Yang Gao

Ryo Iwasaki

Daiki Hamana

Junya Iihama

Sawao Honda

Munni Kumari

Philippe Thomas

Tomokatsu Hayakawa

Samuel Bernard

Yuji Iwamoto
email: iwamoto.yuji@nitech.ac.jp

Green emitting 𝛃-SiAlON:Eu 2+ phosphors derived from chemically modified perhydropolysilazanes

Keywords: europium, oxynitride, photoluminescence, polymer precursor, SiAlON 1

We report the first synthesis of β-SiAlON:Eu 2+ phosphors from singlesource precursors, perhydropolysilazane (PHPS), chemically modified with Al(OCH(CH 3 ) 2 ) 3 , and EuCl 2 . The reactions occurring during the precursor synthesis and the subsequent thermal conversion of polymeric precursors into β-SiAlON:Eu 2+ phosphors have been studied by a complementary set of analytical techniques, including infrared spectroscopy, gas chromatography-mass spectrometry, thermogravimetry-mass spectrometry, X-ray diffraction (XRD), photoluminescence spectroscopy, and scanning electron microscopy. It has been clearly established that Al(OCH(CH 3 ) 2 ) 3 immediately reacted with PHPS to afford N-Al bonds at room temperature, whereas N-Eu bond formation was suggested to proceed above 600 • C accompanied by the elimination of HCl up to 1000 • C in flowing N 2 . The subsequent 1800 • C-heat treatment for 1 h under an N 2 gas pressure at 980 kPa allowed converting the single-source precursors into fine-grained β-SiAlON:Eu 2+ phosphors. XRD analysis revealed that the Al/Si of .09 was the critical atomic ratio in the precursor synthesis to afford singlephase β-SiAlON (z = .55). Moreover, Eu 2+ -doping was found to efficiently reduce the carbon impurity in the host β-SiAlON. The polymer-derived β-SiAlON:Eu 2+ phosphors exhibited green emission under excitation at 460 nm and achieved the highest green emission intensity at the critical dopant Eu 2+ concentration at 1.48 at%.

INTRODUCTION

luminous efficiency, flexible color design, and relatively low cost. [START_REF] Wang | Downconversion nitride materials for solid state lighting: recent advances and perspectives[END_REF] Current high-quality white light has been Solid-state lighting (SSL) solution has been worldwide spread to replace the traditional illumination source, such as incandescent and fluorescent lamps, to save a huge amount of energy and reduce the carbon emission. Among the SSL sources, phosphors-converted white LEDs have attracted extensive attention because of their great developed by assembling the blue LED tips with green and red phosphors (two-pc-wLEDs). The green phosphors applied for the two-pc wLEDs are oxynitride phosphors based on Eu 2+ -activated β-SiAlON, which have been recently developed to be substituted for sulfide-based phosphors.

β-SiAlON with the general formula Si 6-z Al z O z N 8-z (0 ≤ z ≤ 4.2) is a solid solution of β-silicon nitride (Si 3 N 4 ) and aluminum oxide (Al 2 O 3 ) formed by partial replacement of Si-N bond with Al-O or Al-N bond, whereas Eu 2+ -activated β-SiAlON can be expressed by the modified formula of Si 6-z Al z O z -2 y N 8-z +2 y :yEu 2+ , which takes into account the charge compensation by Eu 2+ -doping. [START_REF] Wang | Elucidating structurecomposition-property relationships of the β-SiAlON:Eu 2+ phosphor[END_REF] Photoluminescent (PL) properties of Eu 2+ -activated β-SiAlON (β-SiAlON:Eu 2+ ) phosphors depend on several material factors, including z value, [START_REF] Takahashi | Toward higher color purity and narrower emission band β-sialon:Eu 2+ by reducing the oxygen concentration[END_REF] amount of Eu 3+ as killer site, [START_REF] Li | Achieving high quantum efficiency narrow-band β-sialon:Eu 2+ phosphors for high-brightness LCD backlights by reducing the Eu 3+ luminescence killer[END_REF] and morphology and the phase purity of host β-SiAlON. [START_REF] Zhu | The z value dependence of photoluminescence in Eu 2+ -doped β-SiAlON (Si 6-z Al z O z N 8-z ) with 1≤ z≤ 4[END_REF] Hirosaki et al. fabricated β-SiAlON:Eu 2+ phosphors with several z values and Eu 2+ concentrations. They found that impurity SiAlON polytypoid phases, such as 12H, 21R, and 27R, appeared either by increasing z value or Eu 2+ molar percentage. [START_REF] Xie | Synthesis and photoluminescence properties of β-sialon:Eu 2+ (Si 6-z Al z O z N 8-z : Eu 2+ ): a promising green oxynitride phosphor for white lightemitting diodes[END_REF] Based on these results, they suggested that critical Eu 2+ concentrations are ∼.7 and .5 at% for the host β-SiAlON with z = .1 and .5, respectively, and β-SiAlON:Eu 2+ phosphors with z values ranging from .1 to .5 are suitable for practical applications. [START_REF] Xie | Synthesis and photoluminescence properties of β-sialon:Eu 2+ (Si 6-z Al z O z N 8-z : Eu 2+ ): a promising green oxynitride phosphor for white lightemitting diodes[END_REF] Kikkawa et al. reported that the morphology of β-SiAlON ceramics changed from rod-like to near-equiaxed shape at the z value from 1.0 to 2.0, whereas the green emission color shifted to a lower wavelength consistently with the z value and finally turned to be blue at z = 4. [START_REF] Zhu | The z value dependence of photoluminescence in Eu 2+ -doped β-SiAlON (Si 6-z Al z O z N 8-z ) with 1≤ z≤ 4[END_REF] β-SiAlON ceramics can be fabricated by the conventional powder metallurgy method using starting powders of α-Si 3 N 4 , α-Al 2 O 3 and aluminum nitride (AlN). Generally, fabrication conditions of high-temperature heat treatment at T ≥ 1800 • C under a high nitrogen (N 2 ) pressure at p ≥ .98 MPa are required to afford β-SiAlON as a thermodynamically favorable phase. [START_REF] Takahashi | Toward higher color purity and narrower emission band β-sialon:Eu 2+ by reducing the oxygen concentration[END_REF][START_REF] Xie | Synthesis and photoluminescence properties of β-sialon:Eu 2+ (Si 6-z Al z O z N 8-z : Eu 2+ ): a promising green oxynitride phosphor for white lightemitting diodes[END_REF] On the other hand, a novel polymer-derived ceramics (PDCs) route has been found as convenient and appropriate for the synthesis of silicon (Si)-based non-oxide ceramic systems. [START_REF] Kroke | Si 3 N 4 -TiN-Y 2 O 3 ceramics derived from chemically modified perhydropolysilazane[END_REF][START_REF] Colombo | Polymer-derived ceramics: 40 years of research and innovation in advanced ceramics[END_REF][START_REF] Ionescu | Silicon-containing polymer-titanosilazanes and conversion into Si 3 N 4 -TiN ceramics[END_REF][START_REF] Iwamoto | Preceramic polymers as precursors of advanced ceramics: the polymer-derived ceramics (PDCs) route[END_REF] The PDCs route offers several potential advantages, including compositional homogeneity, purity, and structure tailorability, which can be achieved by an in situ formation of polycrystalline ceramics from a novel metastable amorphous compound produced by pyrolysis of a single-source precursor, in which strictly controlled chemical composition can be established at molecular scale. [START_REF] Sanchez | Molecular engineering of functional inorganic and hybrid materials[END_REF][START_REF] Clément | Sol-gel chemistry: from molecule to func-26[END_REF][13][START_REF] Lale | Polymer-derived ceramics with engineered mesoporosity: from design to application in catalysis[END_REF] In this route, perhydropolysilazane (PHPS) composed of only Si, N, and H has some advantages in high purity for the synthesis of Si 3 N 4 ceramics. [START_REF] Colombo | Polymer-derived ceramics: 40 years of research and innovation in advanced ceramics[END_REF][START_REF] Ionescu | Silicon-containing polymer-titanosilazanes and conversion into Si 3 N 4 -TiN ceramics[END_REF] Moreover, PHPS can be easily modified with various organometallic compounds to afford single-source precursors containing appropriate elements for Si-based multicomponent non-oxide ceramic systems. [START_REF] Funayama | Synthesis of a polyborosilazane and its conversion into inorganic compounds[END_REF][START_REF] Funayama | Synthesis and pyrolysis of polyaluminosilazane[END_REF][START_REF] Iwamoto | Microstructural development of Si 3 N 4 ceramics derived from polymer precursors[END_REF][START_REF] Iwamoto | Microstructural development of Si 3 N 4 -SiC-Y 2 O 3 ceramics derived from polymeric precursors[END_REF][START_REF] Iwamoto | Si 3 N 4 -SiC-Y 2 O 3 ceramics derived from yttrium-modified block copolymer of perhy-2916[END_REF][20][START_REF] Tada | Low temperature in situ formation of cobalt in silicon nitride toward functional nitride nanocomposites[END_REF][22][START_REF] Iwamoto | Crystallization and microstructure development of Si 3 N 4 -Ti(C, N)-Y 2 O 3 ceramics derived from chemically modified perhydropolysilazane[END_REF][START_REF] Lale | Highly active, robust and reusable micro-/mesoporous TiN/Si 3 N 4 nanocomposite-based catalysts for clean energy: understanding the key role of TiN nanoclusters and amorphous Si 3 N 4 matrix in the performance of the catalyst system[END_REF] In Ref. [16], polyaluminosilazanes synthesized by chemical modification of PHPS with (ethyl acetoacetate) aluminum di(isopropoxide) have been successfully converted to appropriate quaternary Si-Al-O-N amorphous compound by pyrolysis at 1000 • C under flowing ammonia (NH 3 ). However, subsequent heat treatment up to 1500 • C in N 2 resulted in the partial crystallization of β-SiAlON or the formation of a mixture of β-SiAlON, J-SiAlON, and mullite (3Al 2 O 3 -2SiO 2 ) phases due to the excess oxygen introduced from the Al alkoxide modifier. [START_REF] Funayama | Synthesis and pyrolysis of polyaluminosilazane[END_REF] On the other hand, polymers containing nanometer-sized filler particles, such as polycarbosilane combined with aluminum hydroxide (Al(OH) 3 ) nanoparticles derived from Al(O-secBu) 3 [START_REF] Soraru | Synthesis and characterization of β'-SiAlON ceramics from organosilicon polymers[END_REF] and polysilazane mixed with nanofillers containing Si 3 N 4 -Al 2 O 3 -AlN 26 or Al 2 O 3 , [START_REF] Parcianello | Optimization of phase purity of β′-sialon ceramics produced from silazanes and nanosized alumina[END_REF] have been reported as useful for the formation of SiAlON-based ceramics. However, as far as we know, no research work has been dedicated to the accurate chemical composition and structure controlling at the nano-to micrometer scale of β-SiAlON:Eu 2+ phosphors through the PDCs route.

Herein, we conducted syntheses of a series of β-SiAlON:Eu 2+ phosphor samples through the PDCs route: single-source polymeric precursors, in which strictly controlled chemical composition is established at molecular scale have been synthesized by chemical modification of a commercially available PHPS ((SiH 2 -NH) n ) with aluminum triisopropoxide (Al(OCH(CH 3 ) 2 ) 3 and europium(II) chloride (EuCl 2 ).

The single-source precursors are converted to Eu 2+doped amorphous SiAlON by pyrolysis at 1000 • C under flowing N 2 . Subsequently, the polymer-derived multicomponent amorphous compounds are converted to β-SiAlON:Eu 2+ phosphors by heat treatment up to 1800 • C under an N 2 gas pressure at 980 kPa. The relationships between the chemical composition in terms of Eu 2+ concentration, phase purity and morphology of host β-SiAlON, and PL properties of the polymer-derived β-SiAlON:Eu 2+ phosphor samples are discussed based on a set of characterization techniques, including infrared spectroscopy, gas chromatography-mass spectrometric (GC-MS) analyses, simultaneous thermogravimetric-MS (TG-MS) analyses, X-ray diffraction (XRD), PL spectroscopy, and scanning electron microscope (SEM) observations. As a typical experiment, we described the synthesis of Al, Eu-modified PHPS labeled as SAE95, and its conversion to β-SiAlON:Eu 2+ phosphor: a 300-ml two-neck roundbottom flask equipped with a magnetic stirrer was charged with 20-wt% xylene solution of PHPS (10 ml, hence Si = 1.36 g, 48.4 mmol), Al(OCH(CH 3 ) 2 ) 3 (.893 g, 4.37 mmol), and EuCl 2 (.542 g, 2.43 mmol) and then refluxed for 1 h. After the reaction mixture was cooled down to room temperature (RT), the solvent was removed under vacuum at RT to give SAE95 as white solid. Subsequently, the SAE95 sample was pyrolyzed by a tube furnace (model ARF60-150-31KC, Asahi Rika, Chiba, Japan) under flowing N 2 at 1000 • C for 1 h with a heating rate of 5 • C min -1 . The pyrolyzed powder was converted to β-SiAlON:Eu 2+ phosphor by heat treatment in a graphite resistance-heated furnace (model High Multi 5000, Fujidempa Kogyo, Co., Ltd., Osaka, Japan) at 1800 • C for 1 h under an N 2 gas pressure at 980 kPa.

