk with t ∈ T and k ∈ I n 0 else (4)
Then, the function

G -→ C g -→ f (g) ϕ π 3 (g)v 3
is invariant by the action of T by left translation and we can do the following computation :

Φ(f ) (v 3 ) = T \G f (g) ϕ π 3 (g)v 3 dg = (T ∩K)\In ϕ π 3 (k)v 3 dk = λ • ϕ(v 3 ).
where λ is a non-zero constant. Thanks to Lemma 1 we know that ϕ(v 3 ) = 0, so Φ(f ) (v 3 ) = 0.

(5)

4.2 From f to F Now, we are going to compute

F = ext(f ) in V 1 ⊗ V 2 .
Let a and b be the numbers

a = µ 1 (̟) √ q b = µ 2 (̟) √ q
They verify (a 2 -1)(b 2 -1) = 0 because π 1 and π 2 are principal series representations. For the sake of simplicity, we shall use the following notation : for any g in G

gv * 1 = π 1 (g), v * 1 and gv 2 = π 2 (g), v 2
Lemma 2. The function F is given by the formula

F = A • v ′ 1 ⊗ v ′ 2 with A = a n (a 2 -1)(b 2 -1) , v ′ 1 = a • γ -(n-1) v 1 -γ -n v 1 and v ′ 2 = b • γ -1 v 2 -v 2 .
Proof : the function f is described by formula (4), and ext(f ) is described by the short exact sequence (1) using the orbits of the action of G on B × B\G × G. The function F must vanish on the closed orbit

∆ B\G = (g, bg) | b ∈ B, g ∈ G .
The open orbit can be identified with T \G via the bijection

     T \G -→ B\G × B\G \ ∆ B\G T g -→ Bg, B 0
1 1 0 g through which, the orbit of the unit in T \G under the action of I n corresponds to

Bk, B 0 1 1 0 k | k ∈ I n . Now, pick any (k, k ′ ) ∈ K × K. If k ′ ∈ Bk, then k ∈ I n ⇐⇒ k ′ ∈ I n and k ∈ I 1 ⇐⇒ k ′ ∈ I 1 . When k ′ ∈ Bk, write k = x y z t and k ′ = x ′ y ′ z ′ t ′ . There exists (b 1 , b 2 ) ∈ B × B such that k = b 1 k 0 and k ′ = b 2 0 1 1 0 k 0 with k 0 = z ′ t ′ z t ∈ M 2 (O F ).
Then (k, k ′ ) is in the orbit of the unit in T \G under the action of I n if and only if k 0 is in T I n . Because k and k ′ are in K, one can see that

k 0 ∈ T I n ⇐⇒ k 0 ∈ I n and k 0 ∈ I n ⇐⇒ z ≡ 0 mod ̟ n and z ′ t ∈ O * F ⇐⇒ z ≡ 0 mod ̟ n and z ′ ∈ O * F ⇐⇒ k ∈ I n and k ′ / ∈ I 1 .
It follows that (k, k ′ ) corresponds to an element of the orbit of the unit in the decomposition of T \G under the action of I n if and only if k ∈ I n and k ′ / ∈ I 1 . Then, it will be enough to check that

F (k, k ′ ) = 1 if k ∈ I n and k ′ ∈ I 1 0 else (6)
This is mere calculation. With a = µ 1 (̟) √ q , we need Lemma 3. Let i be any element of N. The values of the function γ -i v 1 on K are given by the formula

∀k = x y z t ∈ K, γ -i v 1 (k) =            a i if val( z t ) ≤ 0 a i-2 val( z t ) if 1 ≤ val( z t ) ≤ i -1 a -i if i ≤ val( z t ) (7)
Proof : first, recall that the new vector v 1 is a function from G to C such that

∀b ∈ B, ∀k ∈ K, v 1 (bk) = χ 1 (b) • δ(b) 1 2 . Then, for k = x y z t in K, either z or t is in O * F . The other one is in O F . Write kγ -i = x ̟ i y z ̟ i t .
If val z t ≤ 0, then val z = 0, val( ̟ i t z ) ≥ 0 and

kγ -i = xt-yz z x ̟ i 0 z ̟ i 0 -1 1 ̟ i t z with 0 -1 1 ̟ i t z ∈ K and (χ 1 • δ 1 2 ) xt-yz z x ̟ i 0 z ̟ i = µ 1 (̟) √ q i-2val z = a i-2val z = a i .
If 1 ≤ val z t ≤ i -1, then val t = 0, val( ̟ i t z ) ≥ 0 and the computation is quite the same, except that

(χ 1 • δ 1 2 ) xt-yz z x ̟ i 0 z ̟ i = µ 1 (̟) √ q i-2val z = a i-2val z t .
If i ≤ val z t , then val t = 0, val( ̟ i t z ) ≤ 0 and

kγ -i = xt-yz t ̟ i y 0 t 1 0 z t ̟ i 1 with 1 0 z t ̟ i 1 ∈ K and (χ 1 • δ 1 2 )( xt-yz t ̟ i y 0 t ) = µ 1 (̟) √ q -i-2val t = a -i .
NB : the case t = 0 is included in val z t ≤ 0.

