Eduardo Mascarenhas

Dario Gerace

Marcelo França Santos

Alexia Auffèves

Cooperativity of a few quantum emitters in a single-mode cavity

Keywords:

We theoretically investigate the emission properties of a single-mode cavity coupled to a mesoscopic number of incoherently pumped quantum emitters. We propose an operational measure for the medium cooperativity, valid both in the bad and in the good cavity regimes. We show that the opposite regimes of subradiance and superradiance correspond to negative and positive cooperativity, respectively. The lasing regime is shown to be characterized by nonnegative cooperativity. In the bad cavity regime we show that the cooperativity defines the transitions from subradiance to superradiance. In the good cavity regime it helps to define the lasing threshold, also providing distinguishable signatures indicating the lasing regime. Increasing the quality of the cavity mode induces a crossover from the solely superradiant to the lasing regime. Furthermore, we verify that lasing is manifested in a wide range of positive cooperative behavior, showing that stimulated emission and superradiance can coexist. The robustness of the cooperativity is studied with respect to experimental imperfections, such as inhomogeneous broadening and pure dephasing.

I. INTRODUCTION

The optical properties of N emitters interacting with the same electromagnetic environment are drastically different from those of N independent emitters, each interacting with its own reservoir. Signatures of cooperative behavior in the spontaneous emission of an atomic ensemble were first discussed in the context of the celebrated superradiance decay [1], where constructive interference of the atomic dipoles leads to enhanced relaxation of the atomic population, and to the emission of an intense and delayed pulse of light [2].

More recently, the steady state properties of the light field emitted by such a medium have been theoretically investigated, assuming continuous incoherent pumping, both for what concerns its spectral [3,4] and statistical characteristics [5][6][7]. Subradiant and superradiant regimes have been defined and identified, respectively corresponding to the emission of less/more light than in the independent emitters case. In these works, a common electromagnetic environment for the quantum emitters is provided by a leaky cavity, which acts as a collective decay channel. Naturally, increasing the quality factor of the resonator can eventually induce lasing, such that both phenomena could be observed in the same system at different pumping rates, in principle.

In the weak coupling regime, the concept of cooperativity has been implicitly used in the literature by adiabatically eliminating the cavity mode in the theoretical description, as described, e.g., in Ref. 6. However, even in this particular case there was no precise definition to measure such a quantity. In this paper, we propose an operational measure to quantify the cooperativity of a few quantum emitters coupled to the same single-mode resonator. We show that it works well irrespective of the operating regime of the system (weak or strong coupling, good or bad cavity, etc.). In that sense, it is also universal. We also address the spectral and the statistical properties of the emitted radiation supporting our analysis with the cooperativity measure. Specifically, we analyze an ensemble of two-level emitters and study their cooperativity with respect to experimentally addressable parameters. We show that the cooperativity assumes negative values at subradiance and positive values at superradiance, which clearly indicates the change of regime. As the cavity quality factor is increased, we observe a crossover to a lasing regime, where the cooperativity shows distinctive signatures below and above threshold that allow to identify the nonlasinglasing crossover. In the good cavity regime, our measure can reasonably be used to define the lasing threshold (usually not well defined for lasers of few emitters) as the pump rate increases, and the cooperativity changes from negative to non-negative values. Furthermore we show that lasing is manifested in a wide range of positive cooperativity, showing that lasing and superradiance are distinct phenomena that can coexist: when each emitter is strongly coupled to the resonator mode, a cooperative lasing regime manifests by a delayed quenching of the laser at strong incoherent pumping. Finally, we analyze the robustness of cooperativity to experimental imperfections, such as inhomogeneous broadening and dephasing. From an experimental point of view, systems that might realize the present model include small assemblies of artificial atoms, such as semiconductor quantum dots in microcavities [8,9], superconducting qubits coupled to microstrip line resonators [10], or defects in solid state cavities [11].

The paper is organized as follows. In Sec. II we define the hamiltonian and master equation we have used, and discuss the numerical techniques used to solve the problem. In Sec. III we introduce a novel definition of co-operative fraction, which is able to discriminate between the various regimes of the model. In Sec. IV we investigate the signatures of the superradiance-lasing crossover as a function of the incoherent pumping rate, while its robustness to external parameters is discussed in Sec. V, with an emphasis on the role of the cooperativity as a valuable measurable quantity.

II. SYSTEM, MODEL AND METHODS

The system under study is a single-mode electromagnetic cavity coupled to N two level emitters, as schematically pictured in Fig. 1a. The emitters are incoherently pumped and may be dephased or detuned from the cavity frequency. The model describing this system is the well known Tavis-Cummings Hamiltonian [12], which is written in a frame rotating at the cavity frequency as

H = N i [δ i σ † i σ i + g(σ † i a + σ i a † )] , (1)
with σ i being the lowering operator for the ith emitter, a the annihilation operator for the cavity mode, δ i the detuning of the ith atom from the cavity resonance, and g the light-matter coupling constant depending on the two-level system oscillator strength [14]. The incoherent processes in the system, namely cavity losses, incoherent pumping of the emitters, and their pure dephasing are described within a Markovian approximation and Lindblad dynamics, with the Liouville-von Neumann equation of motion being written as (see, e.g., [START_REF] Carmichael | An open systems approach to quantum optics[END_REF] for a rigorous derivation)

ρ = L(ρ) = -i[H, ρ] + kD a (ρ) + i [P D σ † i (ρ) + γD zi (ρ)] ,
(2) with k being the rate of photon leakage from the cavity, P the incoherent pumping rate, γ the pure dephasing rate, and z i = σ † i σ i . The Lindblad expression for an arbitrary operator, x, is given by

D x (ρ) = - 1 2 [x † xρ + ρx † x] + xρx † . (3)
In the following we focus on the stationary properties of the system and numerically compute the steady state values of the cavity population and atomic inversion, respectively. In addition, we calculate the second-order coherence function at zero time delay of the cavity field, defined as [START_REF] Carmichael | An open systems approach to quantum optics[END_REF]

g 2 (0) = a † a † aa a † a 2 , (4)
and the cavity emission spectrum [START_REF] Carmichael | An open systems approach to quantum optics[END_REF] S(ω) = lim which is the Fourier transform of the first-order correlation function. The latter can be calculated for the stationary state as a † (t)a(0) = tr a † e Lt aρ s , with ρ s being the steady state density matrix of the system. The determination of the asymptotic state can be numerically done by standard sparse matrix diagonalization of the total Lindblad operator, such that Lρ s = 0ρ s . In this work, we used a shift-and-invert Arnoldi method, which is coded in the ARPACK library that is builtin the MATLAB environment [START_REF] Saad | Numerical methods for large eigenvalue problems[END_REF]. The time evolution needed to compute the correlation function is done with the Arnoldi method, which is the optimization of the matrix exponential in the Krylov subspace [START_REF] Knap | [END_REF]. A brief description of the method is as follows. One needs to compute propagations of the form ρ(t + ∆t) = e L∆t ρ(t), with the major difficulty being the calculation of the matrix exponential. This is done in the Krylov subspace, which is the subspace generated by the iterative application of the Lindblad matrix K m [L, ρ(t)] = span ρ(t), Lρ(t), L 2 ρ(t), ..., L m-1 ρ(t) .

τ →∞ a † (t + τ )a(τ ) e iωt dt , (5)
(6) This basis for the Krylov space can be ortho-normalized. In fact, the subspace can be constructed in orthonormal form by the modified Gram-Schmidt method. We call V the matrix whose columns are the orthonormal basisvectors of the Krylov space. For the calculations shown in the present work, we have always kept 20 Krylov vectors. Thus, the huge numerical problem can be reduced to a small space by projecting the Lindblad into an upper Hessenberg form, V † LV = H. The eigenvalues of the Hessemberg matrix are Ritz approximate eigenvalues of L, such that H = U DU -1 . Finally, we have the solution (explicitly written in the most effective order of multiplication)

ρ(t + ∆t) = V U e D∆t U -1 V † ρ(t) . (7)

III. A MEASURE OF COOPERATIVITY

To quantify the cooperativity of the ensemble of emitters coupled to the same cavity mode, we compare the two situations that are schematically pictured in Fig. 1. We pump the N quantum emitters at the same rate, P . The emitters are either coupled to the same cavity mode ((a)), or each of them is coupled to its own resonator (b). The output in the photonic channel, which is proportional to the emitted radiation in each case, is compared for the two situations above. The first situation gives rise to an output field (in units of k), that is n(N, {δ i }, γ) = a † a . In the second case, we measure the sum of the outputs from each cavity, where each one contains a single emitter, which is written as

n(1, δ i , γ) = a † i a i .
For a given set of initial conditions, such as pump/dissipation rates, atom-cavity couplings, etc., the system behavior is said to be "cooperative" when the two measurements differ, the difference between them giving direct access to the field that is created or annihilated by cooperativity. Then, a cooperativity parameter, or cooperative fraction, can be defined as

C f = n(N, {δ i }, γ) -i n(1, δ i , γ) n(N, {δ i }, γ) . ( 8
)
By construction, this parameter is positive when cooperativity is constructive, while it necessarily assumes negative values for destructive cooperativity. It is worth stressing here that the absolute value of C f as defined in the last equation is not bounded, i.e. it could be arbitrarily large if i n(1, δ i , γ) is arbitrarily large. However, this will never be the case in situations one is usually concerned with, in particular the ones treated in the present work. The cooperative fraction tends to the limiting value of 1 for maximum constructive cooperativity, and it is ultimately bounded at negative values by the sum of the single emitters output for maximum destructive cooperativity. As a first step, we apply our definition of cooperativity to the bad cavity regime of the Tavis-Cummings model, where subradiant and superradiant regimes have already been theoretically characterized [6,7]. We set the number of emitters to 5 and the coupling g = 0.1k, under which conditions the adiabatic elimination of the cavity mode is appropriate [18]: in this regime, a is proportional to the global atomic mode, defined by the collective dipole operator J = i σ i . Cooperativity is usually evidenced by comparing the total atomic population inversion, Z = i σ † i σ i , to the total atomic dipole n J = J † J [18]: indeed, it directly compares the field emitted by the atomic ensemble to the field that would be emitted by each two-level system in independent reservoirs. Figure 2a shows the evolution of these quantities as a function of pumping rate. As already discussed in Refs. [6,7], a subradiant behavior is manifested at low pumping P < Γ, where Γ = 4g 2 /k is the relaxation rate of a single emitter. It is due to the efficient optical pumping of the atomic ensemble into its dark states, and gives rise to the emission of highly bunched light [5,7]. On the other hand, a superradiant regime is reached when Γ < P < N Γ, because of the preferential population of symmetrical Dicke states showing enhanced coupling to the electromagnetic field. Finally, when P > N Γ, the atomic population is totally inverted, such that each emitter behaves independently from the others [7]. Thus, superradiance (subradiance) is characterized by n J > Z (n J < Z). In Fig. 2b we plot the cooperative fraction, which is negative in the subradiant regime, positive in the superradiant one and goes back to 0 at high pumping rates, as expected. In the same figure, we also plot the quantity (n J -Z)/n J , which provides an intuitive measure of cooperativity in the bad cavity regime, based on the analyses already performed in the literature and described above. As it can be seen, the parameter defined in the present work agrees qualitatively well with former studies, and can be used as a fair marker to describe the transition between subradiant to superradiant regimes, respectively.