EXPERIMENTAL PROCEDURES

Materials and methods

3

St. Joseph, USA) were performed on the 1000 • C-pyrolyzed and subsequent 1800 • C heat-treated samples. The Al and Eu contents in the polymer-derived SiAlON:Eu 2+ samples were analyzed by using an energy-dispersive X-ray spectrometer (EDS) mounted on an SEM (model JSM-6010LA, JEOL Ltd., Tokyo, Japan). In this study, EDS element mapping analysis was performed on the powdered sample surfaces observed within an SEM image area of ∼20 × 25 μm 2 at the magnification and accelerating voltage of 5000x and 15 kV, respectively, to give an atomic percentage for all the constituent elements. For each sample, the Al/Si and Eu/Si ratios were evaluated as an average value of the mapping scans conducted for at least five different areas randomly selected. Then, the chemical composition of the polymer-derived SiAlON-based samples was determined as follows: wt% (Si + Al + Eu) = 100% -wt% (C) -wt% (N) -wt% (2)

Characterizations

Morphology of the polymer-derived β-SiAlON:Eu 2+

The chemical modification reactions of PHPS were monitored by attenuated total reflection Fourier transform infrared (ATR-FTIR) spectroscopy using a FTIR spectrometer (model FT/IR-4200IF, JASCO Corporation, Tokyo, Japan) attached with an ATR equipment (model ATR PRO 550S-S/570S-H, JASCO Corporation, Tokyo, Japan) at a resolution of 4 cm -1 .

The details of the chemical reactions during the precursor synthesis from RT to 140 • C (xylene reflux) and precursor pyrolysis up to 1000 • C were monitored by GC-MS and TG-MS analyses under He flow, respectively (model STA7200, Hitachi, Tokyo, Japan). Commercially available isopropanol (≥99%, Sigma-Aldrich Japan, Tokyo, Japan) was used as a reference for the GC-MS analyses.

XRD analysis for the heat-treated samples was performed (model X' Pert-Pro α1, PANalytical, UK) using a Cu K α radiation.

The β-phase ratio of the polymer-derived SiAlON-based samples was defined as the β/(β + α)-phase ratio evaluated by comparing the diffraction peak intensity (I) of the α-phase (1 0 2), (2 1 0) plane and β-phase (1 0 1), (2 1 0) plane [START_REF] Gazzara | Determination of phase content of Si 3 N 4 by X-ray diffraction analysis[END_REF] :

𝛽 -phase ratio = (𝐼𝛽(101) + 𝐼𝛽(210))∕[(𝐼𝛽(101) + 𝐼𝛽(210)) + (𝐼𝛼(102) + 𝐼𝛼(210))] × 100 (1)
Elemental analyses for oxygen (O), nitrogen (N) by the inert-gas fusion method (model EMGA-930, Horiba, Kyoto, Japan), and carbon (C) by the nondispersive infrared method (model CS844, LECO CORPORATION, samples was observed by using an SEM (model JSM-6010LA, JEOL Ltd., Tokyo, Japan).

The PL emission and excitation spectra were recorded at RT using a fluorescence spectrometer (model F-7000, Hitachi, Tokyo, Japan) with a xenon (Xe) lamp.

RESULTS AND DISCUSSION

Synthesis of single-source precursors and their conversion into inorganic compounds

Single-source precursor syntheses in this study were conducted for the compositions listed in Table 1. The nominal z value of the host β-SiAlON was calculated using the theoretical formula, Si 6-z Al z O z -2 y N 8-z +2 y :yEu 2+ , given by Wang et al. [START_REF] Wang | Elucidating structurecomposition-property relationships of the β-SiAlON:Eu 2+ phosphor[END_REF] To clarify the necessary amount for an in situ formation of host β-SiAlON with z value ≤.5, 6 the amount of the Al(OiPr) 3 was examined at the Al/Si ratios of .03, .06, and .09, and the resulting precursors were labeled as SA3, SA6, and SA9, respectively. On the other hand, to investigate the effect of the amount of Eu-doping on the formation and PL properties of the polymer-derived β-SiAlON:Eu 2+ , the amount of EuCl 2 for the chemical modification was examined at the Eu/Si atomic ratios of .01, .03, .05, and .07, and the synthesized precursors were labeled as SAE91, SAE93, SAE95, and SAE97, respectively. At the polymeric precursor state, the resulting anion contents, O and N need modified expression as Si 6-z Al z O z -2 y+ δ N 8-z +2+ γ :yEu 2+ . The value of differences, δ and γ for each composition are also listed in this table. As typical results, FTIR spectra of as-received PHPS, Almodified PHPS, SA9 sample, and Al, Eu-modified PHPS, SAE95 sample are shown in Figure 1A. In addition to characteristic absorption peaks attributed to PHPS at 3400 (νN-H), 2152 (νSi-H), and 1175 (δN-H) and around 832 (νSi-N-Si) cm -1 , 15-20 SA9 sample presented new peaks at 2900 (νC-H), 1172 (νC-C), 1135 (νC-O), and 1118 (rCH 3 ) cm -1 attributed to OCH(CH 3 ) 2 group 29 originated from Al(OCH(CH 3 ) 2 ) 3 ; moreover, the broadband due to νSi-N-Si showed an apparent red shift (Figure 1B). This peak shift is possibly due to the mass effect on stretching frequencies, that is, Al(OCH(CH 3 ) 2 ) 3 reacted with the NH group in PHPS to form N-Al bond, [START_REF] Funayama | Synthesis and pyrolysis of polyaluminosilazane[END_REF] and the substitution of H by the Al(OCH(CH 3 ) 2 ) 3-x led to the red shift of the Si-N-Si peak:

(SiH 2 -NH) n + Al(OR) 3 -mROH → × (SiH 2 -NH) n-m (SiH 2 -N) m Al(OR) 3-m R = CH(CH 3 ) 2 (3)
On the other hand, the FTIR spectrum of the SAE95 sample was found to be quite similar to that of the SA9 sample.

To study the chemical modification reactions in more details, the reaction of Equation ( 3) was ex situ monitored by GC-MS analyses. The gas chromatograms and mass spectra recorded for the reference sample, isopropanol and reaction solution for the synthesis of the SA6 sample are given in Figure 2. In this study, a vapor sample was taken from the reaction solution for synthesizing the SA6 sample at RT. In addition to the chromatogram peak I due to the background (mass fragments of the m/z ratios of 40 and 20 assigned to Ar applied as an inert reaction atmosphere, 32 and 28 due to O 2 and N 2 , respectively), isopropanol was identified as peak II by detecting mass fragments at the m/z ratios of 59 (C 3 H 7 O + ), 45 (C 2 H 5 O + ), 43 (C 3 H 7 + ), and 27 (C 2 H 3 + ) 30 (Figure 2A). On the other hand, the sample vapor evolved at RT exhibited one broad peak (III) around 65.48 s as well as two distinct peaks at 87.88 s (IV) and 90.64 s (V) (Figure 2B). Careful MS analyses for the peak at the retention time of 65.48 s resulted in the detection of the mass fragments attributed to isopropanol as well as those due to the background (Figure 2C), and a single chromatogram peak composed of major fragments at the m/z ratios of 59 (C 3 H 7 O + ), 45 (C 2 H 5 O + ), 43 (C 3 H 7 + ) was extracted from the broad peak III (Figure 2D). On the other hand, the peaks IV and V were identified as xylene used as reaction solvent by detecting typical mass fragments at the m/z ratios of 106, 91, 77, 65, 51, 39, and 27 and suggested to be assigned to m-and p-xylene, respectively 30 (Figure 2C). These results clearly revealed that Equation (3) immediately proceeded from RT.

High-temperature reactions during pyrolysis up to 1000 • C were further monitored by TG-MS analyses. As typical results, the thermal behaviors of SA9 and SAE95 samples are shown in Figure 3 The SAE95 sample exhibits similar thermal behavior up to 800 • C (Figure 3B(a,b)), and the second weight loss at 150-400 • C was 15%, which was also close to that of SA95 (17.5%, Figure 3A(a)). However, this sample presents an additional weight loss at 800-1000 • C (5%, Figure 3B(a)). The mass spectrum measured during the third weight loss resulted in the detection of the main gaseous component at the m/z ratios of 38 and 36 assigned to HCl (Figure 3B(b)).

As shown in Figure 4, the area of the third TICC peak at 800-1000 • C was found to be increased with the Eu/Si ratio up to .05 (SAE95 sample) in the single-source precursor synthesis at the Al/Si ratio = .09.

Figure 5 shows the thermal behavior under an inert He atmosphere of as-received EuCl Accordingly, the second endothermic peak at 852 • C was suggested to be ascribed to the melting of the thermally decomposed species such as Eu + Cl formed in situ.

These results reveal that the EuCl 2 remained without reacting with the residual alkoxide group, (=N) m Al(OR) 3-m up to ∼600 • C, then began to melt accompanied by Cl -elimination (Equation 4), which triggered the Eu-N bond formations with silylamino moiety derived from PHPS followed by the elimination of HCl as a byproduct (Equation 5).

Table 2 shows the chemical compositions of the 1000 • C pyrolyzed samples. The Al/Si and Eu/Si atomic ratios of each sample were slightly higher than the nominal ones, which was due to the volatilization of low molecular weight silicon species as detected by the TG-MS analyses shown in Figure 3.

Effect of Al content on the formation of polymer-derived β-SiAlON

2(≡ Si) 2 NH + Eu + Cl + Cl -→ (≡ Si) 2 N -Eu -N(Si ≡) 2 + 2HCl
Figure 6A shows the XRD patterns of the polymerderived SiAlON samples after the 1800 In our previous studies, the crystallization of α-Si 3 N 4 from PHPS-derived amorphous silicon nitride (Si-N) was found around 1200 • C, and the α-phase was dominant up to 1800 • C, [START_REF] Iwamoto | Crystallization behavior of amorphous silicon carbonitride ceramics derived from organometallic precursors[END_REF] whereas the αand β-phase transformation was apparently facilitated in the quaternary Si-M-O-N systems derived from M-modified PHPS to afford β-Si 3 N 4 single phase by the 1800 • C heat treatment (M = Y18, Ti 22 ).

It is recognized that the transformation between the α and ß phases of Si 3 N 4 has a reconstructive nature, [START_REF] Cao | α'-Sialon ceramics: a review[END_REF][START_REF] Hampshire | α-Sialon dropolysilazane and hydroxy-polycarbosilane[END_REF] which requires significant amounts of thermal energy for the breaking of Si-N bonds. [START_REF] Weiss | Silicon nitride ceramics: composition, fabrication 43. Lehman RE. Preparation and properties of yttriumparameters, and properties[END_REF][START_REF] Messier | The α/β silicon nitride phase[END_REF][START_REF] Jack | Chapter V -The relationship of phase diagrams to research and development of sialons[END_REF] Therefore, it is generally suggested that the αand ß-phase transformation requires a liquid phase, in which α-Si 3 N 4 grains can dissolve above ∼1400 • C to assist the necessary atomic diffusion for promoting the α-/β-phase transformation. [START_REF] Cao | α'-Sialon ceramics: a review[END_REF][START_REF] Messier | The α/β silicon nitride phase[END_REF][START_REF] Jack | Chapter V -The relationship of phase diagrams to research and development of sialons[END_REF][START_REF] Izhevskiy | Progress in 44. Hampshire S. Oxynitride glasses[END_REF] In the case of the quaternary SiAlON system, the aluminum-silicon oxynitride liquid phase was suggested to be formed in situ, [START_REF] Zhou | Preparation of Eu-doped β-SiAlON phosphors by combustion synthesis[END_REF] which could promote the α-/βphase transformation by analogy with the transformation from αto ß-Si 3 N 4 above 1400 • C and the subsequent solid solution formation to afford stoichiometric β-SiAlON [START_REF] Zhou | Preparation of Eu-doped β-SiAlON phosphors by combustion synthesis[END_REF] Accordingly, an in situ formation of β-SiAlON from the present single-source precursor-derived amorphous compound is suggested to proceed by the nucleation and crystallization of α-Si 3 N 4 or α-SiAlON and subsequent α-/β-phase transformation via the dissolution, diffusion, and re-precipitation processes through the transient liquid phase formed in situ. The amount of the transient liquid phase increased consistently with the Al and O contents, that is, the Al/Si ratio in the single-source precursor synthesis, and the resulting β-phase content defined as the β/(β + α) phase ratio (Equation 1) of the 1800 • C-heat samples was found to increase consistently with the Al/Si ratio (Tables 2 and3). These results reveal that the Al/Si ratio of .09 is required for the β-SiAlON single-phase formation through the PDCs route investigated in this study. However, the amount of carbon impurity in the 1800 • C-NSA9 sample was 6.87 wt% (Table 3), which was due to the formation of the secondary phase assigned to β-silicon carbide (β-SiC, JCPDS card 01-073-1665, Figure 6A).