We can now finish the proof of Lemma 2. The same computation, with v 2 and b instead of v 1 and a, gives the values of γ -i v 2 for any i ∈ N. It is then easy to compute F (k, k ′ ) as given in Lemma 2, for k and k ′ in K, and to check formula (6).

Some test-vectors

On the one hand, from the expression of F given by Lemma 2 we deduce

Ψ(F ) (v 3 ) = A • ℓ(v ′ 1 ⊗ v ′ 2 ⊗ v 3 )
On the other hand, from the relation F = ext(f ) we deduce Ψ(F ) (v 3 ) = Φ(f ) (v 3 ) and from equation 5 in Section 4.1 we get

Ψ(F ) (v 3 ) = 0 Then ℓ(v ′ 1 ⊗ v ′ 2 ⊗ v 3 ) = 0 and v ′ 1 ⊗ v ′
2 ⊗ v 3 is a test vector for ℓ. We are going to simplify it. We can deduce from lemma 2 that

F = A • ab • γ -(n-1) v 1 ⊗ γ -1 v 2 -a • γ -(n-1) v 1 ⊗ v 2 -b • γ -n v 1 ⊗ γ -1 v 2 + γ -n v 1 ⊗ v 2 .
If n ≥ 2, we write

Ψ(F ) (v 3 ) = A • ab • γ -1 ψ n-2 , v 3 -a • ψ n-1 , v 3 -b • γ -1 ψ n-1 , v 3 + ψ n , v 3
where, for m in {n -1, n -2, n}

ψ m V 3 -→ C v -→ ℓ(γ -m v 1 ⊗ v 2 ⊗ v)
Since ℓ is G invariant, ψ m is an element of V 3 which is invariant by the action of

R m ∩ K = I m
But π 3 has conductor n so ψ n-2 = ψ n-1 = 0 and Ψ(F

) (v 3 ) = A • ψ 3 (v 3 ) = A • ℓ(γ -n v 1 ⊗ v 2 ⊗ v 3 ) then ℓ(γ -n v 1 ⊗ v 2 ⊗ v 3 ) = 0
and γ -n v 1 ⊗ v 2 ⊗ v 3 is a test vector for ℓ.

If n = 1, only the two terms in the middle vanish and we get

Ψ(F ) (v 3 ) = A • ab • ℓ(v 1 ⊗ γ -1 v 2 ⊗ v 3 ) + ℓ(γ -1 v 1 ⊗ v 2 ⊗ v 3 )
Now, take g = 0 1 ̟ 0 .

On the one hand gγ -1 = 0 1 1 0 .

and this matrix is in K so gγ -1 v 1 = v 1

On the other hand 0 1 ̟ 0 = ̟ 0 0 ̟ ̟ -1 0 0 1 0 1 1 0 .

The first matrix belongs to the center of G, the second one is precisely γ -1 and the third one is in K, so gv 2 = γ -1 v 2 .

Then

Ψ(F ) (v 3 ) =A • ab • ℓ(gγ -1 v 1 ⊗ gv 2 ⊗ v 3 ) + ℓ(γ -1 v 1 ⊗ v 2 ⊗ v 3 ) =A • ab • ℓ(γ -1 v 1 ⊗ v 2 ⊗ g -1 v 3 ) + ℓ(γ -1 v 1 ⊗ v 2 ⊗ v 3 ) =A • ℓ γ -1 v 1 ⊗ v 2 ⊗ v ′ 3 with v ′ 3 = ab • (g -1 v 3 ) + v 3 The linear form ψ 1 V 3 -→ C v -→ ℓ(γ -1 v 1 ⊗ v 2 ⊗ v)

is a non zero element of V 3 I 1 , so it is a new vector in V 3 . It is known from [B] that a new vector in V 3 does not vanish on on V I 1 3 that is, it does not vanish on a new vector of V 3 :

This concludes the proof of Theorem 5.

NB : it is easy to deduce from Theorem 5 that v 1 ⊗ γ -n v 2 ⊗ v 3 also is a test vector for ℓ. Take

So the linear form

is not zero. Being I n -invariant, it is a new vector in V 3 , which does not vanish on v 3 :


Laurent Goffart

Clara Bourrelly

Julie Quinet

Cerebellar control of visually-guided eye movements by the bilateral mass of activity in the caudal fastigial nuclei

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Laurent Goffart, Clara Bourrelly and Julie Quinet Cerebellar control of visually-guided eye movements by the bilateral mass of activity in the caudal fastigial nuclei Institut de Neurosciences de la Timone, CNRS -Aix-Marseille Université, Marseille, France

The caudal part of both fastigial nuclei (cFN) plays a crucial role in the ability to foveate a static or moving visual target. In the monkey, an unilateral inactivation of cFN by local injection of muscimol impairs the horizontal component of saccadic eye movements. An ipsilesional gaze offset is also observed while a target is being fixated or pursued. In the head-unrestrained monkey (and cat), the head exhibits an ipsilesional deviation relative to the trunk (cervical dystonia).