As recently demonstrated, a huge ensemble of atoms coupled to a bad cavity can produce extremely coherent light, a phenomenon called "steady-state super-radiant laser" [13]. In this situation the coherence is to be attributed to the phase locking of the atomic dipoles, and not to the stimulated emission of one given mode of a high quality factor resonator. This result clearly shows the symmetry between steady-state superradiance and stimulated emission, both physical processes inducing an enhancement of the absorption properties of a medium because of bosonic amplification. Nevertheless, in the bad cavity regime, a lasing-like behavior can only take place if the number of emitters is large enough, such that the matter field can be highly excited. This is not the case in the present few emitters situation. This appears in Figs. 2c andd where the spectrum and second-order coherence function of the cavity field have been plotted. As it can be seen, the second-order coherence does not lock to 1, and increasing the pump only broadens the spectrum. Hence in the case under study, the lasing character can only come from stimulated emission, which requires a high quality factor resonator. This is investigated in the following Section.

IV. SUPERRADIANCE-LASING CROSSOVER

In this Section we consider a regime of parameters in which lasing can take place, focussing our analysis on the respective contributions of stimulated emission and superradiance to positive cooperativity. We start from the case of N = 2, and we progressively increase the light-matter coupling, such that strong coupling regime is eventually reached for each individual atom-cavity system. We have plotted in Fig. 3a,c andd the cavity population, the second-order coherence function of the cavity field, and the total atomic inversion as a function of pumping rate, for g ranging from k/100 to 10k. In addition to these quantities, which provide usual signatures of lasing, we have also plotted the newly defined cooperativity fraction (Fig. 3b). One can mainly distinguish three regimes for cooperativity, corresponding to the cooperative fraction being respectively negative, positive and null. Negative cooperativity still characterizes a subradiant regime, associated with the emission of a bunched light field.

In the case of positive cooperativity, the analysis performed in the bad cavity regime is valid as long as g ≤ k.

On the contrary, when g > k, two different behaviors emerge, as it clearly appears in the plots of Fig. 3. As a first step, the cooperative parameter C f becomes positive and remains quite close to 0. The steady state cavity population increases drastically and the atomic inversion Z is clamped to 1. These are usual signatures of the lasing regime, confirming that stimulated emission takes place and that the non-linear regime is reached. Simultaneously, a plateau for g 2 (0) 1 develops. Indeed, the emission of a Poissonian field induces the locking of the second-order coherence to 1 [START_REF] Loudon | The Quantum Theory of Light[END_REF]. The crossover from nonlasing to lasing regime is captured by C f . In particular, the switch from negative to positive provides a new way to define the lasing threshold in the few emitters case.

In a second step, as we further increase the pump, the cooperativity increases significantly while the system is deeply in the lasing regime. In particular, at the maximum value of cooperativity we have all the typical lasing signatures. In this situation indeed, the medium consisting of a few emitters is still lasing n(N, {δ i }, γ) > 0, while each single-emitter laser has quenched (n(1, δ i , γ) → 0). Quenching takes place when the pumping power starts to overcome the effective light-matter coupling strength [24]. This strength scaling like the number of emitters (see next Section), the few emitters laser quenches for higher pumping power than the individual ones : this is a clear signature of cooperativity, which is captured by the parameter C f defined in this study. This situation can be described in terms of a strongly cooperative laser, pointing towards the coexistence of lasing and superradiant characteristics.

Finally the quenching regime appears at high pumping rates. In fact, for lager values of the pump the quenching of the laser begins, and the cooperativity starts decreasing back to zero. Under such conditions, power broadening spectrally decouples the lasing medium from the cavity mode, leading to a decrease of cavity mode population and the switching to a thermal statistics for the emitted field, i.e. g 2 (0) → 2. With respect to the cooperativity measure, quenching is clearly manifested by the simultaneous condition C f → 0.

The signatures of the crossover from plain superradiance to lasing should also appear in the spectral properties of the radiation emitted through the cavity mode. To confirm this behavior, the results for the calculated spectra are shown in Fig. 4 for a larger number of emitters (4). The lasing crossover is clearly manifested by a visible narrowing of the emission spectrum in the lasing region. For g = 0.5k, at low pumping rates the spectrum is given by the Jaynes-Cummings doublet of each atom-cavity system, under which conditions the lasing narrowing is strikingly evident.

V. INFLUENCE OF EXPERIMENTAL PARAMETERS

In this section we focus on the evolution of the cooperativity fraction with respect to the parameters of the ensemble of emitters. In particular, we investigate the influence of the number of emitters, and the robustness of the cooperativity with respect to dephasing and inhomogeneous broadening.

Cooperativity and number of emitters

The analysis of the emission properties of a mesoscopic number of emitters in a cavity by increasing the number one by one is a fruitful bottom-up approach, which was already taken in [7] to investigate the statistical properties of the emitted radiation, and paves the way to a novel description of the quantum-classical boundary. The results for the system parameters in the bad cavity regime are shown in Fig. 5. The maximum of the cavity population plotted in Fig. 5a increases drastically with the number of emitters, a behavior already emphasized in [6], where such saturation value has been shown to evolve as N 2 . In Fig. 5b, we can appreciate the corresponding evolution of the cooperativity fraction. In the subradiant regime, C f decreases with the number of emitters: this can be interpreted by noting that the low excitation Dicke states are mainly populated on average, for which the decay rate typically scales as N . As a consequence, the larger N , the faster the relaxation, which leads to a lower excitation of the matter field at equal pumping rate and fully justifies the behavior of C f . As expected, C f switches from negative to positive values for P = Γ, which does not depend on the number of emitters. On the contrary, and as it also appears in the Figure , the system returns to an independent-like behavior when P = N Γ, which occurs at larger and larger pumping rates on increasing N .

In the same spirit as above, we revisit now these properties in the case where the resonator has a good quality factor, so that each emitter is individually strongly coupled to the cavity mode (g = 5k). We have investigated the signatures of lasing for a medium consisting of a few identical quantum emitters, ranging from 1 to 3. Usual quantities, namely cavity population n, secondorder correlation at zero time delay, g 2 (0), and population inversion (normalized to the number of emitters, Z/N ) are plotted on the left side of Fig. 6. As previously, the regimes of subradiance, superradiance/lasing and quenching can be clearly identified. Subradiance is manifested in the low pumping regime, by a negative cooperativity fraction and oscillations of the second or- der coherence function between even and odd numbers of emitters. These oscillations have been thoroughly analyzed and already discussed in Ref. 7, essentially related to differences in the Hilbert space geometry of Dicke states participating in the driven/dissipative quantum dynamics, and depend on the parity of N . In particular, these oscillations are the signature of the energy spreading of the Dicke eigenstates, which depends on the parity of the number of atoms [7]. Another signature is the excitation of the matter field even in the low pumping case, as already mentioned: such an effect reflects the optical pumping in the dark states [6] and clearly appears in the plot of the population inversion. It can also be seen that the extension of the lasing regime as a function of the pumping strength is larger the larger is N , as saturation and quenching take place at larger pumping rates. For completeness, we have also reported the results for the single-emitter case, which has been thoroughly investigated in the literature [START_REF] Mu | [END_REF][21][22][23][24][25]. Again, both effects can be accounted for by the increase of the effective light-matter coupling, which scales as √ N . On the right hand side of Fig. 6, we plot the cooperative fraction, the cavity population per emitter, n/N , and the cavity population per excited emitter, which is defined as n/Z. The influence of N on C f is quite straightforward to analyze: increasing N increases the magnitude of the cooperativity fraction. In the subradiant regime, C f is more negative, as in the bad cavity case. In the superradiant regime, the maximum C f occurs at larger pumping, and it reaches larger values: this corresponds to the saturation and quenching of N cooperative emitters occurring at larger pump power as compared to the single-atom laser. The maximum cooperativity is reached in this very region where the cooperative medium is still lasing, while the single emitter medium has quenched. Switching from negative to positive cooperativity does not depend on N , as in the low Q resonator case. The same evolution can be seen in the parameter n/N .

Finally, we have studied the quantity n/Z (see Fig. 6f), which is the typical absorption per emitter. Its meaning in the single atom case is clear: in the spontaneous emission regime, the atomic inversion and the cavity population scale like the pump power, hence the ratio n/Z is constant and locked at the spontaneous emission rate. When the non-linear regime is reached, the population clamps while the cavity population is still increasing. Hence the parameter n/Z increases linearly with the pump power because of stimulated emission, a feature that clearly appears in the plot. Such behavior would also appear for a standard lasing medium made of distinguishable atoms. On the contrary, for the microlaser investigated here, the switching from linear to non-linear behavior is blurred out as soon as the medium involves more than a single emitter. As it can be seen in the plot, the average absorption per emitter continuously increases with respect to the pump power. Indeed, here the increase in the absorption is the result of two contributions: before the lasing threshold, it is due to the displacement of the equilibrium in the Dicke states phase space toward higher excitation states characterized by higher coupling to the light field (steady state superradiance). When lasing takes place, the absorption increase comes from stimulated emission. This behavior is a major difference between standard lasing media made of distinguishable atoms and the specific medium investigated here. At the highest powers the drop-off in n/Z is attributed to quenching, which describes the decrease in the absorption of the atomic medium when the pump power becomes too high. This effect is due to the broadening of the atomic emission line at high excitation power [24].

Cooperativy vs. dephasing/inhomogeneous broadening So far we have assumed the quantum emitters to be identical, on resonance with the cavity mode, and nondephased. Now we relax this latter constraint and analyze how detuning (i.e. inhomogeneous broadening) and dephasing affect the cooperativity. Both of these effects are relevant in solid state implementations of the model studied in this work, such as in a quantum dot laser [8]. We consider a system of two emitters equally detuned from the single-mode cavity frequency, and compute their cooperativity as a function of detuning for different regimes of the pumping rate. The results are plotted in Fig. 7a. As it appears in the plot, increasing the detuning may actually increase the cooperativity, even though the total atom-cavity coupling decreases. This increase is due to the fact that the cavity population in the single emitter is more sensitive to detuning. Thus, while the single emitters are being decoupled from their cavities the cooperative emitters sustain the cavity population. For low pumping regimes the system may be driven from subradiant to superradiant regime. We refer to this positive cooperative regime as superradiant instead of lasing, since it lacks other characteristic features of lasing, such as spectral narrowing (not shown here). For larger values of the pumping rate, the system behaves as a regular laser, as already established. In this regime the cooperativity also presents a maximum as a function of the detuning even though, once again, the effective atom-cavity coupling only decreases. Finally and as expected, for large detuning the cooperativity tends to zero.