(Si 6-z Al z O z N 8-z (0 ≤ z ≤ 4.2)).

Effect of Eu 2+ concentration on the formation and properties of polymer-derived β-SiAlON:Eu 2+ phosphors

XRD patterns of the Eu 2+ -doped samples are shown in Figure 6B. Regardless of the Eu/Si ratio in the single-source precursor synthesis at the Al/Si = .09, the β-SiC formation was successfully suppressed, and all the samples exhibited β-SiAlON single phase. However, at the Eu/Si ratios of .03-.07, an unknown minor impurity phase appeared, and its diffraction pattern (marked by ▼) was found to be similar to the U phase of SiAlON reported for the intergranular crystalline phase of rare-earth (Re)-doped SiAlON polycrystallites (Re = La, Nd). [START_REF] Käll | Preparation and crystal structure of U-phase Ln 3 (Si 3-x Al 3+x ) O 12+x N 2-x (x≈ 0.5, Ln = La, Nd)[END_REF][START_REF] Huang | Microstructural and mechanical characterization of pressure-less sintered AlN-polytypoid based composites by compositional design[END_REF] The chemical compositions of all the samples after the 1800 • C-heat treatment were listed in Table 3. The z and y values were calculated by the modified formula of Si 6-z Al z O z -2 y+ δ N 8-z +2 y+ γ :yEu 2+ . The carbon impurity content was significantly decreased by the doping of Eu 2+ : The carbon content decreased from 6.87 to 1.80 wt% at the Eu 2+ -doping of .41 at% (1800 • C-NSAE91 sample) and further decreased to .13 (1800 • C-NSAE95 sample) and .16 wt% (1800 • C-NSAE97 sample) at the Eu 2+ -doping of 1.48 and 2.11 at%, respectively. On the other hand, the N/Si ratio, that is, the nitrogen content of the 1800 • C-heated samples increased with increasing the amount of Eu 2+ -doping.

During the heat treatment up to 1800 • C for 1 h, alongside the in situ β-SiAlON formation, carbothermal reduction of silica (SiO 2 ) species (Equations 6-8) and subsequent nitridation reactions (Equations 9-10) proceeded as reported for SiO 2 in the presence of free carbon. [START_REF] Weimer | Mechanism and kinetics of the carbothermal nitridation synthesis of α-silicon nitride[END_REF] The results obtained in this study strongly suggest that these reactions in the polymer-derived amorphous compound were facilitated consistently with the amount of Eu 2+ -doping, which avoided the formation of SiC phase (Equation 11).

on the nitrogen content but also varies significantly by the type and amount of the dopant cation. [START_REF] Pomeroy | SiAlON glasses: effects of nitrogen on structure and properties[END_REF] As shown in Table 2, the nitrogen content of the 1000 • C-pyrolyzed samples increased with the amount of the Eu 2+ -doping. Accordingly, it is difficult to expect that the formation onset temperature of the transient liquid phase decreases consistently with the amount of Eu 2+ -doping. To clarify the dominant mechanism, further investigations are necessary; however, one possible suggestion is the significant decrease in the viscosity of the transient liquid phase by the Eu 2+ -doping, which accelerates the initial solid-state carbothermal reduction (Equation 6), and then subsequent reactions (Equations 7 and 8) to release gaseous species such as CO 2 and CO followed by nitridation reactions (Equations 10 and 11) to afford the Eu 2+ -doped β-SiAlON with the chemical composition close to the theoretical one expressed as Si 6-z Al z O z -2 y N 8-z +2 y :yEu 2+ .

As a typical microstructure observation result, SEM images of 1800 • C-NSAE91 sample (z = .55, Eu 2+ .41 at%) and 1800 • C-NSAE95 sample (z = .55, Eu 2+ 1.48 at%) are shown in Figure 7. All of the polymer-derived β-SiAlON:Eu 2+ samples exhibited typical rod-like β-SiAlON grains, and it was found that the increase in the Eu 2+ concentration did not affect the morphology of the matrix β-SiAlON.

C (s) +SiO 2 (s) → SiO (g) +CO (g) (6)
SiO 2 (s) +CO (g) → SiO (g) +CO 2 (g)

C (s) +CO 2 (g) → 2CO (g)

3C (s) +3SiO (g) +2N 2 (g) → Si 3 N 4 (s) +3CO (g) ( 9)

3SiO (g) +3CO (g) +2N 2 (g) → Si 3 N 4 (s) +3CO 2 (g) (10) 2SiO 2 (s) +6C (s) → 2SiC (s) +4CO (g) (11)
However, as shown in Figure 8, some equiaxed grains embedded within the rod-like β-SiAlON grains were observed for the 1800 • C-NSAE97 sample, and the morphology was quite similar to those previously reported for the SiAlON polytypoid phases such as 12H and 15R. [START_REF] Hou | Synthesis, phase transformation and mechanical property of SiAlON ceramics from coal gangue[END_REF] These Al-rich polytypoid phases were found to be formed in the case of excess Eu 2+ concentration in the Eu 2+ -doped SiAlON ceramics. [START_REF] Xie | Synthesis and photoluminescence properties of β-sialon:Eu 2+ (Si 6-z Al z O z N 8-z : Eu 2+ ): a promising green oxynitride phosphor for white lightemitting diodes[END_REF] In addition to the unknown phase formation (Figure 6B), phase purity of the host β-SiAlON was suggested to be decreased due to the 2.11 at% Eu 2+ -doping.

The RT PL excitation and emission spectra of the polymer-derived β-SiAlON:Eu 2+ phosphor samples are shown in Figure 9. The excitation spectra (Figure 9A) were monitored at the appropriate maximum emission wavelengths of each sample shown in Figure 9B. The emission spectra shown in Figure 9B,C were monitored under exci-Loehmann reported that the glass transition temperature (T g ) of the series of Y-Si-Al-O-N glasses varied from ∼810 to 920 • C and suggested that the T g increased consistently with the nitrogen content in the range of 1.5-7 at%. 43 Pomeroy and Hampshire 44,45 also examined the properties of various metal cation-doped SiAlON and SiM-gON glasses and suggested that, in addition to the T g , the Littleton softening temperatures [START_REF] Lofaj | SiAlON ceramics[END_REF] and the dilatometric softening temperatures also increase linearly with nitrogen content in the range of 0-8 at%, [START_REF] Pomeroy | SiAlON glasses: effects of nitrogen on structure and properties[END_REF] whereas the viscosity of these glasses measured at 875-930 • C depends not only tation at 410 and 460 nm, respectively. All the samples except the 1800 • C-NSAE97 sample show a similar shape of excitation spectrum with a dominant broad peak around 290 to 300 nm and several peaks at around 410 to 500 nm attributed to the absorptions of the host lattice and the 4f 7 -4f 6 5d 1 transitions of dopant Eu 2+ cations, respectively. [START_REF] Xie | Synthesis and photoluminescence properties of β-sialon:Eu 2+ (Si 6-z Al z O z N 8-z : Eu 2+ ): a promising green oxynitride phosphor for white lightemitting diodes[END_REF] These polymer-derived β-SiAlON:Eu 2+ phosphor samples exhibit a typical green emission under violet light excitation (λ ex = 410 nm, Figure 9B) or blue light excitation (λ ex = 460 nm, Figure 9C). Moreover, the green emission intensity was found to increase consistently 7, the 1800 • C-NSAE95 sample was composed of fine-grained polycrystallites similar to those of the 1800 • C-NSAE91 sample, whereas the green emission intensity of the 1800 • C-NSAE95 sample was found to be approximately two times higher than that of the 1800 • C-NSAE91 sample (Figure 9B,C).

On the other hand, the 1800 • C-NSAE97 sample (z = .59, Eu 2+ 2.11 at%) showed an apparent decrease in the green emission intensity and exhibited an additional weak broad peak centered around 450 nm, which reveals the different surrounding environment of Eu 2+ cations. As discussed earlier, the formation of impurity phases, such as U-phase and SiAlON polytypoid phases, were suggested for the 1800 • C-NSAE97 sample, and excess Eu 2+ cations in this sample were thought to be located in these impurity phases with lower crystal field strength, which led to the blue shift of the green emission peak due to the weaker split of the 5d energy level of Eu 2+ ions.

In terms of the green emission intensity, the critical Eu 2+ concentration of the polymer-derived host β-SiAlON with z = .55 was found to be governed by the impurity phase formation prior to the concentration quenching. However, F I G U R E 9 (A) Excitation and (B), (C) emission spectra of polymer-derived β-SiAlON:Eu 2+ phosphor samples measured at room temperature (RT) the green emission intensity was successfully enhanced by the Eu 2+ -doping up to 1.48 at%, and this value was approximately three times higher than that (.5 at%) previously suggested for the Eu 2+ -activated β-SiAlON with z = .5 fabricated by the conventional powder metallurgy method. These results suggest that the PDCs route investigated in this study offers some potential benefits for developing high-performance β-SiAlON:Eu 2+ green phosphors such as higher critical Eu 2+ concentration up to approximately 1.5 mol% for the host β-SiAlON (z = .55), and the host grain refinement easy for assembling w-LEDs without intensive pulverization process. The results can be summarized as follows:

F I G U R E 1

 1 Attenuated total reflection infrared (ATR-IR) spectra for as-received perhydropolysilazane (PHPS), Al-modified PHPS (SA9 sample), and Al, Eu-modified PHPS (SAE95 sample): (A) 4000-600 cm -1 and (B) 1240-1080 cm -1

  . The TG-curve of the SA9 sample shows two main weight loss regions at RT-150 • C (1.25%) and 150-400 • C (17.5%), and the total ion

F I G U R E 2

 2 Results of gas chromatography-mass spectrometric (GC-MS) analyses for chemical modification reaction of perhydropolysilazane (PHPS) with Al(OCH(CH 3 ) 2 ) 3 . Total ion chromatograms of (A) reference sample, isopropanol and (B) vapor sample taken from reaction solution for synthesis of SA6 sample at room temperature (RT). (C) Mass fragments recorded for each peak shown in (B) and (D) continuous in situ monitoring of gaseous species evolved from reaction solution for synthesis of SA6 sample current chromatogram (TICC) exhibits two broad peaks centered at 100 and 250 • C, respectively (Figure 3A(a)). The mass spectrum measured at 100 • C was composed of main fragments at the m/z ratios of 106, 91, 77, 65, 63, 51, 39, and 27, which are attributed to xylene, whereas those at 133 (Si 3 N 3 H 7 + :N-SiH 2 -NH-SiH 2 -NH-SiH + ), 75 (H 2 SiNHSiH 2+ ), 45 (H 2 SiNH + ), and 16 (NH 2+ ) are thought to be due to the partial decomposition of low molecular weight fraction of PHPS originally existed or formed in situ via the cleavage of the perhydrosilazane linkages 18 partially proceeded during the chemical modification of PHPS with Al(OCH(CH 3 ) 2 ) 3 (Figure 3A(b)). As the molecular ion (C 6 H 4 (CH 3 ) 2 + , m/z = 106) and the tropylium ion (C 7 H 7 + , m/z = 91) were only detected up to ∼200 • C (Figure 3A(c)), the first weight loss from RT to 150 • C was due mainly to the residual xylene used as the reaction solvent. The mass spectrum measured at 250 • C (Figure 3A(b)) exhibits three main fragments with m/z ratios at 41 ((CH 3 ) 2 CH + ), 29 (CH 3 CH 2 + ), and 16 (CH 4 + ), which reveals that the second weight loss is attributed to