From the demonstration that larger current enhances the size and velocity of saccades evoked by electrical cFN microstimulation (Fig. 3), we propose that the variability of saccade endpoints reflects the variable size of the active population of cFN neurons contributing to the movement generation. Indeed, if during the pharmacological experiments, the perturbation is not exactly centered in cFN, the number of inactivated neurons should increase as the muscimol diffuses, resulting in a dysmetria that increases with time. from Goffart, Chen & Sparks J Neurophysiol (2004)

Spatial distribution of dwell times before (A) and after (B) inactivation of the right FOR.

Each area encloses the 5, 68, and 98% (from darkest to lightest) of the total fixation time during an experiment. After muscimol injection in the right FOR, the primary saccade (either hypometric or hypermetric) brings the eyes to the ipsilesional side. A contralesional saccade is prompted for improving target foveation, but its amplitude is smaller than required to accurately place the target image on the same region of the fovea as before the injection. Ipsilesional saccades can also be observed even before gaze reaches the target. Consequently, gaze dwells longer in the ipsilesional side.

from Guerrasio, Quinet, Büttner & Goffart J Neurophysiol (2010) Fig. 1 Table 1 from Goffart, Chen & Sparks Journal of Neurophysiology (2004) In a computational study, Eggert, Robinson & Straube (2016) reported more variability in saccade endpoints after cFN inactivation than before., and interpreted this effect as altered putative "noises" in brain activity: planning noise and/or signal dependent motor noise. Unfortunately, they did not consider the fact that after cFN inactivation, the monkeys have difficulties to foveate a visual target. The variability in saccade endpoints can indeed result from the fact that saccades did not start from the same starting positions (Table 1 and Fig. 1). Moreover, when we look at the literature, we do not find any particular increase in the variability of amplitudes. However, we find differences between experiments: the amount of dysmetria varies between the experiments (Fig. 2).

BILATERAL HYPOTHESIS

For every saccade (horizontal, oblique or vertical), the left and right cFN regulate the balance of activity between the excitatory and inhibitory input (from EBNs and IBNs) to the motor and internuclear neurons in the abducens nucleus.

EXAMPLE OF EXPERIMENT WITH TIME-DEPENDENT HYPERMETRIA

For some injection, the magnitude of hypermetria increased with the trial number. EXPLANATION : The drug diffuses with the trial number. As the drug diffuses, the number of inactivated neurons increases (= the number of active neurons diminishes). The reduced number of active neurons removes the drive that cFN neurons exert upon neurons (IBNs) inhibiting the agonist motoneurons. This disinhibiting effect increases the peak velocity and/or does not prevent the excitatory EBN input from continuing to drive the motor neurons, leading to hypermetric saccades.

For some injection, the size of hypometria increased with the trial number. EXPLANATION : The drug diffuses with the trial number. As the drug diffuses, the number of inactivated neurons increases (= the number of active neurons diminishes). The reduced number of active neurons diminishes the peak velocity without compensatory increase in saccade duration, leading to hypometric saccades.

EXAMPLE OF EXPERIMENT WITH TIME-DEPENDENT HYPOMETRIA

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There is no need to appeal to various kinds of noise in the brain to explain the variability of saccades after cFN inactivation. The number of active neurons contributes to the dynamics of saccades The different evolutions of ipsilesional and contralesional dysmetria suggest that these two deficits are unrelated Between the topographical and temporal "codes", instead of appealing to a negative feedback loop that reduces a mismatch between internal signals encoding kinematic parameters (desired versus current displacement as estimated from the integration of velocity commandes), we propose that a saccade is the outcome of a process that restores symmetry of activities between commands exerting opposite movement tendencies. The orienting movement is saccadic for no teleological purpose; it is saccadic merely because the flow of activity from the retinal to the motor neurons involves neurons whose pattern of firing is bursting and also because of fast-twitch properties of extraocular muscles.

Spatio-temporal transformation

Fig. 3 from

 3 Fig. 3

  Fig. 2

  The endpoints of contralesional saccades (positive values) and ipsilesional saccades (negative values) do not change across trials. muscimol injection in the left caudal fastigial nucleus, the horizontal component is hypermetric for ipsilateral (leftward) saccades and hypometric for contralateral (rightward) saccades. Vertical saccades also exhibit a leftward deflection in their trajectory.Endpoints of saccades before ( ) and after muscimol injection in the left (same injection as in the previous panel) or in the right cFN ( ) of the same monkey. The next panel to the right show that the hypometria of contralesional saccades increased during the experiment (top).

  TOPOGRAPHICAL "code" (e.g. deep Superior Colliculus) TEMPORAL "code" (e.g. motoneurons) " (e.g. deep Superior Colliculus) TEMPORAL "code" (e.g. motoneurons)