Interestingly, a similar behavior is found by adding pure dephasing to the emitters, as shown in Fig. 7b. However, it is clear that the cooperativity is much more sensitive to detuning rather than dephasing. We also point out that detuning the emitters asymmetrically (not shown here) with respect to the cavity frequency, or non equally detuning them, usually leads to less cooperativity as compared to the case in which the emitters are symmetrically detuned around the cavity mode frequency. This sensitivity is due to the fact that the resonator couples to the symmetric "brilliant" state of the two emitters, which remains identical if the two emitters are detuned symmetrically with respect to the mode.

VI. CONCLUSION

In conclusion, we have introduced a new operational quantity to measure the degree of cooperativity in the emission characteristics of a system of few incoherently pumped emitters coupled to a single-mode cavity. It is defined by comparing the photon emission of an ensemble of emitters coupled to the same cavity mode with the overall emission of the same emitters each one individually coupled to its own resonator mode. We have shown FIG. 7: Cooperativity fraction in the two-emitter case: (a) as a function of (symmetric) detuning from the cavity resonance frequency, (b) as a function of pure dephasing rate (and zero detuning). We considered the strong coupling regime g = 5k.

how such a quantity is able to quantitatively describe the crossover between steady state subradiance, superradiance, and lasing. We have analyzed the effects of inhomogeneous broadening and pure dephasing on coopera-tivity, which might be relevant in solid state implementations of this model. We have shown that in the good cavity regime, a few indistinguishable two-level systems provide a new type of lasing medium as compared to an ensemble of distinguishable emitters. These results are quite promising for emerging experiments in mesoscopic quantum optics, where a bottom up approach allows the possibility to sequentially add an increasing number of emitters coupled to the same cavity mode.

FIG. 1 :

 1 FIG. 1: The system under investigation, and the operational definition of cooperativity. N emitters being incoherently pumped, two situations are compared. (a) A detector captures the output of a cavity containing the ensemble of emitters. (b) Each emitter is coupled to an individual cavity, and each output signal is assumed to be detected and summed.

FIG. 2 :

 2 FIG. 2: Subradiance and superradiance for 5 emitters in the bad cavity regime, g = 0.1k. (a) Total atomic inversion, Z, and average value of the total atomic dipole, J † J ; (b) Cooperative fraction compared to (nJ -Z)/nJ ; (c ) Cavity spectrum; (d) Second-order coherence function g 2 (0). All the data are plotted as a function of the pump rate.

FIG. 3 :

 3 FIG. 3: The superradiance-lasing crossover is shown for 2 emitters coupled to the same cavity mode: (a) cavity population, (b) cooperative fraction, (c ) second-order coherence function, and (d) atomic inversion. The data are plotted as a function of pumping rate for different values of the emittercavity coupling constant.

FIG. 4 :

 4 FIG. 4: The spectral signature of superradiance-lasing crossover for 4 emitters coupled to the same cavity mode: spectrum of the emitted radiation vs. pumping rate for (a) g = 0.2k, and (b) g = 0.5k, respectively.

FIG. 5 :

 5 FIG. 5: Influence of the number of emitters in the bad cavity regime, g = 0.3k: (a) cavity population, and (b) cooperativity fraction.

FIG. 6 :

 6 FIG.6: Subradiance and lasing in the strong coupling regime for increasing number of emitters, i.e. g = 5k: (a) cavity population, (b) second-order correlation function, (c) population inversion per emitter, (d) cooperative fraction, (e) cavity population per emitter, (f) cavity population per excited emitter. The data are shown for one emitter (blue continuous), two (green dashed) and three (red dashed-dotted) emitters, respectively. All the data are plotted as a function of the pumping rate.

Acknowledgments

The authors acknowledge Brazilian Agencies CNPq and Fapemig, the NanoSci EraNet grant "LECSIN," the Fondation Nanosciences of Grenoble and ANR-PNano "CAFE", and the Italian Ministry of University and Research through Fondo Investimenti Ricerca di Base (FIRB) "Futuro in Ricerca" project RBFR12RPD1 for financial support.


Keywords: « Energy Storage System», « Solid State Power Controller (SSPC)», « Bidirectional converter « Silicon MOSFET»

This paper deals with the design of a solid state power controller (SSPC) for a DC-DC converter in an aeronautic application. First, the specifications are drawn appropriately with the aeronautic environment. Then the design and experimental validation are described.

A Design of Solid State Power Controller for a bidirectional DC-DC Converter in an aeronautic context

Hassan Cheaito (1) , Bruno Allard (1) , Guy Clerc (1) , Joris Pallier (2) , Pascal Pommier-Petit (2) (1)

Univ Lyon, INSA Lyon, Univ Claude Bernard, Ecole Centrale Lyon, CNRS, F-69621 Villeurbanne, France (2) CentumAdeneo Ecully, France cheaito.hasan@gmail.com, bruno.allard@insa-lyon.fr, guy.clerc@univ-lyon1.fr

Introduction

The protection against the reversibility of the supply polarity, inrush current or short-circuit become inevitable in high power application. Moreover in an aeronautic application, the power density is one of the most important priorities [START_REF] Molligoda | Review of design and challenges of DC SSPC in more electric aircraft[END_REF]- [START_REF] Boukari | Solid state power controller (SSPC) for protection of continuous embedded network[END_REF]. The electromechanical circuit breakers that have historically been used for aeronautic application have to be challenged regarding the mass and volume [START_REF] Molligoda | Review of design and challenges of DC SSPC in more electric aircraft[END_REF]. Instead, power transistors can be connected back-to-back as shown in Fig. 1 either in common source or drain to completely isolate the power source from the load, or from another power source. This solution based on semiconductor devices is called Solid State Power Controller (SSPC). In addition, the SSPC requires a minimum maintenance and provides fast interruption of the current thanks to the fast commutation of MOSFETs (a few microseconds) [START_REF] Feehally | A MOSFET based solid-state power controller for aero DC networks[END_REF]. This protection have been integrated in the SUNSET equipment detailed in [START_REF] Cheaito | Preliminary Design of Energy Storage System and Bidirectional DC-DC Converter for Aircraft application[END_REF]. The SUNSET equipment targets energy recovery from the final breaking of an aircraft into a battery pack through the aircraft high-voltage network. The battery management system (BMS) monitors permanently the voltage, temperature and current of the battery pack. Depending on data collect, the BMS turns on (safe mode) or cuts off (security mode) the SSPC. This paper deals with the design of a SSPC board that will be implemented between a DC-DC converter and a battery pack (350 V). A parallel array of 4 Si-MOSFETs has been used to supply 35 A nominal current and 350A as short-circuit current. Results analysing on-state losses, thermal performance, inrush current limiting, inductive load breaking, and over-current response times will be discussed.

SSPC specifications

This section describes the main environment characteristics to draw the mandatory specifications of the SSPC. The voltage level of the battery is around 350 V with a nominal current of 35 A. Hence the breakdown voltage (V BR ) of the SSPC transistors must be larger. Therefore, the V BR taking into account a safety margin (0.7), will be at least 500 V. Regarding the heat sink, the thermal study has defined 20 W as the maximum dissipation for each array of transistors. It is worth to be noted that the switching losses appear only one time at the beginning and the end of charging thus they are negligible relatively to conduction losses. Table 1 summarizes the main specifications to be met in the design.

Table I: The specifications of the SSPC

SOA curves

The most important characteristic of an SSPC is its capability of dissipating energy during opening. The current limit during the switching time is given by datasheet in Safe Operation Area (SOA). Several datasheets of 650 V MOSFET have been compared. The best candidates show the ability of conducting 200 A under 650 V for 1μs. In our application, the short-circuit current (350 A) has been calculated based on the assumption of 1 Ohm as internal impedance of the battery pack (350 V). The batteries as well as the SSPC must be able to carry this current until the default is eliminated. Thus, at least two parallel transistors are needed. The circuit loop in the case of SSPC opening is equivalent to an inductive circuit whose current rise depends on the values of τ = L / R. The inductance of connection and cables is estimated to 1 µH which means τ = 1µs with the latter impedance assumption. Thus, the steady state current will reach 350 A in only 5 µs. Therefore, in a real case the fault must be eliminated in 10 μs maximum. So, all the transistors connected in parallel must stand 350 A for 10 μs at reduced V DS (ohmic region). This leads us to compare the maximum current per MOSFET to deduce the number of devices to be connected in parallel. However, the most dimensioning parameter is the power dissipated during the switching ON-OFF because of the dynamic voltages and currents that MOSFETs must handle during the switching. Thus, the energy to be dissipated during the turn-off of the MOSFET is equivalent to the product of the power dissipated by the switching time

Hypothesis

Choice of the MOSFET

As mentioned, several 650 V MOSFET have been compared. Figure 3 shows the continuous power losses due only to R DS-on . It shows the decrease in power losses as function of the number of parallel MOSFETs. Even if STW77N65M5 exhibits the highest conduction losses, it is still within the specifications if '4' MOSFETs are in parallel. Therefore, the choice of 4 MOSFETs (STW77N65M5) seems to make a good trade-off by minimizing the cost / mass / losses while guaranteeing robustness in case of opening the circuit in short-circuit.

PCB design

As mentioned before, the SSPC board is inserted between the converter and the battery pack. So, it has a particular form (like a T) imposed by the space environment (see Figure 4).

Layout PCB

For EMI issues, the ground potential is linked to some layers of the PCB. This reduces the loop of the current return. In order to limit the current density at 15 A/mm² in the cupper track, two layers have been used for the return current and for the drain connection. As a result, six layers have been used in total.

Fig. 4 3D modeling and layer decomposition of the SSPC board

Overvoltage protection

The SSPC MOSFETs should interrupt current in case of a short circuit or overcurrent. While opening an inductive circuit, an overvoltage is unavoidable (L di/dt). In order to prevent breakdown and false turn-on, two solutions have been implemented as shown in Figure 5: Active Clamping and Advanced Active Clamping [START_REF] Garcia | Safe Driving of Multi-Level Converters Using Sophisticated Gate Driver Technology[END_REF]. These solutions can work both or separately. However, this paper discusses only the first one.