4 )F I G U R E 3 GAO

 43 2 . The TG and DTA curves indicate the gradual weight loss of 16% at 600-1000 • C associated with two endothermic peaks at 744 and 852 • C. Weight loss was found to be explained by the elimination of one chlorine (Equation 4), Cl (at. wt = 35.45)/EuCl 2 (mw = 222.86) x 100 ≈ 16%), whereas the first endothermic peak at 744 • C was found close to the m.p. of EuCl 2 (738 • C 31 ). the thermal decomposition of the residual isopropoxy (OCH 2 (CH 3 ) 2 ) group. EuCl 2 → Eu + Cl + Cl - (et al. Thermal behavior under the flow of Al, (Eu)-modified perhydropolysilazane (PHPS) samples, (A) SA9 and (B) SAE95. (a) Thermogravimetric (TG) curve and total ion current chromatogram (TICC), (b) mass fragments monitored at specific temperatures of 100, 250, and 850 • C, and (c) selected MS fragments continuously monitored up to 1000 • C

F I G U R E 4

 4 The third total ion current chromatogram (TICC) peak at 800-1000 • C detected for SAE91 (Eu/Si = .01), SAE93 (Eu/Si = .03), SAE95 (Eu/Si = .05), and SAE97 (Eu/Si = .07) samples synthesized at the Al/Si = .09 F I G U R E 5 Thermogravimetric (TG) and DTA curves of as-received EuCl 2 used in this study no. 00-048-1616) and α-SiAlON (Ca 0.8 Si 9.2 Al 2.8 O 1.2 N 14,8 , JCPDS card no. 00-033-0261) are referred to label βand α-SiAlON phase. The 1800 • C-NSA3 (Al/Si = .03) and 1800 • C-NSA6 (Al/Si = .06) samples present a mixture of αand βphases, whereas the 1800 • C-NSA9 (Al/Si = .09) sample exhibits β-SiAlON single phase. F I G U R E 6 X-ray diffraction (XRD) patterns of polymer-derived (A) SiAlON and (B) Eu 2+ -doped SiAlON samples TA B L E 3 Chemical compositions of 1800 • C-heated samples Sample name Al/Si Eu/Si O N C Empirical formula z y δ γ ( r r ) ( %)

F

  electron microscope (SEM) images showing morphology of (A) 1800 • C-NSAE91 sample and (B) 1800 • C-NSAE95 F I G U R E 8 Scanning electron microscope (SEM) image of 1800 • C-NSAE97 sample showing an equiaxed grain (marked by red arrows) embedded within β-SiAlON rod-like grains with Eu 2+ concentration from .41 (1800 • C-NSAE91 sample) to 1.48 at% (1800 • C-NSAE95 sample). As shown in Figure

( 6 )

 6 The polymer-derived host β-SiAlON (z = .55) kept the fine-gained polycrystallites with rod-like morphology at the Eu 2+ -doping up to 1.48 at%, whereas the green emission intensity under violet or blue light excitation increased consistently with the amount of Eu 2+ -doping and reached the highest at the suggested critical concentration of 1.48 at%.A C K N O W L E D G M E N T SThis work was supported by the Japan Science and Technology Agency (JST) SPRING, Grant no. JPMJSP 2112. Dr. Samuel Bernard, Dr. Philippe Thomas, and Prof. Yuji Iwamoto would like to thank CNRS who financially supported the present work via the International Research 4 CONCLUSIONS Project (IRP) "Ceramics materials for societal challenges." In this work, β-SiAlON:Eu 2+ phosphors were synthesized from single-source precursors, chemically modified PHPS with Al(OCH(CH 3 ) 2 ) 3 , and EuCl 2 . The chemical modification reactions were intensively studied by a set of characterization techniques, including IR spectroscopy, ex situ reaction monitoring by GC-MS analyses up to 140 • C, and in situ thermal behavior analysis up to 1000 • C of the single-source precursors by TG-MS analyses. The single source precursors were successfully converted to β-SiAlON:Eu 2+ phosphors by 1000 • C-pyrolysis in N 2 and subsequent 1800 • C-heat treatment for 1 h under an N 2 gas pressure at 980 kPa. The relationships between the amount of Eu 2+ -doping, phase purity, grain growth and morphology of host β-SiAlON, and PL properties were discussed.

Calculated chemical composition Si in (SiH 2 -NH) n Name (PHPS) M/Si ratio Al(OCH(CH 3 ) 2 ) 3 EuCl 2 Formula Si 6-z Al z O z

  Nominal molar ratios for synthesis of single-source precursors for β-SiAlON and Eu 2+ -activated β-SiAlON phosphors

									2+
										Eu 2+
						z	y	δ	γ	(at%)
	SA3	1	.03	.00	SiAl 0.03 O 0.09 N 1.00	.17	.000	.35	-2.00	.00
	SA6	1	.06	.00	SiAl 0.06 O 0.18 N 1.00	.34	.000	.68	-2.00	.00
	SA9	1	.09	.00	SiAl 0.09 O 0.27 N 1.00	.50	.000	.99	-2.00	.00
	SAE91	1	.09	.01	SiAl 0.09 O 0.27 N 1.00 Eu 0.01 .50	.055	1.10	-2.11	.42
	SAE93	1	.09	.03	SiAl 0.09 O 0.27 N 1.00 Eu 0.03 .50	.165	1.32	-2.33	1.27
	SAE95	1	.09	.05	SiAl 0.09 O 0.27 N 1.00 Eu 0.05 .50	.275	1.54	-2.55	2.07
	SAE97	1	.09	.07	SiAl 0.09 O 0.27 N 1.00 Eu 0.07 .50	.385	1.76	-2.77	2.88

TA B L E 1 -2 y + δ N 8-z +2 y+ γ :yEu Abbreviation: PHPS, perhydropolysilazane.

  Chemical compositions of 1000 • C as-pyrolyzed samples

		Pyrolyzed	EDS (molar ratio)	Element analysis (wt%)
	Precursor	sample	Al/Si	Eu/Si	O	N	C	Empirical formula
	SA3	NSA3	.03	.00	3.08	24.92	2.16	SiAl 0.03 O 0.08 N 0.74 C 0.07
	SA6	NSA6	.07	.00	5.43	25.59	2.98	SiAl 0.07 O 0.15 N 0.83 C 0.11
	SA9	NSA9	.11	.00	6.46	21.51	9.46	SiAl 0.11 O 0.26 N 1.03 C 0.38
	SAE91	NSAE91	.11	.02	5.70	22.25	6.25	SiAl 0.11 O 0.19 N 0.84 C 0.28 Eu 0.02
	SAE93	NSAE93	.12	.03	7.60	28.11	5.08	SiAl 0.12 O 0.29 N 1.22 C 0.26 Eu 0.03
	SAE95	NSAE95	.10	.06	7.12	29.00	4.57	SiAl 0.10 O 0.30 N 1.39 C 0.25 Eu 0.06
	SAE97	NSAE97	.13	.08	7.93	26.81	4.61	SiAl 0.13 O 0.36 N 1.38 C 0.27 Eu 0.08

• C-heat treatment. (5) β-SiAlON (Si 6-z Al z O z N 8-z , z = 2:Si 4 Al 2 O 2 N 6 , JCPDS card TA B L E 2

A U T H O R C O N T R I B U T I O N S

Samuel Bernard https://orcid.org/0000-0001-5865-0047 Yuji Iwamoto https://orcid.org/0000-0002-6927-5446


Daniel Juteau

AND Édric Lecouvey

Karine Sorlin

SPRINGER BASIC SETS AND MODULAR SPRINGER CORRESPONDENCE FOR CLASSICAL TYPES

We define the notion of basic set data for finite groups (building on the notion of basic set, but including an order on the irreducible characters as part of the structure), and we prove that the Springer correspondence provides basic set data for Weyl groups. Then we use this to determine explicitly the modular Springer correspondence for classical types (defined over a base field of odd characteristic p, and with coefficients in a field of odd characteristic ℓ = p): the modular case is obtained as a restriction of the ordinary case to a basic set. In order to do so, we compare the order on bipartitions introduced by Dipper and James with the order induced by the Springer correspondence. We also provide a quicker proof, by sorting characters according to the dimension of the corresponding Springer fiber, an invariant which is directly computable from symbols.

To the memory of Tonny Springer.

1. Introduction 1.1. Context. Let G be a connected reductive algebraic group defined over F p , and let W be its Weyl group. Assume p is good. If O is a nilpotent orbit, we denote by B O the fibre of the Springer resolution of the nilpotent cone [START_REF] Springer | The unipotent variety of a semi-simple group[END_REF] at an element x ∈ O. In the landmark paper [START_REF] Springer | Trigonometric sums, Green functions of finite groups and representations of Weyl groups[END_REF], Springer defined an action of W on the ℓ-adic cohomology of B O , and a correspondence which associates to each irreducible representation of W a pair consisting of a nilpotent orbit and a G-equivariant ℓ-adic local system on that orbit.

The Springer correspondence is an injection rather than a bijection: some pairs are missing; it was later generalized by Lusztig, to include all pairs [START_REF] Lusztig | Intersection cohomology complexes on a reductive group[END_REF]. Actually he works in the group case (with unipotent classes rather than nilpotent orbits), and for arbitrary p. Note that when p is good, the classification of unipotent classes or nilpotent orbits is independent of p, and we have a G-equivariant Springer homeomorphism from the unipotent variety to the nilpotent cone [START_REF] Springer | The unipotent variety of a semi-simple group[END_REF]; for those p, the Springer correspondence is independent of p.

The non-generalized correspondence (for the group) was explicitly determined in all types [Spr76, HS77, Sho79, Sho80, AL82], and also the generalized correspondence [Lus84, LS85, Spa85, Lus86, Lus19] (there is one last ambiguity in type E 8 for p = 3). For p bad, besides the group case one can consider the Lie algebra or its dual, and all three situations are different [START_REF] Xue | Combinatorics of the Springer correspondence for classical Lie algebras and their duals in characteristic 2[END_REF].

The Springer correspondence and its generalization are closely related to Green functions. The orthogonality relations for generalized Green functions led to an algorithm, due to Lusztig and Shoji, to compute the stalks of the intersection cohomology complexes of all nilpotent orbits, with coefficients in any equivariant local system [START_REF] Lusztig | Character sheaves V[END_REF]§24].

In [START_REF] Juteau | Modular Springer correspondence and decomposition matrices[END_REF], a modular Springer correspondence was defined, following the Fourier-Deligne transform approach of [START_REF] Brylinski | Transformations canoniques, dualité projective, théorie de Lefschetz, transformations de Fourier et sommes trigonométriques[END_REF]. That is, one considers irreducible representations of the Weyl groups over a field F of characteristic ℓ = p, and local systems of vector spaces over F. Besides, one can see the decomposition matrix of a Weyl group as a submatrix of a decomposition matrix for equivariant perverse sheaves on the nilpotent cone defined in [START_REF] Juteau | Decomposition numbers for perverse sheaves[END_REF]. The modular Springer correspondence was explicitly determined in [START_REF] Juteau | Modular Springer correspondence and decomposition matrices[END_REF] in type A n , and also in other types of rank at most three. These results are recalled in [START_REF] Juteau | Modular Springer correspondence, decomposition matrices and basic sets[END_REF], together with the explicit determination of the modular Springer correspondence for exceptional Weyl groups. See also [START_REF] Achar | Weyl group actions on the Springer sheaf[END_REF] for a construction of the modular Springer correspondence for complex reductive Le algebras, and [AHJR16, [START_REF] Achar | Modular generalized Springer correspondence II: classical groups[END_REF][START_REF] Achar | Modular generalized Springer correspondence III: exceptional groups[END_REF] for the generalized case.

In the present paper, we determine the modular Springer correspondence for classical types when ℓ = 2. Since the case ℓ = 2 (generalized) was solved in [AHJR17a, Corollary 9.8] (for g over C rather than F p but it does not really matter), and the exceptional cases were done in [START_REF] Juteau | Modular Springer correspondence, decomposition matrices and basic sets[END_REF], the modular Springer correspondence is therefore completely determined in all types (for p good, but for any ℓ = p). As for the combinatorics of the generalized modular Springer correspondence for classical Lie algebras in the case ℓ = 2, the implicit conjectures in [AHJR17a, §7-8] (i.e. in each case, the combinatorial bijection which is used to count cuspidals should be the generalized Springer correspondence) remain to be proved. 1.2. A diagram. Our strategy is to use the known results in characteristic zero [START_REF] Shoji | On the Springer representations of the Weyl groups of classical algebraic groups[END_REF][START_REF] Lusztig | Intersection cohomology complexes on a reductive group[END_REF], and unitriangularity properties of the decomposition matrices (both for the Weyl groups and for the perverse sheaves) so that there is only one possibility for the correspondence. The whole paper is summed up in the following diagram, whose meaning will be gradually explained.