Active clamping is a technique that keeps a transient overvoltage below the critical limits when the MOSFET turns off. TVS voltage should be higher than nominal voltage and lower than breakdown voltage. As shown in Figure 5, the standard approach for active clamping is to use TVS Diodes connected between the drain and the gate. When the V DS voltage exceeds the TVS breakdown voltage, the TVS starts to conduct and this current will cause the gate-source voltage to increase. Thus, the MOSFET is still held in an active mode and the turnoff process is prolonged. As a result, voltage overshoot and turn-off overvoltage ΔV DS are reduced. An LtSpice simulation of the turn-off of 35 Amps has been done using SPICE model of the used component: TVS (4 SMAJ100 in series) and MOSFET (4 STW77N65M5 in parallel). As seen in Figure 6, no overvoltage V DS has been detected and voltage is maintained less than 600 V during turnoff. During the 8 µs for turning-off, the total dissipated energy is 92 mJ for each array of transistors which is in compliant with the specifications in table 1.

Experimental results

The PCB has been manufactured as shown in the Fig. 7-b following the thermal study done in the paragraph III. In order to test the performances of the PCB, the synopsis in the Fig. 1 has been applied. At this stage, both the converter and the EMC filter are not connected. As seen in the Fig. 7-a, the battery pack has been replaced by an inductive load (R= 10 Ω, L=7.5 µH); the HVDC is represented by the power supply (350V). The measurements show a very good agreement with the simulations. The turn-off is as simulated before around 10 µs; the overshoot voltage is limited by the TVS up to 500 V which is lower than the breakdown voltage of the MOSFET. The dissipated switching energy is is found to be 11 mJ. Indeed, the current falls to zero much shorter in the experience.

Conclusion

For better power density, SSPC replaces advantageously heavy circuit breaker dedicated to high power application. This paper proposes a methodology to draw the specifications regarding environmental conditions. At the same time, it shows a design of SSPC which is able to cut off current up to 350 A under 650 V nominal voltage. Active clamping is integrated to reduce drain-source over-voltage during cut-off. A PCB has been manufactured and tested. A very good agreement has been shown between the simulation and the measurements at the nominal current (35 A). The next step would be the test at the short-circuit current (350 A).

Fig. 1 :

 1 Fig. 1: Synopsis of the SSPC environment

  (1 µs). The following equation calculates roughly the maximum energy by getting the maximum current and voltage divided by √3 to compensate the decrease in current during the switching. E = U * I * t = (670V * 350A / √3) * 1μs = 123mJ.

Fig. 2 :

 2 Fig. 2: Safe operation Area (SOA) for SCTH90N65G2V device (left) and SCT3017AL device (right)

Fig. 3

 3 Fig. 3 Power losses during conduction mode for several MOSFETs

Fig. 5

 5 Fig. 5 Circuit design of TVS and driver clamping to prevent both: false turn-on and overvoltage

Fig. 6

 6 Fig. 6 Simulation of turn-off MOSFET showing: V DS (green), I D (red) and V GS (purple).

Fig. 7

 7 Fig. 7 The SSPC: (a) Setup of the test, (b) picture of the PCB designed

Fig. 8

 8 Fig. 8 Measurement of the turn-off MOSFET showing: VDS (green), ID (red) and VGS (blue).

Acknowledgements

This project has received funding from the Clean Sky 2 Joint Undertaking under the European Union's Horizon 2020 research and innovation programme under grant agreement n°785585.


Nicolas Jacon

Abel Lacabanne
email: abel.lacabanne@uca.fr

Gpl

ON CALOGERO-MOSER CELLULAR CHARACTERS FOR IMPRIMITIVE COMPLEX REFLECTION GROUPS

We study the relationship between Calogero-Moser cellular characters and characters defined from vectors of a Fock space of type A 8 . Using this interpretation, we show that Lusztig's constructible characters of the Weyl group of type B are sums of Calogero-Moser cellular characters. We also give an explicit construction of the character of minimal b-invariant of a given Calogero-Moser family of the complex reflection group

Introduction

The representation theory of finite Coxeter groups, and more precisely the Kazhdan-Lusztig theory, is of crucial importance for the understanding of various problems in Lie theory. For example, in his classification of unipotent characters of finite groups of Lie type, Lusztig has underlined the importance of a partition of irreducible characters of the associated Weyl group into families. The starting point of Kazhdan-Lusztig theory is the construction of a basis of Hecke algebras in [START_REF] Kazhdan | Representations of Coxeter groups and Hecke algebras[END_REF], known nowadays as the Kazhdan-Lusztig basis. From this basis, one can define partitions of the Coxeter group in the so-called right, left or two-sided cells. To every right cell is attached a module over the Hecke algebra and a character (not necessarily irreducible) of the Coxeter group. These characters are the cellular characters 1 . Lusztig has then realized that these constructions are also possible with Hecke algebras at unequal parameters [START_REF] Lusztig | Hecke algebras with unequal parameters[END_REF]. Many questions around Kazhdan-Lusztig theory for Hecke algebra with unequal parameters still remain open, see [START_REF] Bonnafé | Kazhdan-Lusztig cells with unequal parameters[END_REF] for a recent monograph about this subject.

Since the Coxeter groups are real reflection groups, one may ask the following question: does there exist a Kazhdan-Lusztig theory for finite complex reflection groups? Of course, one first need the notion of Hecke algebras for complex reflection groups. These objects have been introduced by Broué-Malle-Rouquier in [START_REF] Broué | Complex reflection groups, braid groups, Hecke algebras[END_REF], where presentations in terms of braid relations are also given. In the particular case of the complex reflection group Gpl, 1, nq, they recover an algebra already introduced in [START_REF] Ariki | A Hecke algebra of pZ{rZq S n and construction of its irreducible representations[END_REF], nowadays known as the Ariki-Koike algebras. Unfortunately, we do not know a basis of the Hecke algebra associated with a finite complex reflection group which has similar properties as the Kazhdan-Lusztig basis for Hecke algebras associated with Coxeter groups.

But this story does not stop there. Rouquier [START_REF] Rouquier | Familles et blocs d'algèbres de Hecke[END_REF] managed to give an interpretation of the Kazhdan-Lusztig families of cristallographic Coxeter groups in terms of blocks of

1
The associated modules have not to be confused with the cell modules in the sense of Graham and Lehrer, even if there are closed relationships between these two objects [START_REF] Geck | Hecke algebras of finite type are cellular[END_REF] the Hecke algebra, which extends to the complex reflection group case. These Rouquier families have been computed for all finite complex reflection groups [START_REF] Chlouveraki | Rouquier blocks of the cyclotomic Ariki-Koike algebras[END_REF]. The next major step in a tentative of the generalization of the Kazhdan-Lusztig theory has been made by Gordon, via the study of the representation theory of restricted rational Cherednik algebras at t " 0 [START_REF] Gordon | Baby Verma modules for rational Cherednik algebras[END_REF]. He introduced the notion of a Calogero-Moser family, which is a partition of the irreducible characters of a given complex reflection group. If the complex reflection group is a Coxeter group, it is worth to notice that the construction of the Calogero-Moser families and the Kazhdan-Lusztig families are very different. But it has been shown, via explicit computations on both sides, that these two notions coincide for most finite irreducible Coxeter groups [START_REF] Bellamy | Cuspidal Calogero-Moser and Lusztig families for Coxeter groups[END_REF]Theorem 2.4].

Finally, Bonnafé and Rouquier [START_REF] Bonnafé | Cherednik algebras and Calogero-Moser cells[END_REF] defined the notion of Calogero-Moser right, left and two-sided Calogero-Moser cells by studying the geometry of the Calogero-Moser space associated with a complex reflection group. Their approach is partially motivated by the ideas of Gordon in which the rational Cherednik algebra at t " 0 plays an essential role. To each right cell is also associated a Calogero-Moser cellular character. If the complex reflection group is a Coxeter group, they conjecture that the various output of this Calogero-Moser theory is the same as the output of the Kazhdan-Lusztig theory (families, cells, cellular characters). However, only the some particular cases are well understood: the symmetric group [START_REF] Brochier | RSK and Calogero-Moser Cells in Type A[END_REF], partially the dihedral groups [START_REF] Bonnafé | On the Calogero-Moser space associated with dihedral groups[END_REF][START_REF] Bonnafé | Calogero-Moser cells of dihedral groups at equal parameters[END_REF] or some exemples in small ranks [START_REF] Bonnafé | Cherednik algebras and Calogero-Moser cells[END_REF]Chapter 18 and 19].

In this paper, we are mainly interested in the case of the complex reflection group Gpl, 1, nq and more specifically in the cellular characters for Gpl, 1, nq. The second author has stated in [START_REF] Lacabanne | On a conjecture about cellular characters for the complex reflection group Gpd, 1, nq[END_REF] some conjectures relating these Calogero-Moser cellular characters with the expressions of the canonical basis of the level l Fock space for U q pgl 8 q. Thanks to this interpretation of constructible characters for the Weyl group of type B n settled by Leclerc and Miyachi [START_REF] Leclerc | Constructible characters and canonical bases[END_REF], one of our main result is the following, see Theorem 4.6 for the precise details on the parameters involved.

Theorem 1.1. Lusztig's constructible characters for the Weyl group of type B n are sums of Calogero-Moser cellular characters.

Our second main result concerns the computation of certain remarkable elements in the Calogero-Moser families. We provide an explicit proof of a result concerning the binvariants of the characters of a Calogero-Moser family. Namely, it has been proven that for every such family, there exists a unique irreducible character of minimal invariant among the family [START_REF] Bonnafé | Cherednik algebras and Calogero-Moser cells[END_REF]Theorem 7.4.1(b)]. We here give another proof of this result, in the specific case of Gpl, 1, nq, and explicitly provide a description in terms of l-symbols of the character of minimal b-invariant in a Calogero-Moser family. We refer to Section 5 for more details, and especially to Theorem 5.2.

The organisation of the paper is as follows. In Section 2, we review the definition of Calogero-Moser cellular characters via Gaudin algebras, as in [START_REF] Bonnafé | Cherednik algebras and Calogero-Moser cells[END_REF]Chapter 8], and focus then in the specific case of the complex reflection group Gpl, 1, nq. Section 3 concerns the Fock spaces, and the quasimonomial vectors, that we relate to some approximations of the Calogero-Moser cellular characters. Then, in Section 4, we study in details the case of the Weyl group of type B and the relationship between Calogero-Moser cellular characters and Lusztig's constructible characters. Before the last section, we go back to the case of the complex reflection group Gpl, 1, nq and describe the symbol of minimal b-invariant in a Calogero-Moser family using the combinatorics of l-symbols. Finally, we end in Section 6 with some various conjectures, mostly related with the b-invariant.