Irr KW Bip W λ → λ * / / r Ψ G K $ $ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ Bip W _ Φ G K Φ G co / / Symb G Π G v v dist δ co Irr FW ? β ℓ = γ ℓ O O Bip (ℓ) W ? O O Ψ G F / / P G pr δ + + | ❙ ❙ ❙ ❚ ❚ ❯ ❯ ❯ ❱ ❱ ❲ ❲ ❲ DSymb G
Π G 1 y y s s s s s s s s s s s s s s s s s

? O O δ co $ $ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ N G ? triv O O δ : O -→ dim B O 2 2 N
Let us first briefly describe the objects in the diagram. First, W is the Weyl group of a classical group G over F p with p odd. In the first column, we see its ordinary and modular characters (K has characteristic 0, while F has characteristic ℓ = 2, p); the second column contains their parametrizations by bipartitions (there are variants according to the types, but we use the generic notation Bip W ). In the next column the geometry appears: P G represents the set of pairs consisting of a nilpotent orbit together with an irreducible G-equivariant local system; we identify the ordinary and modular cases, because all component groups of centralizers are 2-groups and ℓ is odd. The first projection to the set of nilpotent orbits N G has a section assigning the trivial local system to each nilpotent orbit. In the next column we see a set of symbols associated to G, which are very useful to describe the Springer correspondence, as they can parametrize both characters and local systems [START_REF] Lusztig | Intersection cohomology complexes on a reductive group[END_REF][START_REF] Geck | On the existence of a unipotent support for the irreducible characters of a finite group of Lie type[END_REF]. There is an equivalence relation on symbols such that similar symbols correspond to different local systems on the same nilpotent orbit; the subset DSymb G of distinguished symbols corresponds to the trivial local systems.

Let us now comment on the various Springer correspondences that appear. The arrow Ψ G K stands for the ordinary correspondence obtained by Fourier transform, as in [START_REF] Springer | Trigonometric sums, Green functions of finite groups and representations of Weyl groups[END_REF][START_REF] Brylinski | Transformations canoniques, dualité projective, théorie de Lefschetz, transformations de Fourier et sommes trigonométriques[END_REF], and Ψ G F is its modular analogue studied in [START_REF] Juteau | Modular Springer correspondence and decomposition matrices[END_REF][START_REF] Juteau | Modular Springer correspondence, decomposition matrices and basic sets[END_REF][START_REF] Achar | Weyl group actions on the Springer sheaf[END_REF]. However, the ordinary Springer correspondence is most conveniently expressed (via Φ G co ) in its other version Φ G K , which can be obtained by restricting intersection cohomology complexes to the nilpotent cone [START_REF] Lusztig | Green polynomials and singularities of unipotent classes[END_REF][START_REF] Borho | Représentations des groupes de Weyl et homologie d'intersection pour les variétés nilpotentes[END_REF]. The two versions differ by the tensoring with the sign character ε W , which is reflected combinatorially by transposing the bipartitions. Only some pairs and symbols will be relevant for us:

P Spr G := Im Φ G K ⊃ Im triv and Symb Spr G := Im Φ G co ⊃ DSymb G .
1.3. Outline. The unitriangularity properties of the decomposition matrices are related to the notion of basic sets, which are subsets of the set of ordinary irreducible characters. Usually an order on the characters is fixed. We define the more precise notion of basic set datum, where the order is the first half of the structure, the other half being an injection from the modular characters to the ordinary characters. This is because we will consider several orders simultaneously. The definition and first properties of basic set data are the subject of §2. The main point is that if we have two basic set data whose order relations are comparable (i.e. one is finer than the other), then the injection has to be the same.

In §3, we will see that the Springer correspondence gives rise to basic set data for Weyl groups. The situation is much simpler when the characteristic ℓ does not divide the orders of the components groups of the centralizers of the nilpotent elements: one may identify ordinary and modular local systems on nilpotent orbits with a common indexing set P G , and the Springer order is directly induced by the closure inclusion of nilpotent orbits (local systems on the same nilpotent orbit are not comparable). This defines a canonical injection β ℓ : Irr FW ֒→ Irr KW . As an illustration, we re-derive the modular Springer correspondence for general linear groups using this language, noting that the order on nilpotent orbits and the order relevant to the modular representation theory of the symmetric group are the same: the dominance order on partitions.

From §4, we consider only groups of classical types (over F p with p odd). We recall results of Dipper and James, which in particular provide a basic set datum, for a very coarse order. In particular, we get an injection γ ℓ : Irr FW ֒→ Irr KW . At the combinatorial level, this is the inclusion of ℓ-regular bipartitions into all regular bipartitions (for type D n , one has to take unordered bipartitions, possibly with decorations; to have a uniform notation, we call Bip W the set of bipartitions which is relevant to W ).

We will see that the modular Springer correspondence Ψ G F can be described by composing β ℓ with the ordinary Springer correspondence Ψ G K . Our main theorem says that β ℓ = γ ℓ , which gives a concrete description of the modular Springer correspondence. In order to prove this, we want to compare the Dipper-James order with the one induced by the Springer correspondence. However, it is easier to compare both of them to a third order, given by the dimensions of the Springer fibres, which have a combinatorial description by [START_REF] Geck | On the existence of a unipotent support for the irreducible characters of a finite group of Lie type[END_REF]. In §5, we recall the combinatorics of symbols, and the function δ co giving the dimensions of the Springer fibres in terms of symbols. Then in §6, we prove that the Dipper-James order is compatible with the δ co function. Since the Springer order also satisfies this property, we can conclude.

So the trick of using δ co allowed us to prove our main theorem with a minimal amount of combinatorics, bypassing the link between symbols and nilpotent orbits. However, it was still an interesting question to determine whether there is a direct relationship between the Dipper-James and Springer orders. We are able to do so in §7.

Acknowledgements. During our investigations, it was very helpful to use the development version [START_REF] Michel | The development version of the CHEVIE package of GAP3[END_REF] of the CHEVIE [GHL + 96] package of GAP3 [S + 97]: it contains all the data of the ordinary (generalized) Springer correspondence. We thank Jean Michel for his help with programming.

Basic set data

Let W be a finite group, ℓ a prime number, and (K, O, F) a sufficiently large ℓ-modular system, e.g. we can take for K a finite extension of Q ℓ containing m-th roots of unity, where m is the exponent of W , with ring of integers O and residue field F. We will denote by EW the group algebra of W over E, and for E = K or F, we write Irr EW for a set of representatives of isomorphism classes of simple EW -modules. We have an ℓ-modular decomposition matrix

D W := (d W E,F ) E∈Irr KW, F ∈Irr FW where d W E,F is the composition multiplicity of the simple module F in F ⊗ O E O , where E O is some integral form of E. This is independent of the choice of E O [Ser67, §15.2].
Definition 2.1. A basic set datum (for W ) is a pair B = (≤, β) consisting of a partial order ≤ on Irr KW , and an injection β : Irr FW ֒→ Irr KW such that the following properties hold:

d W β(F ),F = 1 for all F ∈ Irr FW ; (2.1) d W E,F = 0 ⇒ E ≤ β(F ) for all E ∈ Irr KW, F ∈ Irr FW. (2.2)
We will need to compare different basic set data. We adopt the convention that an order relation on a set X is a subset of the product X × X. Thus we can consider the intersection of two orders, say that one order is included in another one, and so on.

Proposition 2.2. Let us consider a fixed injection β : Irr FW ֒→ Irr KW .

(1) If (≤ 1 , β) is a basic set datum and ≤ 1 ⊆≤ 2 , then (≤ 2 , β) is also a basic set datum.

(2) If (≤ 1 , β) and (≤ 2 , β) are two basic set data, then (≤ 1 ∩ ≤ 2 , β) is also a basic set datum.

Proof. This follows directly from the definitions.

Proposition 2.3. Let B 1 = (≤ 1 , β 1 ) and B 2 = (≤ 2 , β 2 ) be two basic set data for W . Then, for any F ∈ Irr FW , we have

β 1 (F ) ≤ 2 β 2 (F ) ≤ 1 β 1 (F ). If moreover ≤ 1 ⊆≤ 2 , then β 1 = β 2 .
Proof. Let F be an element of Irr FW . By the property (2.1) for B 1 , we have d W β 1 (F ),F = 1 = 0, hence by the property (2.2) for B 2 , we have β 1 (F ) ≤ 2 β 2 (F ). Similarly, by the property (2.1) for B 2 , we have d W β 2 (F ),F = 1 = 0, hence by the property (2.2) for B 1 , we have β 2 (F ) ≤ 1 β 1 (F ). Thus

β 1 (F ) ≤ 2 β 2 (F ) ≤ 1 β 1 (F ). If moreover ≤ 1 ⊆≤ 2 , then β 1 (F ) ≤ 2 β 2 (F ) ≤ 2 β 1 (F ), hence β 1 (F ) = β 2 (F ), for any F .
Corollary 2.4. Given a partial order ≤ on Irr KW , there is at most one injection β : Irr FW ֒→ Irr KW such that (≤, β) is a basic set datum.

Springer basic set data

3.1. Modular Springer correspondence. Let us consider a reductive group G over F p . We assume that p is very good for G [Car93, p.29] (thus for classical groups of type BCD, we only have to assume p = 2). The group G acts by the adjoint action on its Lie algebra g, and there are finitely many orbits in the nilpotent cone N ⊆ g. For x in N , we denote by O x the G-orbit of x. The set N G of nilpotent orbits is partially ordered by the following relation:

O 1 ≤ O 2 if and only if O 1 ⊆ O 2 .
For each orbit O ⊆ N , we choose a representative x O , and we denote by

A G (O) the group of components of the centralizer C G (x O ), that is, A G (O) := C G (x O )/C G (x O ) 0 .
As in the last section, we choose an ℓ-modular system (K, O, F), and we assume ℓ = p.

For E = K or F, let P G E (or P E if G is clear from the context) denote the set of all pairs (O, L)
where O is a nilpotent orbit in g and L is an irreducible G-equivariant E-local system on O. Moreover, we will denote by L ρ the local system corresponding to some ρ ∈ Irr EA G (O). The simple G-equivariant perverse sheaves on N with E-coefficients are intersection cohomology complexes IC(O, L), for (O, L) ∈ P E . Similarly to the case of a finite group, one can define a decomposition matrix [Jut09]

D N := (d N (O,Lρ),(O ′ ,Lσ) ) (O,Lρ)∈P K , (O ′ ,Lσ)∈P F for G-equivariant perverse sheaves on N , where d N (O,L),(O ′ ,L ′ ) is the composition multiplicity of IC(O ′ , L ′ ) in F ⊗ L O IC(O, L O )
for any choice of an integral form L O of the local system L. Let W be the Weyl group of G. If E = K or F, one can naturally define an injective map

Ψ E = Ψ G E : Irr EW ֒→ P E (3.1)
using a Fourier-Deligne transform. In the ordinary case, it is the Springer correspondence of [START_REF] Brylinski | Transformations canoniques, dualité projective, théorie de Lefschetz, transformations de Fourier et sommes trigonométriques[END_REF] (and coincides with the one of [START_REF] Springer | Trigonometric sums, Green functions of finite groups and representations of Weyl groups[END_REF]), and in the modular case it was first introduced in [START_REF] Juteau | Modular Springer correspondence and decomposition matrices[END_REF]. See also [START_REF] Achar | Weyl group actions on the Springer sheaf[END_REF] for complex reductive Lie algebras.

Remark 3.1. The injectivity of Ψ E comes from the fact that there is an inverse Fourier transform.

Remark 3.2. We have Ψ E (E) = ({0}, E) because the Fourier-Deligne transform of the constant sheaf is a sky-scraper sheaf supported at the origin, just as the Fourier transform of a constant function is a Dirac distribution supported at the origin.

Remark 3.3. What is usually called Springer correspondence (but only in the case E = K) is the one defined in [START_REF] Lusztig | Green polynomials and singularities of unipotent classes[END_REF][START_REF] Borho | Représentations des groupes de Weyl et homologie d'intersection pour les variétés nilpotentes[END_REF], and for which one can find a description in all types in [START_REF] Carter | Finite groups of Lie type[END_REF]. We will denote this map by Φ K = Φ G K : Irr KW ֒→ P K . Note that this construction makes sense both for the group G and for its Lie algebra g. Since the characteristic p is very good for G, we can also use a Springer isomorphism [START_REF] Springer | The unipotent variety of a semi-simple group[END_REF] from the unipotent variety to the nilpotent cone to transfer the contruction for the group to the Lie algebra, but by [START_REF] Shoji | Geometry of orbits and Springer correspondence[END_REF]5.3] this gives the same result as working directly with the Lie algebra.

The maps Φ K and Ψ K differ by the sign character ε W of the Weyl group: Proposition 3.4. [START_REF] Shoji | Geometry of orbits and Springer correspondence[END_REF]17.7] For any E ∈ Irr KW , we have

Ψ K (E) = Φ K (ε W ⊗E).
See also [START_REF] Achar | Weyl group actions on the Springer sheaf[END_REF] for the same result in the modular case. The following theorem from [START_REF] Juteau | Modular Springer correspondence and decomposition matrices[END_REF] shows that the ordinary and modular Springer correspondences respect decomposition numbers.

Theorem 3.5. For E ∈ Irr KW and F ∈ Irr FW , we have

d W E,F = d N Ψ K (E),Ψ F (F ) .
Hence D W can be seen as a submatrix of D N . Now D N has the following easy property, as follows from considering supports [Jut14, Corollary 2.6 and Proposition 2.7].

Proposition 3.6. For (O, L ρ ) ∈ P K and (O ′ , L σ ) ∈ P F , we have d N (O,Lρ),(O ′ ,Lσ) = 0 unless O ′ ≤ O, (3.2) d N (O,Lρ),(O ′ ,Lσ) = d A G (O) ρ,σ if O = O ′ . (3.3)
In the preceding formula, we have denoted by d

A G (O) ρ,σ
the decomposition numbers for the finite group A G (O). In order to define basic sets using the Springer correspondence, the following result plays a crucial role [Jut14, Proposition 5.4]. ) for W , the Springer basic set datum. Remark 3.8. If G is simple and of adjoint type (which we may assume for the purpose of describing the correspondence, since we are working in a modular, but not generalized setting, in the sense of Lusztig), then the A G (O) are as follows: trivial in type A n , elementary abelian 2-groups in other classical types, and symmetric groups of rank at most 5 in exceptional types. We know that these groups admit basic sets. In the sequel it will be combinatorially more convenient to work with special orthogonal and symplectic groups (rather than the adjoint groups), but for those the A G (O) are also elementary abelian 2-groups.

Proposition 3.7. If (O ′ , L σ ) ∈ Im Ψ F and d N (O,Lρ),(O ′ ,Lσ) = 0, then (O, L ρ ) ∈ Im Ψ K . 3.
Remark 3.9. In the case where ℓ does not divide the orders of the groups A G (x), we will see that much of the discussion simplifies (see §3.3). In particular, there will be no need to choose basic set data for each orbit: in that case, we may assume that we take the trivial order everywhere, and the β O are canonical bijections (in that case, the Brauer characters are just the ordinary characters). Definition 3.10. We define the partial order ≤ N B• on Irr KW by the following rule. For i ∈ {1, 2}, let E i ∈ Irr KW , and let us write

Ψ K (E i ) = (O i , L ρ i ). Then E 1 ≤ N B• E 2 ⇐⇒ either O 2 < O 1 , or (O 1 = O 2 =: O and ρ 1 ≤ O ρ 2 ).
It is clear that this defines indeed an order relation on Irr KW (for the antisymmetry, one needs to use the injectivity of Ψ K ).

Proposition 3.11. Let F ∈ Irr FW , and let us write

Ψ F (F ) = (O, L σ ). Then there exists a unique E ∈ Irr KW such that Ψ K (E) = (O, L β O (σ) ).
Proof. We have (O, L σ ) = Ψ F (F ) ∈ Im Ψ F , and

d N (O,L β O (σ) ),(O,Lσ) = d A G (O) β O (σ),σ = 1 = 0, hence (O, L β O (σ) ) ∈ Im Ψ K
Irr FW Ψ F / / β N B• P F β• ∼ / / ⊔ O Irr FA G (O) ⊔ O β O Irr KW Ψ K / / P K ∼ / / ⊔ O Irr KA G (O)
where β • is defined by the commutativity of the right square. The β O are injective, thus β • is injective, and the composition β

• • Ψ F = Ψ K • β N B• is injective as well. It follows that β N B• is injective.
F := β N B• (F ). By definition, we have Ψ K (E F ) = (O ′ , L β O ′ (σ) ). Thus d W β N B• (F ),F = d W E F ,F = d N (O ′ ,L β O (σ) ),(O ′ ,Lσ) = d A G (O ′ )
β O ′ (σ),σ = 1, using Theorem 3.5 and Proposition 3.6. Now, in addition, let E ∈ KW , and assume d W E,F = 0. Let us write (O, L ρ ) := Ψ K (E). By Theorem 3.5 again, we have

d N (O,Lρ),(O ′ ,Lσ) = d W E,F = 0 hence O ≥ O ′ by Proposition 3.6. Now, if O = O ′ , then we have 0 = d N (O,Lρ),(O,Lσ) = d A G (O) ρ,σ again by Proposition 3.6, hence ρ ≤ O β O (σ) since B O = (≤ O , β O ) is a basic set for A G (O). This proves that E ≤ N B• E F = β N B• (F ). Thus (≤ N B• , β N B• ) is a basic set datum for W . The pair (≤ N B• , β N B•
) is the Springer basic set datum for W associated to the choice of the family B • , and we call the image of β N B• a Springer basic set. 3.3. Good case. In this paragraph, we assume that ℓ does not divide the orders of the groups A G (O), where O runs over the nilpotent orbits in g. Then the algebras FA G (O) are semisimple and the decomposition matrices D A G (O) are equal to the identity matrix. Thus we can take the trivial order (that is, the identity relation) = O on Irr KA G (O), and β O is a canonical bijection. By abuse of notation, we will identify the sets P K and P F with a common index set P. Let us reformulate the definition of Springer basic sets in this favorable situation. The order ≤ N B• , in this case, is the order ≤ N triv defined by

E 1 ≤ N triv E 2 ⇐⇒ (E 1 = E 2 or O 2 < O 1 ) where E i ∈ Irr KW and (O i , L i ) := Ψ K (E i ), for i = 1, 2.
The injection β N B• is the map β ℓ : Irr FW ֒→ Irr KW defined in the following way: to F ∈ Irr FW , we associate the unique E := β ℓ (F ) ∈ Irr KW such that Ψ K (E) = Ψ F (F ) ∈ P.

Then the pair B ℓ := (≤ N triv , β ℓ ) is a basic set datum for W . Moreover, any choice of a family B • of basic set data for the A G (O) will give rise to a basic set datum for W with the same injection β ℓ , since the trivial order = O is contained in any chosen order ≤ O , and thus the order ≤ N triv is contained in ≤ N B• , and Proposition 2.3 applies. Remark 3.14. Basic sets for Weyl groups (and actually for Hecke algebras) in the good prime case have already been constructed, for example in [START_REF] Geck | Filtrations on projective modules for Iwahori-Hecke algebras[END_REF]. For symmetric groups, this goes back to James [START_REF] James | The irreducible representations of the symmetric groups[END_REF].

3.4. The case of the general linear group. Although the modular Springer correspondence for GL n has been determined in [START_REF] Juteau | Modular Springer correspondence and decomposition matrices[END_REF], we will do it again here in the language of Springer basic sets, as this case is much simpler than the other classical types, for which it will serve as a model.

Let Part n denote the set of all partitions of n. The prime ℓ being fixed, the subset Part n . Nilpotent orbits for GL n are parametrized by Part n via the Jordan normal form. We will denote them by O λ , for λ ∈ Part n . The orbit closure order is given by the classical dominance order on partitions.

It turns out that the simple modules of KS n , called Specht modules, are also parametrized by Part n . In [START_REF] James | The irreducible representations of the symmetric groups[END_REF], James classifies the simple modules for FS n as follows. The Specht module S λ is defined as a certain submodule of the permutation module M λ = K[S n /S λ ], where S λ is the parabolic subgroup corresponding to the partition λ. The permutation module M λ is endowed with a natural scalar product, which restricts to a scalar product on S λ . The module and the bilinear form are defined over O (actually over Z), and thus one can reduce them to get a module for FS n , still called a Specht module, endowed with a bilinear form, which no longer needs to be non-degenerate. Then the quotient of the Specht module S λ by the radical of the form is either zero or a simple module denoted by D λ . The partitions giving a non-zero result are exactly the ℓ-regular partitions. The D λ , for λ ∈ Part (ℓ) n , form a complete set of representatives of FS nmodules. Moreover, James shows that (in our language) the pair consisting of the dominance order on partitions, and the injection sending D λ (λ ∈ Part Again, those modules are defined over Z and endowed with a bilinear form; factoring out the radical of the form over F yields 0 or a simple module; and one obtains a complete collection of simple FW n -modules in this way. In our case (ℓ = 2), the result is that we have a parametrization

Irr FW n = {D λ | λ ∈ Bip (ℓ) n }, where Bip (ℓ)
n denotes the set of ℓ-regular bipartitions of n, that is, the bipartitions λ = (λ (1) , λ (2) ) such that λ (1) and λ (2) are ℓ-regular. Naming the modular simple modules in this way implicitly amounts to define an injection γ ℓ from Irr FW n to Irr KW n , sending D λ to S λ , for λ ∈ Bip

d Wn λ,µ = d Sr λ (1) ,µ (1) • d S n-r λ (2) ,µ (2)
and otherwise we have d Wn λ,µ = 0. This is good motivation to introduce the following very coarse order relation. Definition 4.2 (Dipper-James order on bipartitions). Let λ = (λ (1) , λ (2) ) and µ = (µ (1) , µ (2) ) be two bipartitions of n. We say that λ ≤ DJ µ if, for i ∈ {1, 2}, we have |λ (i) | = |µ (i) | and λ (i) ≤ µ (i) . This induces a partial order on Irr KW n that we still denote by ≤ DJ .

By the results for the symmetric group, the preceding proposition implies: Proposition 4.3. The pair (≤ DJ , γ ℓ ) is a basic set datum for W n .

We will call it the Dipper-James basic set datum. Since the two versions of the ordinary Springer correspondence are related by tensoring with the sign character ε W , it is useful to recall how this translates combinatorially. If λ = (λ (1) , λ (2) ) is a bipartition, we set λ * := (λ (2) * , λ (1) * ). A classification of the simple KW ′ n -modules is obtained as follows:

• if (λ (1) , λ (2) ) ∈ Bip n satisfies λ (1) = λ (2)
, then the KW n -modules S (λ (1) ,λ (2) ) and S (λ (2) ,λ (1) )

have the same restriction to W ′ n , a simple KW ′ n -module which we will denote by S [λ (1) ,λ (2) ] ; • the restriction of S (λ,λ) splits into a direct sum of two non-isomorphic simple KW ′ n -modules which are denoted by S [λ,+] and S [λ,-] (this case arises only for even n). Moreover, every simple KW ′ n -module arises exactly once in this way. Note that we use the notation (a, b) for an ordered pair, and [a, b] for an unordered pair. Let Bip n denote the set of unordered bipartitions of n. Thus Irr KW ′ n is parametrized by

Bip n = Bip n if n is odd, {[λ 1 , λ 2 ] ∈ Bip n | λ 1 = λ 2 } ∪ {[λ, ±] | λ ∈ Part n
2 } if n is even. We denote by Bip

) (1) There is a parametrization

Irr FW ′ n = {E λ | λ ∈ Bip (ℓ) n } such that, if γ ℓ : Irr FW ′ n → Irr KW ′ n is induced by the natural inclusion Bip (ℓ)
n → Bip n , the following properties hold:

• for all F ∈ Irr FW ′ n , we have d

W ′ n γ ℓ (F ),F = 1 and a γ ℓ (F ) = a F ; • given E ∈ Irr KW ′ n and F ∈ Irr FW ′ n , we have d W ′ n E,F = 0 ⇒ a F < a E or E = γ ℓ (F ).
(2) Moreover, we have the following relations between the simple modules for FW n and for FW ′ n :

• if λ (1) = λ (2) , then D (λ (1) ,λ (2)
) and D (λ (2) ,λ (1) ) have the same restriction to W ′ n and this restriction is the simple FW ′ n -module E [λ (1) ,λ (2) ] ;

• the restriction of D (λ,λ) splits into a direct sum of two non-isomorphic simple FW ′ nmodules which are E [λ,+] and E [λ,-] . Definition 4.7 (Dipper-James order on unordered bipartitions). We define a partial order on the set Bip n of unordered bipartitions of n by:

[λ (1) , λ (2) ] ≤ DJ [µ (1) , µ (2) ] (λ (1) , λ (2) ) ≤ DJ (µ (1) , µ (2) ) or (λ (1) , λ (2) ) ≤ DJ (µ (2) , µ (1) )
(It is easy to check that this is indeed a partial order.)

We have a natural projection ϕ :

Bip n → Bip n , sending [λ, µ] to [λ, µ] and [λ, ±] to [λ, λ]
. The order ≤ DJ on Bip n induces an order that we still denote by ≤ DJ on Bip n , hence on Irr KW ′ n , such that λ < µ if and only if ϕ(λ) < ϕ(µ) for λ, µ ∈ Bip n . So the irreducible modules S [λ,+] and S [λ,-] are not comparable.