Calogero-Moser cellular characters

The Calogero-Moser cellular characters are a conjectural extension to complex reflection groups of cellular characters in Kazhdan-Lusztig theory. The definition of cells usually requires the study of representation theory of Cherednik algebras, but we will use the definition of the Calogero-Moser characters using the notion of Gaudin algebras. We recall now their definition and then focus on the case of Gpl, 1, nq.

2.1. Set-up. Let V be a finite dimensional C-vector space with dual V ˚. The duality pairing is denoted by x¨, ¨y : V ˆV ˚Ñ C and the determinant map by det : GLpV q Ñ C ˚.

Let W Ă GLpV q be a complex reflection group. We denote by RefpW q the set of pseudoreflections of W , that is elements of W which fix an hyperplane. For each s P RefpW q, we choose α s P V ˚and α _ s P V such that kerps ´Idq " kerpα s q and Imps ´Idq " Cα _ s . In other terms, we have spvq " v ´p1 ´detpsqq xv,αsy xα _ s ,αsy α _ s for any v P V . We denote by IrrpW q the set of irreducible complex characters of W and for any χ P IrrpW q, we fix V χ a simple representation of W affording the character χ.

Finally, we also fix a function c : RefpW q Ñ C, s Þ Ñ c s which is invariant by conjugation of W on RefpW q.

2.2. Definition via Gaudin algebras. For any x P V , we consider the following element of CpV qrW s

D x " ÿ sPRefpW q c s detpsq xx, α s y α s s.
Definition 2.1. The Gaudin algebra with parameter c, denoted by Gau c pW q is the CpV qsubalgebra of CpV qrW s generated by p D x q xPV . Since x Þ Ñ D x is clearly linear, the Gaudin algebra Gau c pW q is generated by pD x q xPB , where B is a basis of V . Proposition 2.2 ([7, Proposition 8.3.1]). For any x, y P V , the elements D x and D y commute in CpV qrW s. Therefore Gau c pW q is a commutative subalgebra of CpV qrW s.

Even though Gau c pW q is commutative, its representation theory is far form being wellunderstood: it depends heavily on c and the field CpV q is almost never a splitting field. Some simple representations are not absolutely simple and therefore a simple representation of Gau c pW q is not automatically of dimension 1.

Definition 2.3. Let L be a simple representation of Gau c pW q. The Calogero-Moser cellular character associated with L is

γ CM L " ÿ χPIrrpW q " Res CpV qrW s GaucpW q pCpV q b C V χ q : L ı χ.
The name Calogero-Moser cellular characters comes from another definition of these characters, which is related with the Calogero-Moser partition of W into cells, see [START_REF] Bonnafé | Cherednik algebras and Calogero-Moser cells[END_REF].

2.3. The case of Gpl, 1, nq and Jucys-Murphy elements. We now focus on the case of the complex reflection group Gpl, 1, nq. We fix ζ " expp2iπ{lq, µ l " xζy the group of l-th root of unity in C ˚and suppose that V is of dimension n with basis py i q 1ďiďn and dual basis px i q 1ďiďn . Identifying GLpV q with GL n pCq through the basis py i q 1ďiďn , the group Gpl, 1, nq consist of monomial matrices with entries in µ l . By convention, Gpl, 1, 0q will denote the trivial group.

We denote by s i,j the reflection exchanging y i and y j and fixing y k for k R ti, ju and by σ i the reflection fixing y k for k ‰ i and sending y i to ζy i . The group Gpl, 1, nq is then generated by ts i,i`1 | 1 ď i ă nu Y tσ 1 u. The reflections of Gpl, 1, nq fall into l conjugacy classes, namely S 0 " σ r j s i,j σ ´r j ˇˇ1 ď i ‰ j ď n, 0 ď r ă l ( and S 1 , . . . , S l´1 given by S k " σ k i ˇˇ1 ď i ď n ( . Finally, we set c i " c |S i as a shorthand for the parameter c. Another set of parameters appears also in the context of Cherednik and Gaudin algebras, namely k 0 , . . . , k l´1 given by

k i " 1 l l´1 ÿ j"1 ζ jp1´iq c j .
We will consider the indices modulo l and note that k 0 `¨¨¨`k l´1 " 0.

The following Jucys-Murphy elements have been introduced by the second author in [START_REF] Lacabanne | On a conjecture about cellular characters for the complex reflection group Gpd, 1, nq[END_REF] J i "

ÿ sPRefpGpl,1,iqq sRRefpGpl,1,i´1qq c s detpsqs.
These elements generate a commutative subalgebra JM c pl, nq of CrGpl, 1, nqs. Similarly to Definition 2.3, we define the Jucys-Murphy cellular characters. Definition 2.4. Let L be an irreducible representation of JM c pl, nq. The Jucys-Murphy cellular character associated with L is

γ JM L " ÿ χPIrrpW q " Res CrGpl,1,nqs JMcpl,nq pV χ q : L ı χ.
The Jucys-Murphy cellular characters approximate the Calogero-Moser cellular characters in the following sense: We now aim to give an inductive characterization of these Jucys-Murphy cellular characters. We first recall the combinatorics of the l-partitions of n which govern the representation theory of Gpl, 1, nq, see [START_REF] Geck | Representations of Hecke algebras at roots of unity[END_REF]Section 5.1].

A partition λ of n is a decreasing sequence of non-negative integers λ 1 ě λ 2 ě ¨¨¨ě λ k ě ¨¨¨such that λ i " 0 for i " 0 and |λ| :"

ř 8 i"1 λ i " n.
A l-partition of n is then a l-tuple of partitions pλ p1q , . . . , λ plq q such that ř l i"1 |λ piq | " n. The isomorphism classes of irreducible representations of Gpl, 1, nq are parametrized by the set of l-partitions of n. For every l-partition λ of n, we denote by V λ the corresponding irreducible representation and by χ λ its character.

Restriction and induction can be described using by the Young graph of l-partitions of n. Given a l-partition λ, its Young diagram rλs is the set pa, b, cq P Z ą0 ˆZą0 ˆt1, . . . , lu ˇˇ1 ď b ď λ pcq a ( whose elements are called boxes. A box γ of the Young diagram of a l-partition λ of n is said to be removable if rλsztγu is the Young diagram of a l-partition µ of n ´1. The box γ is also said to be addable to the Young diagram of µ.

Proposition 2.6 ([13, Proposition 5.1.8]). Let λ be a l-partition of n. Then

Res

Gpl,1,nq Gpl,1,n´1q pV λ q " ' µ V µ where the sum is taken over the l-partitions µ of n ´1 whose Young diagram is obtained by removing a box of rλs and Ind Gpl,1,n`1q Gpl,1,nq pV λ q " ' µ V µ where the sum is taken over the l-partitions µ of n `1 whose Young diagram is obtained by adding a box to rλs.

Using these decompositions, we obtain a basis of V λ indexed by the so-called standard tableaux of shape λ. Given a l-partition λ of n, a standard tableau of shape λ is a bijection t : rλs Ñ t1, . . . , nu such that for all boxes γ " pa, b, cq and γ 1 " pa 1 , b 1 , cq of λ, we have tpγq ă tpγ 1 q if a " a 1 and b ă b 1 or a ă a 1 and b " b 1 . The datum of a standard tableau of shape λ is equivalent to the datum of a finite sequence pλrisq 1ďiďn such that λris is a l-partition of i, λrns " λ and rλri `1ss is obtained from rλriss by adding a box. We denote by λ t the sequence of l-partitions obtained from the standard tableau t, that is rλ t riss " t ´1t1, . . . , iu.

Therefore, using the branching rule for restriction, we have V λ " À t D t where the sum is over the standard tableaux of shape λ and D t is a one dimensional space contained in the irreducible component V λ t ris of Res Gpl,1,nq Gpl,1,iq pV λ q for all 1 ď i ď n. The Jucys-Murphy elements act diagonally on V λ with respect to the decomposition V λ " À t D t . On D t , the Jucys-Murphy element J i acts by multiplication by lpk 1´c ´c0 pb ´aqq, where tpa, b, cq " i, see [START_REF] Lacabanne | On a conjecture about cellular characters for the complex reflection group Gpd, 1, nq[END_REF]Corollary 1.8].

Given ξ P C and λ a l-partition of n ´1, we refine the induction as follows:

ξ ´Ind Gpl,1,nq Gpl,1,n´1q pV λ q " à µ V µ ,
the sum being over the l-partitions µ of n whose Young diagram is obtained by adding a box to rλs and such that J n acts on V µ by multiplication by ξ. Using this truncated induction, we recursively define a set of representations of Gpl, 1, nq as follows:

E 0 " t1u and E n " ! ξ ´Ind Gpl,1,nq Gpl,1,n´1q pV q ˇˇV P E n´1 , ξ P C ) zt0u,
where 1 denotes the trivial representation of the trivial group. From the above discussion, we immediately deduce the following new way to define the Jucys-Murphy cellular characters.

Proposition 2.7. The set of characters of representations in E n coincides with the set of Jucys-Murphy constructible characters.

Remark 2.8. Instead of considering the Jucys-Murphy elements J i , we might consider the shifts J i `k by any scalar k. The set E n does not change if we replace the Jucys-Murphy elements by these shifts.

In terms of parameters, this is equivalent to replace the condition k 0 `¨¨¨`k l´1 " 0 by k 0 `¨¨¨`k l´1 " k. Therefore, we may and will remove the condition k 0 `¨¨¨`k l´1 " 0 on the parameters without changing the set of Jucys-Murphy constructible characters.

Quasimonomial vectors of Fock spaces in type A 8

Following [13, Section 6.2], we introduce the Fock space as a representation of U q psl 8 q together with quasimonomial vectors. We then relate evaluations at q " 1 of these vectors with the Jucys-Murphy cellular characters introduced in the previous section.

We fix q an indeterminate and define the usual quantum integers and factorials in Zrq, q ´1s by rns "

q n ´q´n q ´q´1 and rns! "

n ź k"1 rks,
for n P N.

3.1. The algebra U q psl 8 q and the Fock space F s . Let U q psl 8 q be the Qpqq-algebra with generators E i , F i and K ˘1 i for i P Z subject to

K i K ´1 i " 1 " K ´1 i K i , K i K j " K j K i , K i E j " q ´δi´1,j `2δ i,j ´δi`1,j E j K i , K i F j " q δ i´1,j ´2δ i,j `δi`1,j F j K i , rE i , F j s " δ i,j K i ´K´1 i q ´q´1
, and the Serre relations

E 2 i E j ´r2sE i E j E i `Ej E 2 i " 0, F 2 i F j ´r2sF i F j F i `Fj F 2 i " 0, if |i ´j| " 1, rE i , E j s " 0, rF i , F j s " 0, if |i ´j| ą 1.
We define the divided powers E prq i

and

F prq i by E prq i " E i rrs! and F prq i " F i rrs! .
We endow U q psl 8 q with a structure of Hopf algebra. The coproduct ∆, the antipode S and the counit ε are given on the generators by

∆pE i q " E i b 1 `K´1 i b E i , SpE i q " ´Ei K i , εpE i q " 0, ∆pF i q " F i b K i `1 b F i , SpF i q " ´K´1 i F i , εpF i q " 0, ∆pK i q " K i b K i , SpK i q " K ´1 i , εpK i q " 1.
The fundamental weights of U q psl 8 q are denoted by pΛ k q kPZ . We now fix l ě 0 and s " ps 1 , . . . , s l q P Z l . We define the Fock space F s of charge s as the Qpqq vector space with basis p|λ, syq where λ runs in the set of l-partitions of integers. We set

F n s " à λ Qpqq|λ, sy
where λ runs in the set of l-partitions of n so that F s " À nPN F n s . In order to define an action of U q psl 8 q on the Fock space F s , we first need the notion of charged content of a box. Given λ a l-partition and γ " pa, b, cq a box of its Young diagram, the charged content (relative to s) is the integer b ´a `sc . If γ " pa, b, cq and γ 1 " pa 1 , b 1 , c 1 q are two removable or addable boxes of λ with same content, we set γ ă s γ 1 (resp. γ ą s γ 1 ) if c ă c 1 (resp. c ą c 1 ). Remark 3.1. Note that two removable (resp. addable) boxes of same content must lie in different components of rλs. Thus, for each i P Z, we have at most l removable (resp. addable) boxes with content i in rλs.

Finally, if λ and µ are two l-partitions respectively of n and n `1 with rµs " rλs Y tγu and γ of charged content i, we set

N ą i pλ, µq "|taddable nodes γ 1 to λ of charged content i with γ 1 ą s γu| ´|tremovable nodes γ 1 of λ of charged content i with γ 1 ą s γu| N ă
i pλ, µq "|taddable nodes γ 1 to λ of charged content i with γ 1 ă s γu| ´|tremovable nodes γ 1 of λ of charged content i with γ 1 ă s γu| N i pλq "|taddable nodes to λ of charged content iu| ´|tremovable nodes of λ of charged content iu| Proposition 3.2 ([16, Proposition 3.5]). The Fock space F s is an integrable U q psl 8 q-module with action given by E i ¨|λ, sy " ÿ µ,rλszrµs"tγu γ of charged content i q ´N ă i pµ,λq |µ, sy F i ¨|λ, sy " ÿ µ,rµszrλs"tγu γ of charged content i q N ą i pλ,µq |µ, sy K i ¨|λ, sy " q N i pλq |λ, sy.

Therefore, the vector |H, sy is a highest weight vector of weight

ř l i"1 Λ s i . A non-zero vector v of F n
s is said to be quasimonomial if it is obtained from |H, sy by successive applications of F i 's, that is if there exists a sequence of integers pi 1 , . . . , i n q such that v " pF in ¨¨¨F i 1 q ¨|H, sy. A non-zero vector v of F n s is said to be monomial if it is obtained from |H, sy by successive applications of divided powers of the F i 's, that is if there exists a sequence of integers pi 1 , . . . , i k q with i j`1 ‰ i j and pr 1 , . . . , r k q integers with r 1 `¨¨¨`r k " n such that v " pF

pr k q i k ¨¨¨F pr 1 q i 1 q ¨|H, sy.
From the formulas of the action in Proposition 3.2 we easily see that monomial and quasimonomial vectors are elements of À Nrq, q ´1s|λ, sy. To a quasimonomial vector v P F n s , we associate a character γ v of Gpl, 1, nq as follows. We first express v along the basis p|λ, syq of F n s : v " ÿ λ a λ pqq|λ, sy, where λ runs in the set of s-partitions of n and a λ pqq P Nrq, q ´1s. We then obtain the character γ v by evaluation at q " 1:

γ v " ÿ λ a λ p1qχ λ . 3.2. Canonical basis. Let x Þ Ñ x be the Q-linear algebra involution of U q psl 8 q given by q " q ´1, K i " K ´1 i , E i " E i , F i " F i .
We consider the submodule V s of F s generated by the highest weight vector |H, sy, which is then isomorphic to the integrable irreducible module of highest weight Λ " ř l i"1 Λ s i . Any element v P V s can then be expressed as v " x ¨|H, sy for some x P U q psl 8 q and we set v " x ¨|H, sy. This gives a well-defined involution on V s .

Let R be the subring of Qpqq of rational functions regular at q " 0 and consider F s,R the R-sublattice of F s generated by p|λ, syq λ for λ running in the l-partitions of integers. In order to state the existence of canonical basis, we need the definition of a cylindrical multipartition. Definition 3.3. Suppose that s " ps 1 , . . . , s l q is such that s 1 ě s 2 ě ¨¨¨ě s l . A l-partition λ " pλ p1q , . . . , λ plq q is said to be cylindrical if λ pkq i`s k ´sk`1 ě λ pk`1q i for all 1 ď k ď l and i P N ˚.

Theorem 3.4 ([21, 17]

). There exists a unique basis tGpλ, sq | λ cylindrical l ´partitionu of V s such that ' Gpλ, sq " Gpλ, sq, ' Gpλ, sq " |λ, sy mod qF s,R .

The basis pGpλ, sqq of V s is the canonical basis of V s . As for quasimonomial vectors, we can evaluate at q " 1 and obtain (virtual) characters of Gpl, 1, nq. It is expected that these characters should coincide with the Calogero-Moser cellular characters.

Remark 3.5. The cylindrical multipartitions are the multipartitions which naturally indexed the crystal basis of V s in the case where s 1 ě s 2 ě ¨¨¨ě s l . In the general case where s P Z l , the analogous multipartitions are the images of the cylindrical multipartitions by certain combinatorial maps described in [START_REF] Jacon | Crystal isomorphisms for irreducible highest weight U q p p slpeq -modules of higher level[END_REF].

3.3.

Another description of the Fock space. We follow the description of the Fock space given in [START_REF] Leclerc | Constructible characters and canonical bases[END_REF]. Let V pΛ k q be the irreducible integrable representation of U q psl 8 q of highest weight Λ k . A basis of V pΛ k q is given by pv β q β where β runs in the set of sequences pβ j q jďk such that β j´1 ă β j for all j ď k and β j " j for j ! 0. The action of U q psl 8 q on V pΛ k q is given by

E i v β " # v γ if i R β and i `1 P β, with γ " pβzti `1uq Y tiu, 0 otherwise, F i v β " # v γ if i P β and i `1 R β, with γ " pβztiuq Y ti `1u, 0 otherwise, K i v β " $ ' & ' % qv β if i P β and i `1 R β, q ´1v β if i R β and i `1 P β, v β otherwise.
Given s " ps 1 , . . . , s l q P Z l , the Fock space F 1 s is the tensor product F 1 s " V pΛ s 1 q b ¨¨¨b V pΛ s l q, the U q psl 8 q-action is given by the coproduct. A basis of F 1 s is given by pv S q S , with S a l-symbol S "

¨β1

. . .

β l ',
where β i " pβ i,j q jďs i is a sequence of integers such that β i,j´1 ă β i,j for all j ď s i and β i,j " j for j ! 0. The vector v S is the tensor product v β 1 b ¨¨¨b v β l .

Such symbols are in bijection with l-partitions. Given such a symbol S we associate a l-partition λ S " pλ p1q , . . . , λ plq q by setting λ pkq i " β k,s k ´i`1 ´sk `i ´1.

Thanks to the assumptions on the entries of the symbol S, we do have λ pkq i ě λ pkq i`1 and λ pkq i " 0 for i " 0. It is then easy to see that S Þ Ñ λ S is a bijective map. We can then detect the addable and removable nodes of the l-partition λ S from the symbol S. Indeed, a node γ " pa, b, cq is removable if and only if β c,sc´a`1 " b ´a `sc `1 and b ´a `sc does not appear in β c . Similarly, a node γ " pa, b, cq is addable to λ if and only if β c,sc´a`1 " b ´a `sc and b ´a `sc `1 does not appear in β c . Since b ´a `sc is nothing more than the charged content of γ, we can also recover the content of a removable or addable node from the symbol S.

'.

It corresponds to the 3-partition λ S " p4.3.1, 3, 1 3 q whose Young diagram is

rλ S s " ¨3 4 5 6 2 3 4 1 , 1 2 3 , 2 1 0 ‹ ‹ ‹ ‹ ‹ '
the entries corresponding to the charged content. Removable nodes are in gray and correspond to the entries 2, 5 and 7 in the first row of S, to the entry 4 in the second row of S and to the entry 1 in the last row of S.

Proposition 3.7. The above bijection between l-symbols and l-partitions induces an isomorphism of U q psl 8 q-modules between F 1 s and F s given by v S Þ Ñ |λ S , sy. Proof. It suffices to check that the isomorphism of vector spaces v S Þ Ñ |λ S , sy is U q psl 8 qequivariant, which is easily checked.

The vector |H, sy then corresponds to the vector v S 0 where S 0 is the symbol such that its i-th row is the sequence pjq jďs i .

Finally, note that detecting a cylindrical multipartition is easy from its corresponding symbol: the l-partition λ s is cylindrical if and only if the symbol S is standard, that is if β i,k ě β i,k`1 for all 1 ď k ă l and i ď s k`1 .

Comparison with cellular characters.

Hypothesis: We suppose that the parameter c is such that there exists k P C with lk i P Z `k for every i and that c 0 " ´1 l .

We set k 1 " pk 1 1 , . . . , k 1 l q P Z l where k 1 c " lk 1´c ´k. Therefore, given a standard tableau t of shape λ, the shift of the Jucys-Murphy element J i ´k acts on the line D t by multiplication by the charged content of t ´1piq.

Theorem 3.8. The set of Jucys-Murphy cellular characters of Gpl, 1, nq for the parameter c coincide with the characters obtained by evaluation at q " 1 of quasimonomial vectors of

F n k 1 . Proof. If v P F n´1 k 1
and v 1 " F i ¨v P F n k 1 are quasimonomial vectors, then it follows from the definition of the action and the particular choice of parameter that γ v 1 " i Índ

Gpl,1,nq Gpl,1,n´1q pγ v q. Therefore the characters of Gpl, 1, nq obtained from specializations at q " 1 of quasimonomial vectors coincide with the set E n introduced in Section 2.3, since the only quasimonomial vector of F 0 k 1 is |H, k 1 y. Therefore, we conclude using Proposition 2.7.

Using Proposition 2.5, we deduce that the characters obtained by evaluation at q " 1 of quasimonomial vectors of F n k 1 are sums of Calogero-Moser c-cellular characters. The second author has also formulated the following conjecture concerning vectors of the canonical basis of V s : Conjecture 3.9. The set of Calogero-Moser cellular characters of Gpl, 1, nq for the parameter c coincide with the characters obtained by evaluation at q " 1 of the vectors of height n of the canonical basis of V s .