Proposition 4.8. The pair (≤ DJ , γ ℓ ) is a basic set datum for W ′ n . To prove the proposition, we will need the fact that the restriction from W n to W ′ n commutes with decomposition maps. Let us introduce some notation. If A is a finite dimensional algebra over a field, we denote by R 0 (A) the Grothendieck group of the category of finite dimensional A-modules. The class of an

A-module V in R 0 (A) is denoted by [V ].
Lemma 4.9 ([Gec00], lemma 5.2). The restriction of modules from Irr KW n to Irr KW ′ n (resp. from Irr FW n to Irr FW ′ n ) induces maps between Grothendieck groups fitting into the following commutative diagram

R 0 (KW n ) Res ----→ R 0 (KW ′ n )   d   d ′ R 0 (FW n ) Res ----→ R 0 (FW ′
n ) where d and d ′ denote the decomposition maps.

Proof of Proposition 4.8. By the lemma, for λ (1) = λ (2) , we have ,-] ] + (larger terms for ≤ DJ ) By Proposition 4.6 (1), we get d -] and S [λ,+] have the same a-invariant. This proves that (≤ DJ , γ ℓ ) is a basic set datum.

d ′ ([S [λ (1) ,λ (2) ] ]) = Res(d([S (λ (1) ,λ (2) ) ])) = Res([D (λ (1) ,λ (2) ) ]) + Res(larger terms for ≤ DJ ) = [E [λ (1) ,λ (2) ] ] + (larger terms for ≤ DJ ) For λ (1) = λ (2) = λ, we have d ′ ([S [λ,+] ] + [S [λ,-] ]) = [E [λ,+] ] + [E [λ
W ′ n S [λ,+] ,E [λ,-] = 0 as S [λ,
We define the transposition * on Bip n by [λ (1) , λ

Symbols and dimensions of Springer fibers

From now on, G is a classical group defined over F p , with p odd. In this section, we recall the bare minimum that we need from the combinatorics of the ordinary Springer correspondence for classical types, in order to prove our main theorem saying that the modular Springer correspondence can be deduced by restricting to ℓ-regular bipartitions. Namely, we introduce symbols, and describe the map Φ G co from bipartitions to symbols (which is very easy). To complete the description, one would then have to explain the correspondence between symbols and local systems on nilpotent orbits. However, we only need the combinatorial formula computing the dimension of the corresponding Springer fibre, directly from the symbol, in order to prove our main theorem. We will come back to the complete description in §7.

Since we only study the non-generalized case, the isogeny type is irrelevant to the determination of the correspondence, and we may assume that G is a symplectic or special orthogonal group. We work with the Lie algebra g, but since p is good for G, it gives the same results as for the group. 5.1. Generalized symbols. Following [LS85, GM00], we will use generalized symbols. Let n, r and s be positive integers, and d ∈ {0, 1}. We denote by X r,s n,d the set of ordered pairs A B of finite sequences A = (a 1 , . . . , a m+d ), B = (b 1 , . . . , b m ), such that:

(1)

a i -a i-1 ≥ r + s for 1 < i ≤ m + d, (2) b i -b i-1 ≥ r + s for 1 < i ≤ m, (3) b 1 ≥ s, (4)
m+d i=1 a i + m i=1 b i = n + rm(m + d -1) + sm(m + d), taken modulo the shift operation A B → 0, a 1 +r+s, ..., a m+d +r+s
s, b 1 +r+s, ..., bm+r+s .

If s = d = 0, we denote by Y r n the quotient of X r,0 n,0 by the operation of swapping the two rows of a symbol. The class of A B will be denoted A B . Finally, let Y r n be the set similar to Y r n , but where each symbol with identical rows has a ± decoration.

Varying r and s, these symbols are useful for parametrizing both irreducible representations of classical Weyl groups, and local systems on nilpotent orbits, for the different classical types. It is remarkable that all instances of the Springer correspondence (generalized case, bad prime p = 2. . . ) for classical types can be expressed via generalized symbols. The integer n will be the rank, and here the "defect" d is limited to {0, 1} because we only consider the non-generalized case.

Note that the set X r,s 0,d consists of a single element,

Λ r,s 0,d =    ∅ ∅ = 0 s = 0 r+s s r+2s = 0 r+s 2r+2s s r+2s 2r+3s = • • • if d = 0, 0 ∅ = 0 r+s s = 0 r+s 2r+2s s r+2s = • • • if d = 1.
Adding representatives of the same length componentwise defines an addition

X r,s n,d × X r ′ ,s ′ n ′ ,d -→ X r+r ′ ,s+s ′ n+n ′ ,d .
We define the reading of a symbol Λ = A B by the sequence w(Λ) := (a 1 , b 1 , a 2 , b 2 , a 3 . . .) obtained by alternating the entries of A and B. A symbol is called distinguished if the entries of its reading weakly increase from left to right. Two symbols are similar if the multisets of entries of suitable representatives coincide. Any symbol Λ is similar to a (clearly unique) distinguished symbol dist(Λ): indeed, one can keep exchanging pairs of consecutive entries which are in the wrong order, until one reaches a distinguished symbol [GM00, §2.A]. Similar symbols will correspond to different local systems on the same nilpotent orbit. 5.2. From bipartitions to symbols. Recall that the irreducible characters of the Weyl group W n of type B n or C n are parametrized by the set Bip n of bipartitions of n; the latter may be identified with X 0,0 n,1 by writing entries of each partition in weakly increasing order, and adding 0 entries if necessary to get symbols of defect 1. Similarly, for type D n , we identify Irr KW ′ n with Bip n = Y 0 n (keeping track of the decorations). It will be convenient to let

Bip W denote either Bip n if W is of type B n /C n , or Bip n if W is of type D n .
The pairs in P Spr G,K = Im Φ G K can be parametrized by a set of symbols that we will denote by Symb Spr G (see §7); it is X 1,1 n,1 for type C n , X 2,0 n,1 for type B n , and

Y 2 n for type D n . Then Φ G K = Ψ G K • (ε W ⊗-) is deduced from the map Φ G co : Bip W → Symb Spr G defined as follows: Type C n : Bip n = X 0,0 n,1 -→ X 1,1 n,1 , α β -→ α β + Λ 1,1 0,1 ; (5.1) Type B n : Bip n = X 0,0 n,1 -→ X 2,0 n,1 , α β -→ α β + Λ 2,0 0,1 ; (5.2) Type D n : Bip n = Y 0 n -→ Y 2 n , α β -→ α β + Λ 2,0 0,0 . (5.3)
(keeping track of the decorations for degenerate bipartitions or symbols in type D n with n even), so that characters sent to similar symbols have the same associated nilpotent orbit (but with different local systems). Let us call Λ G the symbol that is added in the description of Φ G co . Note that Φ G co describes Φ G K rather than Ψ G K , whereas the Springer basic set datum is defined with respect to the latter. But tensoring by the sign character amounts to transposing bipartitions, which just reverses the order on bipartitions. Of course one should take representatives of Λ and Λ G of the same size: here m is the length of the second line. We can be more explicit: for symbols in Symb Spr G (i.e. d = 0, 1), we have

σ(w(Λ G )) =      m(4m 2 -1)/3 in type C n ; m(m -1)(4m + 1)/3 in type B n ; m(m -1)(4m -5)/3 in type D n .
Definition 5.1. Let ≤ δ be the order relation on Irr KW = Bip W defined by

λ < δ µ ⇐⇒ δ Ψ K (λ) < δ Ψ K (µ) ⇐⇒ δ co Φ G co (λ * ) < δ co Φ G co (µ * ) .
Note that we have used the fact that the two versions of the Springer correspondence differ by tensoring by the sign character, which amounts to transposing the bipartitions before applying the correspondence.

Let λ, µ ∈ Bip W , and set

Ψ K (λ) = (O, L ρ ) and Ψ K (µ) = (O ′ , L ρ ′ ). Note that λ < N triv µ ⇐⇒ O > O ′ =⇒ dim O > dim O ′ ⇐⇒ dim B O < dim B O ′ ⇐⇒ λ < δ µ
, where we have used the fact that dim B O is equal to half the codimension of O in N [START_REF] Springer | Trigonometric sums, Green functions of finite groups and representations of Weyl groups[END_REF][START_REF] Steinberg | On the desingularization of the unipotent variety[END_REF]. Thus we have ≤ N triv ⊂ ≤ δ ; since (≤ N triv , β ℓ ) is a basic set datum, we deduce that thus (≤ δ , β ℓ ) is a basic set datum as well, by Proposition 2.3. Remarkably, this very limited part of the combinatorics is enough for the proof of our main theorem in the next section.

Modular Springer correspondence for classical types

Proposition 6.1. We have ≤ DJ ⊂ ≤ δ .

Proof. Taking into account the fact that transposing the bipartitions reverses the Dipper-James order on Bip W , we need to show that for λ and µ in Bip W , we have

λ < DJ µ =⇒ δ co Φ G co (µ) < δ co Φ G co (λ)
. The proof is similar for symplectic and special orthogonal groups. We give the proof for G = Sp 2n . Moreover, it is sufficient to prove the result when µ covers λ, in which case we may assume for example that λ (2) = µ (2) , while λ (1) is obtained from µ (1) by increasing one part by 1 and decreasing another part by 1 (the case where λ (1) = µ (1) is similar). Then the symbols Φ G co (λ) and Φ G co (µ) have the same relationship: their second lines are identical, while the first line of Φ G co (λ) is obtained from the first line of Φ G co (µ) by increasing one part by 1 and decreasing another part by 1.

Let us reorder the entries of Φ G co (λ) and Φ G co (µ) into the respective sequences

x 1 ≤ x 2 ≤ • • • ≤ x 2m+1 and y 1 ≤ y 2 ≤ • • • ≤ y 2m+1 .
Then there exist indices a < b such that y b = x b + 1 and y a = x a -1, while y i = x i for all i = a, b. We get:

δ co Φ G co (µ) -δ co Φ G co (λ) = 1≤i<j≤2m+1 min(y i , y j ) - 0≤i<j≤2m min(x i , x j ) = 1≤i≤2m+1 (2m -i)(y i -x i ) = (2m -a)(y a -x a ) + (2m -b)(y b -x b ) = a -b < 0.
Corollary 6.2. The inclusion β ℓ : Irr FW ֒→ Irr KW , which was introduced in §3.3, coincides with the inclusion γ ℓ defined in §4.1 and §4.2.

Proof. Recall that (≤ DJ , γ ℓ ) is a basic set datum by Proposition 4.3. By Propositions 6.1 and 2.3, (≤ δ , γ ℓ ) is a basic set datum as well. We have seen at the end of §5 that (≤ δ , β ℓ ) is also a basic set datum. It follows from Corollary 2.4 that β ℓ = γ ℓ .

Reformulating, we get the main result of the paper.

Theorem 6.3. Let p be an odd prime number, and let G be a classical group over F p , with Weyl group W . For an odd prime ℓ = p, the modular Springer correspondence for G is described as follows: for λ ∈ Bip

(ℓ) W , Ψ G F (D λ ) = Ψ G K (S λ ) = Φ G K (S λ * ).
For the statement to make sense, recall that we may identify P K with P F because ℓ = 2. 7. Compatibility between Dipper-James and Springer orders 7.1. Classical nilpotent orbits. For G a classical group with natural representation V , with p good, a nilpotent orbit is determined by the Jordan type of its elements acting on V , with the exception of the so-called "very even" case in type D n . More precisely, we have a set of partitions P(G) (with decorations for the very even case), described below according to the type, such that the nilpotent orbits can be naturally called O G λ , λ ∈ P(G): (1) P(C n ) is the set of partitions of 2n in which each odd part appears with an even multiplicity (we ensure that the total number of parts is odd by adding a 0 entry if necessary: this will be useful later);

(2) P(B n ) is the set of partitions of 2n + 1 in which each even part appears with an even multiplicity (the total number of parts is necessarily odd); (3) P(D n ) is the set of partitions of 2n in which each even part appears with an even multiplicity, except that a partition with no odd part is decorated with a ± sign (the total number of parts is necessarily even).