The type B and Lusztig's constructible characters

In this Section, we take l " 2 so that the complex reflection group Gpl, 1, nq is isomorphic to the Weyl group of type B n . We aim to compare, using results of Leclerc-Miyachi [START_REF] Leclerc | Constructible characters and canonical bases[END_REF], the Jucys-Murphy cellular characters and Lusztig's constructible characters.

Lusztig's constructible characters depend also on a choice of parameters for each conjugacy class of reflections. We will always take the parameter equal to 1 for the conjugacy class S 0 and the parameter for the conjugacy class S 1 is denoted by r and will be explicitely related to the multicharge s. Theorem 4.1 ([20, Theorem 10 and Proposition 4]). Let s " ps 1 , s 2 q P Z 2 with s 1 ě s 2 . Then the characters of the Weyl group of type B obtained from specializations at q " 1 of the vectors of the canonical basis of V s coincide with Lusztig's constructible characters at parameter r " s 1 ´s2 .

Moreover, every element in the canonical basis of V s is of the following form:

Gpλ, sq " F pr k q i k ¨¨¨F pr 1 q i 1 ¨vS 0 ,
where i j P Z and r j P t1, 2u for all 1 ď j ď k.

Because of the divided powers, it is not clear if the vectors of the canonical basis are monomial or not. Proposition 4.2. Let s " ps 1 , s 2 q P Z 2 with s 1 ą s 2 . For any n P N, any k P N, any i 1 , . . . , i k P Z and any r 1 , . . . , r k P t1, 2u such that ř k i"1 r i " n there exist j 1 , . . . , j n such that F

pr k q i k ¨¨¨F pr 1 q i 1
¨vS 0 " F jn ¨¨¨F j 1 ¨vS 0 . Proof. We proceed by induction on n. There is nothing to prove if n " 0 or if n " 1. Note that since s 1 ą s 2 , we have F p2q i ¨vS 0 " 0, which also settles the case n " 2. Now suppose that n ě 3 and that the result is proven for every l ď n´1. Fix i 1 , . . . , i k P Z and r 1 , . . . , r k P t1, 2u such that ř k i"1 r i " n. Of course, we suppose that F

pr k q i k ¨¨¨F pr 1 q
i 1 ¨vS 0 is non-zero, otherwise there is nothing to show.

If r k " 1, then the induction hypothesis gives us j 1 , . . . , j n´1 such that

F pr k´1 q i k´1 ¨¨¨F pr 1 q i 1 ¨vS 0 " F j n´1 ¨¨¨F j 1 ¨vS 0 ,
and acting by F i k shows that

F i k F pr k´1 q i k´1 ¨¨¨F pr 1 q i 1 ¨vS 0 " F i k F j n´1 ¨¨¨F j 1 ¨vS 0
has the desired form. Finally, suppose that r k " 2. We use the induction hypothesis to obtain j 1 , . . . , j n´2 such that F

pr k´1 q i k´1 ¨¨¨F pr 1 q i 1 ¨vS 0 " F j n´2 ¨¨¨F j 1 ¨vS 0 ,
and we now discuss on the value of j n´2 . If

j n´2 R ti k ´1, i k , i k `1u then F p2q
i k and F j n´1 commute and then

F p2q i k F pr k´1 q i k´1 ¨¨¨F pr 1 q i 1 ¨vS 0 " F j n´2 F p2q i k F j n´3 ¨¨¨F j 1 ¨vS 0 .
We once again apply the induction hypothesis to F p2q i k F j n´3 ¨¨¨F j 1 ¨vS 0 to reach the conclusion.

Since we have supposed that

F p2q i k F pr k´1 q i k´1 ¨¨¨F pr 1 q i 1
¨vS 0 is non-zero, we have j n´2 ‰ i k . Indeed, F 3 i k acts by zero on the Fock space by Remark 3.1. If j n´2 " i k ´1, we want to use the quantum Serre relation

F p2q i k F j n´2 " F i k F j n´2 F i k Fj n´2 F p2q
i k and we show that F p2q i k acts by zero on F j n´3 ¨¨¨F j 1 ¨vS 0 . If a symbol S " `β γ ȋs such that v S appears in F j n´3 ¨¨¨F j 1 ¨vS 0 with a non zero coefficient, then the value of β X γ and β Y γ does not depend on the chosen symbol. Then i k R β X γ, otherwise F i k´1 would act by zero on F j n´3 ¨¨¨F j 1 ¨vS 0 . Therefore F p2q i k acts by zero on F j n´3 ¨¨¨F j 1 ¨vS 0 and we have

F p2q i k F j n´2 F j n´3 ¨¨¨F j 1 ¨vS 0 " F i k F j n´2 F i k F j n´3
¨¨¨F j 1 ¨vS 0 , which has the expected form.

If j n´2 " i k `1, we can proceed similarly: we show that i k `1 P β Y γ and then

F p2q i k
acts by zero on F j n´3 ¨¨¨F j 1 ¨vS 0 .

Example 4.3. If we take the multicharge s " p1, 0q then the vector F p2q 0 F 1 ¨vS 0 is an element of the canonical basis. Since F p2q 0 acts by zero on v S 0 , we have F p2q 0 F 1 ¨vS 0 " F 0 F 1 F 0 ¨vS 0 , and we obtain a monomial vector.

If the multicharge s is of the form ps, sq, we have almost the same result. Proposition 4.4. Let s " ps, sq P Z 2 . For any n P N, any k P N, any i 1 , . . . , i k P Z and any r 1 , . . . , r k P t1, 2u such that ř k i"1 r i " n and pr 1 , . . . , r k q ‰ p2, . . . , 2q, there exist j 1 , . . . , j n such that F

pr k q i k ¨¨¨F pr 1 q i 1 ¨vS 0 " F jn ¨¨¨F j 1 ¨vS 0 .
If n " 2k is even and pr 1 , . . . , r k q " p2, . . . , 2q then the vector F p2q i k ¨¨¨F p2q i 1 ¨vS 0 is either zero or of the form v S for S a symbol of the form `β β ˘.

Proof. The strategy is the same as in the proof of Proposition 4.2. We proceed by induction on n and the only difference being the case of a non-zero vector of the form

F i k F p2q i k´1 ¨¨¨F p2q i 1 vS
0 . Indeed, we cannot apply the induction hypothesis to

F p2q i k´1 ¨¨¨F p2q i 1 ¨vS 0 . As F i k F p2q i k´1 ¨¨¨F p2q i 1 ¨vS 0 is non-zero, we have i k ‰ i k´1 . If i k ‰ i k´1 ˘1, then F i k and F p2q i k´1 commute and we rewrite F p2q i k´1 F i k F p2q i k´2 ¨¨¨F p2q i 1 ¨vS 0 as in the proof of Proposition 4.2. If i k " i k´1 ´1, as we supposed that F i k F p2q i k´1 ¨¨¨F p2q i 1 ¨vS 0 , the vector F p2q i k´2 ¨¨¨F p2q i 1 ¨vS 0 is equal to v S with S " `β β ˘with i k `1 " i k´1 P β. Therefore F p2q i k ¨vS " 0 and F i k F p2q i k´1 ¨¨¨F p2q i 1 ¨vS 0 " F i k´1 F i k F i k´1 F p2q i k´2 ¨¨¨F p2q i 1 ¨vS 0 ,
and we once again conclude by induction. The case i k " i k´1 `1 is similar.

The two previous propositions enable us te get rid of the divided powers in almost all cases.

Corollary 4.5. Let s " ps 1 , s 2 q P Z 2 be a multicharge with s 1 ě s 2 .

(1) If s 1 ą s 2 then any vector of the canonical basis of V s is monomial.

(2) If s 1 " s 2 and then any vector of the canonical basis of V s different from v S with S " `β β ˘is monomial.

Theorem 4.6. Lusztig's constructible characters in type B at parameter r are sums of Calogero-Moser cellular characters at parameters c with c 0 " ´1{2 and c 1 " ´r{2.

Proof. In order to be consistent with the hypothesis of Section 3.4, we take k " ´r{2 so that pk 1 1 , k 1 2 q " pr, 0q. If γ is a Lusztig's constructible character, then thanks to Theorem 4.1, γ is obtained from the evaluation at q " 1 of a vector of the canonical basis of V s .

If r ‰ 0 Corollary 4.5 ensures that γ is obtained from the evaluation at q " 1 of a monomial vector of the Fock space. Therefore, thanks to Theorem 3.8, γ is a Jucys-Murphy constructible character, which is a sum of Calogero-Moser c-cellular characters, see Proposition 2.5.

If r " 0, then the same argument shows that any Lusztig's constructible character different from χ pλ,λq is a sum of Calogero-Moser c-cellular characters. But the character χ pλ,λq is alone in its Calogero-Moser family. Indeed, the two rows of the 2-symbol S λ corresponding to the bipartition pλ, λq are equal: there exists only one symbol with the same content of S λ and the Calogero-Moser family is determined by the content, see [25, Theorem 3.9 and 3.13]. Hence, using [7, Proposition 7.7.1 and Theorem 13.5.1], we deduce that χ pλ,λq is also a Calogero-Moser c-cellular character.

5. Calogero-Moser families and b-invariant 5.1. Calogero-Moser families. Using the representation theory of restricted rational Cherednik algebras at t " 0, Gordon [START_REF] Gordon | Baby Verma modules for rational Cherednik algebras[END_REF] defines the notion of Calogero-Moser families of characters for a complex reflection group W . It consists in a partition of the set of irreducible representations for W , which, in the case of Weyl group, conjecturally correspond to the notion of Lusztig family. This conjecture is known to hold in the case where W is a classical Weyl group thanks to a case by case analysis. Now, for each irreducible representation E of W , one can define its fake degree f E pqq which is a polynomial in one indeterminate q. This object is connected with the invariant theory of W . The valuation of f E pqq is denoted by b E and is known as the b-invariant of E. In [START_REF] Bonnafé | Cherednik algebras and Calogero-Moser cells[END_REF], Bonnafé and Rouquier have proved that in each Calogero-Moser family, there is a unique element with minimal b-invariant. Our aim is to recover this result and to give the explicit form of this element in the case where W " Gpl, 1, nq.

5.2.

The case W " Gpl, 1, nq. We first describe the Calogero-Moser family in this case and give an explicit formula for the b-invariant of an irreducible character in terms of symbols. Let l P N ą0 and let s " ps 1 , . . . , s l q P N l . We will suppose that s 1 ě s 2 ě ¨¨¨ě s l and consider the set of l-symbols Symbpsq of multicharge s. The symbols considered in this section are slightly diffrent from the ones introduces in Section 3, and we stress the difference of conventions. First, the multicharge is allowed to have only non-negative values. Then, the i-th row of an l-symbol will be a finite increasing sequence of s i integers. Infinite symbols as in Section 3 have the advantage to be in bijection with all l-partitions of all integers, whereas there are some restriction for finite symbols: we can only recover partitions of integers with the i-th component of length at most s i . Therefore, the irreducible representations of Gpl, 1, nq may be parametrized by Symbpsq as soon as s l ě n. In the following, we will need this assuption on s l every time we want to describe an irreducible representation by its associated symbol. We now write a symbol S P Symbpsq as follows:

S " ¨β1 1 . . . β 1 s 1 ´s2 β 1 s 1 ´s2 `1 . . . β 1 s 1 ´sl β 1 s 1 ´sl `1 . . . β 1 s 1 β 2 1 . . . β 2 s 2 ´sl β 2 s 2 ´sl `1 . . . β 1 s 2 . . . . . . . . . . . . . . . β l 1 . . . β l s l ‹ ‹ ' .
Given S P Symbpsq, we define bpSq P N and b 1 pSq P N by bpSq "

n ÿ i"1 s i ÿ j"1 plps i ´jq `i ´1qpβ i j ´j `1q and b 1 pSq " n ÿ i"1 s i ÿ j"1 plps i ´jq `i ´1qβ i j .
Then bpSq is the b-invariant associated with the simple representation parametrized by S. This can be deduced from the computation of the fake degrees in [23, Bemerkung 2.10]. This result can be thought as an extension of a similar property for Lusztig's constructible characters for Coxeter groups, see [START_REF] Bonnafé | Constructible characters and b-invariant[END_REF].

Thanks to computer calculations, we conjecture some similar properties for the vectors of the canonical basis. Conjecture 6.1. We fix s " ps 1 , . . . , s l q P Z l and consider λ a cylindric l-partition. We expand Gpλ, sq along the standard basis: Gpλ, sq " ÿ µ a λ,µ pqq|µ, sy, with a λ,µ pqq P Zrq, q ´1s.

Then there exists a unique ν with minimal b-invariant among the µ such that a λ,µ pqq ‰ 0.

If one considers Lusztig's constructible characters at equal parameters, given a Lusztig's family of characters F, every constructible character whose underlying family F share an irreducible component, which is the character of minimal b-invariant in the family F. We also conjecture a similar behavior for the canonical basis, for specific parameters s. Let us denote by mpλq the l-partition ν of the previous conjecture. Conjecture 6.2. Suppose that the parameter s is of the following form: s " ps `1, . . . , s 1, s, . . . , sq. Let λ be a cylindrical l-partition. Then mpλq is the symbol of minimal content in the Calogero-Moser family containing λ. This conjecture asserts that the constituent with minimal b-invariant is the same if the two vectors of the canonical basis correspond to the same family of symbols.

We finally give a last conjecture, which corresponds to [START_REF] Geck | Left cells and constructible representations[END_REF]Lemma 4.6] and [24, Remarque 5.7]. This conjecture is related with the so-called spetsial parameters. These parameters should be the correct generalization of the equal parameters for Hecke algebras associated with equal parameters, see [START_REF] Bonnafé | Cherednik algebras and Calogero-Moser cells[END_REF]Chapter 8] for the correspondance of parameters between rational Cherednik algebras and cyclotomic Hecke algebras. In terms of the multicharge s, these spetsial parameters correspond to the case s " ps `1, s, . . . , sq. Note that symbols of shape given by s are considered in [START_REF] Malle | Unipotente Grade imprimitiver komplexer Spiegelungsgruppen[END_REF], where they parametrize the unipotent degrees of the principal family for the imprimitive complex reflection group Gpl, 1, nq.

Let s E be the Schur element associated with the representation E of the Ariki-Koike algebra at spetsial parameters. These elements have been explicitly computed, see [START_REF] Chlouveraki | Rouquier blocks of the cyclotomic Ariki-Koike algebras[END_REF]Theorem 2.4] for a factorization of these Schur elements. The leading coefficient of s E is denoted by ξ E . Since the Ariki-Koike algebra is a flat deformation of the group algebra

Proposition 2 . 5 (

 25 [START_REF] Lacabanne | On a conjecture about cellular characters for the complex reflection group Gpd, 1, nq[END_REF] Theorem 1.10]). Every Jucys-Murphy cellular character is a sum of Calogero-Moser cellular characters.

Example 3 . 6 .

 36 Consider the multicharge p3, 1, 2q and the following symbol S S "

6. 2 .

 2 Canonical basis and b-invariant. In [7, Theorem 11.4.2(c)], Bonnafé and Rouquier have proved that every irreducible constituent of a Calogero-Moser c-cellular character belongs to the same Calogero-Moser family. Therefore, one can define the Calogero-Moser family associated to a Calogero-Moser c-cellular character. Note that vectors of the standard basis appearing in the expression of the a vector of the canonical basis have a similar behaviour. As for Calogero-Moser families, Bonnafé and Rouquier have proven a result in relation with the b-invariant: every Calogero-Moser c-cellular characters have a unique irreducible constituent of minimal b-invariant.

A program for the computation of monomial and canonical basis elements is available in http://ftp. math.rwth-aachen.de/homes/CHEVIE/contr/arikidec.g

Acknowledgements. A.L. is supported by a PEPS JCJC grant from INSMI (CNRS), N.J. is supported by ANR AHA 18-CE40-0001 and ANR CORTIPOM 21-CE40-001. The authors would like to thank Cédric Bonnafé, Maria Chlouveraki, Jérémie Guilhot and Dario Mathia for useful discussions.

It is clear that for two symbols S and S 1 associated with multipartitions with the same length, we have: bpSq ď bpS 1 q ðñ b 1 pSq ď b 1 pS 1 q.

Given a symbol S P Symbpsq, we associate a sequence zpSq by zpSq " pβ 1 1 , . . . , β 1 s1´s2 , β 2 1 , β 1 s1´s2`1 , . . . , β 2 s2´s3 , β 1 s1´s3 , β 3 1 , β 2 s2´s3`1 , β 1 s1´s3`1 , . . . , β l 1 , β l´1 1 , . . . , β 1 1 q. This sequence is obtained from the symbol S by reading the entries from bottom to top and from left to right.

Let W be the tuple pw 1 , . . . , w l q such that b 1 pSq "

‹ '

Note that zpW q is a strictly decreasing sequence (eventhough W in not a symbol, the definition of zpW q still makes sense).

Then the two symbols S and T in Symbpsq belong to the same Calogero-Moser family if the underlying multiset is the same; this defines a partition of Symbpsq into families of symbols.

Fix F a family of symbols with underlying multiset of entries E. We define a symbol S F P F through the sequence zpS F q:

(1) we set zpS F q 1 to be the minimum of E, (2) suppose that we have defined zpS F q 1 , . . . , zpS F q i ; then we set zpS F q i`1 to be the minimal element x of EztzpS F q 1 , . . . , zpS F q i u such that there exists a symbol S P F with associated sequence zpSq starting with pzpS F q 1 , . . . , zpS F q i , xq.

Example 5.1. Here, we take the multicharge p5, 5, 3q and the family with underlying multiset t0, 0, 0, 1, 1, 2, 2, 5, 7, 8, 9, 11, 12u. Then we have zpS F q " p0, 0, 1, 1, 0, 2, 2, 5, 7, 8, 9, 11, 12q and the corresponding symbol is

Theorem 5.2. Let F Ă Symbpsq be a family of symbols. Then bpS F q ď bpSq for all S P F with equality if and only if S " S F .

Let be the order on Symbpsq induced by the dominance order on the associated sequences:

From the definition of b 1 , it is easy to see that b 1 pSq "

Therefore if S T then b 1 pSq ď b 1 pT q and if S T and b 1 pSq " b 1 pT q then S " T . Hence Theorem 5.2 will follow immediately from the next proposition.

Proposition 5.3. Let F be a family of Symbpsq. Then S F S for all S P F.

Proof. Let S P F different from S F and consider 1 ď j ď s 1 `¨¨¨`s l minimal such that zpSq j ‰ zpS F q j . We construct a symbol T such that T S and zpT q i " zpS F q i for all 1 ď i ď j. By induction, this implies that S F T S.

Let j ă k ď s 1 `. . . `sl be minimal such that zpSq k " zpS F q j . We consider the two rows S a and S b of S such that β a p is the entry zpSq j and β b q the entry zpSq k . Note that, by definition of S F , we must have zpSq j ą zpS F q j and zpS F q j does not appear in S a .

Since k ą j, we have s b ´q `1 ď s a ´p `1. Therefore there exists p 1 ě p such that β a p 1 ‰ β b r for all r ě q. Indeed, if β a p 1 belongs to S b for all p 1 ě p, then these s a ´p `1 different integers all bigger that zpSq k " β b q appear at the right of β p q . But there are s b ´q ă s a ´p `1 integers to the right of β b q . Now that we have find an entry in S a at the right of zpSq j (zpSq j included) which does not appear in S b , we now define the symbol T from S. Denote by q 0 " q, q 1 , . . . , q p 1 ´p´1 the integers such that β a p`r " β b q r`1 for all 0 ď r ď p 1 ´p ´1. Note that β b q 0 " β b q cannot appear in S a otherwise, it would appear at the left of β a p and thus it would appear with multiplicity 2 in row S a F . Let T be the symbol obtained by exchanging β a p`r with β b qr for 0 ď r ď p 1 ´p and increasingly reordering the row in position b. Since zpT q is obtained from zpSq by a successive exchange of pairs px, yq with x ą y and x appearing before y, we have T S. Moreover, by construction, the j ´1 first entries of zpT q and zpSq coincide with those of zpS F q and zpT q j " zpS F q j . 6. Conjectures 6.1. Monomial and canonical bases. In section 4, we have seen that every canonical basis elements are in fact monomial vectors in the case where l " 2.

We conjecture that this is also the case when l " 3 even if a closed formula for them might been more difficult to obtain than in the case l " 2. When l ą 2, the canonical basis elements are not monomial in general, but such counter-examples are not easy to find.

From explicit computer calculations 2 , we found the following counter-example for l " 5 with the multicharge s " p3, 2, 2, 1, 0q and the multipartition λ " pp3q, p3q, p1q, H, Hq. The corresponding element of the canonical basis Gpλ, sq is not a monomial vector of the Fock space.

of Gpl, 1, nq, the irreducible representations are parametrized by l-partitions of n, and we write ξ λ instead of ξ E if E is the representation indexed by λ. Conjecture 6.3. Let s be ps `1, s, . . . , sq the spetsial parameters. Let λ be a cylindrical l-partition. Then a λ,mpλq p1q " ÿ µ a λ,µ p1q ξ µ .