7.2. From nilpotent orbits to bipartitions and symbols. Let λ ∈ P(G). Partition the sequence λ = (λ 1 ≤ . . . ≤ λ t ) into blocks of lengths 1 or 2 such that if λ is symplectic (resp. orthogonal) then all even (resp. odd) λ i lie in a block of length 1, and all odd (resp. even) λ i lie in a block of length 2. By [START_REF] Geck | On the existence of a unipotent support for the irreducible characters of a finite group of Lie type[END_REF], the symbol associated to λ (i.e. corresponding to the pair (O λ , E)) is the unique distinguished symbol with reading (c 1 , . . . , c t ) given by

c i = ⌊λ i /2⌋ + i -1 if{λ i } is a block, c i = c i+1 = ⌈λ i /2⌉ + i -1 if {λ i , λ i+1 } is a block.
One obtains the corresponding bipartition (written as a symbol) by subtracting Λ G . It has reading (a 1 , . . . , a t ), where in the symplectic case:

a i = λ i /2 if {λ i } is a block, (a i , a i+1 ) = (⌈λ i /2⌉ , ⌊λ i /2⌋) if {λ i , λ i+1 } is a block;
and in the special orthogonal case:

     a i = (λ i + (-1) i )/2 if {λ i } is a block, (a i , a i+1 ) = (λ i /2, λ i /2) if {λ i , λ i+1 } is a block with i odd, (a i , a i+1 ) = (λ i /2 + 1, λ i /2 -1) if {λ i , λ i+1 } is a block with i even.
The resulting bipartition is actually the 2-quotient of λ, as defined in [START_REF] Macdonald | Symmetric functions and orthogonal polynomials[END_REF]. Here we have explicit formulas because λ ∈ P(B n ) ∪ P(C n ) ∪ P(D n ) has a special form.

In the following examples, we give a partition corresponding to a nilpotent orbit, then the corresponding bipartition, then the corresponding symbol. We highlight the nontrivial blocks. We have found it convenient to display the partition on two lines as well, in the same shape as the corresponding symbol.

Example 7.1. Type C 28 , λ = (9 2 , 7 2 , 6 2 , 4, 3 2 , 2). 0 3 4 6 7 9 2 3 6 7 9 -→ 0 2 2 3 3 4 1 1 3 4 5 -→ 0 4 6 9 11 14 2 4 8 11 14

Example 7.2. Type B 25 , λ = (8 2 , 7, 6 2 , 4 2 , 3, 2 2 , 1).

1 2 4 6 7 8 2 3 4 6 8 -→ 0 0 2 3 3 3 2 2 2 3 5 -→ 0 2 6 9 11 13 3 5 7 10 14

Example 7.3. Type D 25 , λ = (8 2 , 7, 6 2 , 4 2 , 3, 2 2 ).

2 3 4 6 8 2 4 6 7 8 -→ 1 1 1 2 4 1 3 4 4 4 -→ 1 3 5 8 12 1 5 8 10 12

We will not need the precise formulation of the correspondence for nontrivial local systems: the only thing we need to know is that local systems on the same orbit have similar symbols. So we will only give a summary of the description: a local system is given by a sequence of signs; starting from the symbol of the trivial local system on the same orbit, one defines some "intervals" inside the symbol, and one swaps the two lines of some of the intervals, according to those signs (see [START_REF] Lusztig | Intersection cohomology complexes on a reductive group[END_REF] 1 ). This may change the defect of the symbol. Only those with defect 1 are in the image of the nongeneralized Springer correspondence in types B n or C n , 2 and only those with defect 0 in type D n . 7.3. From bipartitions and symbols to nilpotent orbits. Consider a bipartition λ ∈ Bip W corresponding to a distinguished symbol Λ = S T or S T . Write S = (s 1 , . . . , s m+d ) and T = (t 1 , . . . , t m ). Let us describe how to recover the partition λ ∈ P(G) of the corresponding nilpotent orbit. Let x = w(Λ) be the reading of Λ. Then λ = 2 xw(Λ G ) , where x is a modification of x, which we will now describe according to the type, and Λ G was defined in §5.2. Type C n . A distinguished pair in Λ is a pair (s i , t i ) with s i = t i or a pair (t i , s i+1 ) with t i = s i+1 . Since the rows of the symbols strictly increase, the distinguished pairs are pairwise disjoint. Then x is the sequence obtained from x by changing each distinguished pair (s i , s i ) into (s i -1/2, s i + 1/2).

Type B n or D n . A distinguished pair is a pair (t i , s i+1 ) such that t i = s i+1 . A frozen pair is a pair (s i , t i ) with s i = t i . Since the rows of the symbols strictly increase, distinguished and frozen pairs are disjoint. Then x is the sequence obtained from x by changing each s i into s i + 1/2 and each t i into t i -1/2 provided they do not appear in a distinguished or frozen pair, and each distinguished pair (t i , t i ) into (t i -1, t i + 1), One sees easily this procedure is an inverse to that of the preceding section, hence we have a bijection between distinguished symbols and nilpotent orbits, which was denoted by Π G 1 in the diagram of §1.2.

If we start with a non-distinguished symbol Λ, we first have to reorder its entries to get a distinguished symbol dist(Λ), and then apply the procedure above to get the corresponding nilpotent orbit. The local system is encoded in what happens during that sorting, but we do not need a precise description to obtain the main result of this section: local systems are not comparable for our Springer order ≤ N triv anyway (this is possible only because ℓ = 2). 7.4. Compatibility between Dipper-James and Springer orders. Recall the Dipper-James order ≤ DJ from Definition 4.2, and the Springer order (for the good characteristic case) ≤ N triv defined in §3.3. We will also need the dominance order on compositions: for x ∈ N t , we let S i (x) = i≤k≤t x k ; for x, y ∈ N t , we write x ≤ y if and only if S i (x) ≤ S i (y) for all i.

Proposition 7.4. We have ≤ DJ ⊂ ≤ N triv . Proof. Let λ, µ ∈ Irr KW = Bip W , and let λ, µ ∈ P(G) be the partitions of the corresponding nilpotent orbits (we may ignore decorations for very even orbits). We want to show λ < DJ µ =⇒ λ < µ.

We write λ (i) and µ (i) for the components of λ and µ. In the G = SO 2n case, by definition of the Dipper-James order on unordered bipartitions, we can assume (up to flipping λ) that (λ (1) , λ (2) ) ≤ DJ (µ (1) , µ (2) ) (in Bip n rather than Bip n ).

As in the proof of Proposition 6.1, we may assume that λ (2) = µ (2) , and that λ (1) is obtained from µ (1) by increasing one part by 1 and decreasing another part by 1. Again, if we reorder the 1 Actually there is a mistake in the description of the generalized correspondence in type C in [Lus84, Theorem 12.3], which was corrected in [START_REF] Shoji | Unipotent characters of finite classical groups[END_REF] (see his Remark 5.8). As pointed out in the final remark of [GM00, §2.B], this does not affect the description of the nongeneralized Springer correspondence.

2 Actually for type Bn, the "right" notion is that of symbol of defect ±1 up to swapping the two lines, but it does not really matter for us since we don't need to describe precisely the local systems in the correspondence.

  by Proposition 3.7. This proves the existence of E. The uniqueness of E follows from the injectivity of Ψ K . Definition 3.12. We define the map β N B• : Irr FW → Irr KW by associating to each F ∈ Irr FW the module E ∈ Irr KW provided by Proposition 3.11. In other words, β N B• completes the following commutative diagram :

  Theorem 3.13. Assume we have chosen a family B • = (B O ) of basic set data for the groups A G (O), where O runs over all nilpotent orbits. Then the pair (≤ N B• , β N B• ) defined above is a basic set datum for W . Proof. Let F ∈ Irr FW . Let us write (O ′ , L σ ) := Ψ F (F ), and let E

  of ℓ-regular partitions, that is, those which do not contain entries repeated at least ℓ times. The symbols λ ⊢ n and λ ⊢ (ℓ) n mean respectively λ ∈ Part n and λ ∈ Part

n

  ) to S λ , is a basic set datum, which we will call the James basic set datum. Now, the ordinary Springer correspondence sends the Specht module S λ to the orbit O λ * (with the trivial local system), where λ * denotes the conjugate partition of λ. It follows that the Springer and James basic set data involve the same order relation, hence they coincide by Corollary 2.4. We have deduced the modular Springer correspondence for GL n .Theorem 3.15. For µ ∈ Part (ℓ) n , we have Ψ GLn F (D µ ) = (O µ * , F). To simplify the notation, we set d Sn λ,µ := d Sn S λ ,D µ and d N λ,µ := d N (O λ ,K),(Oµ,F) . Theorems 3.5 and 3.15 imply that d Sn λ,µ = d N λ * ,µ * for λ ⊢ n and µ ⊢ (ℓ) n. 4. Dipper-James basic set data 4.1. Type B n /C n . Let W n be the Weyl group of type B n . We can identify it with the group of signed permutations of the set {±1, . . . , ±n}. The classification of ordinary and modular simple modules is completely analogous to the case of the symmetric group [DJ92]. We have a parametrization Irr KW n = {S λ | λ ∈ Bip n }, where Bip n denotes the set of all bipartitions of n, that is, the set of pairs of partitions λ = (λ (1) , λ (2) ) such that |λ (1) | + |λ (2) | = n. For example, the trivial representation is labeled by the pair ((n), ∅) and the sign representation by ((∅, (1 n )).

  ], a Morita equivalence allows to express the decomposition numbers d Wn λ,µ := d Wn S λ ,D µ of W n in terms of decomposition numbers for the symmetric groups S r for 0 ≤ r ≤ n. Proposition 4.1. [DJ92] (Recall that ℓ = 2.) Let λ = (λ (1) , λ (2) ) and µ = (µ (1) , µ (2) ) be two bipartitions of n, the latter being ℓ-regular. If |λ (1) | = |µ (1) | =: r, then

  Proposition 4.4 ([GP00], Theorem 5.5.6). For λ in Bip n , we have ε Wn ⊗S λ = S λ * . Remark 4.5. We have λ ≤ DJ µ ⇔ µ * ≤ DJ λ * . 4.2. Type D n . For n ≥ 2, let W ′ n be the index two subgroup of W n consisting of signed permutations with an even number of sign flips. It is a Weyl group of type D n .

  Bip n the subset of ℓ-regular unordered bipartitions (those where each partition is ℓ-regular), and similarly for Bip

  any E ∈ Irr KW ′ n we denote by a E the a-invariant of E (see Definitions 6.5.7 and 9.4.8 in [GP00]). For any F ∈ Irr FW ′ n , we define a F := min{a E | E ∈ Irr KW ′ n and d W ′ n E,F = 0}. Proposition 4.6. (Relation between Irr FW n and Irr FW ′ n [Gec00, §6].

  (2) ] * = [λ (2) * , λ (1) * ], and [λ, ±] * = [λ * , ±]. Proposition 4.10 ([GP00], Remark 5.6.5). For λ ∈ Bip n , we have ε W ′ n ⊗S λ = S λ * . Remark 4.11. For λ, µ ∈ Bip n , we have λ ≤ DJ µ ⇔ µ * ≤ DJ λ * .

5. 3 .

 3 Dimensions of Springer fibres. For (O, L) ∈ P G , let us denote δ((O, L)) = dim B O . There is a combinatorially defined function δ co : Symb G → N describing δ (see [GM00, §2.E]). Since this quantity depends only on the nilpotent orbit and not on the local system, the function δ co needs to be constant on similarity classes. For Λ ∈ Symb G , we have δ co (Λ) = σ(w(Λ))σ(w(Λ G )), where σ(s 1 , . . . , s 2m+d ) = 1≤i<j≤2m+d min(s i , s j ).

  2. Definition of Springer basic sets. Suppose that we have a family B • = (B O ) of basic set data for the groups A G (O). For each nilpotent orbit O, we write B O = (≤ O , β O ). To such a family B • , we will associate a basic set datum (≤ N B• , β N B•

This research was partially supported by the ANR projects REPRED (ANR-09-JCJC-0102-01) and GEREPMOD (ANR-16-CE40-0010-01). The first author is gratedul for the support of the Charles Simonyi Endowment at the Institute for Advanced Study, where the last stages of the writing were accomplished.

entries of the symbols Φ G co (λ) and Φ G co (µ) into the respective sequences x = (x 1 ≤ x 2 ≤ • • • ≤ x t ) and y = (y 1 ≤ y 2 ≤ • • • ≤ y t ), i.e. x = w(dist Φ G co (λ)) and y = w(dist Φ G co (µ)), then there exist indices a < b such that y b = x b + 1 and y a = x a -1, while y i = x i for all i = a, b. This implies that for 1 ≤ i ≤ t, we have

be the sequences obtained from x and y as in §7.3, so that λ = 2(xw(Λ G )) and µ = 2(ỹw(Λ G )). Hence to prove that λ < µ, it is enough to prove that x < ỹ. Moreover, proving x ≤ ỹ is enough: we know that that λ and µ cannot be equal, since by §6, dim O G λ < dim O G µ . First consider the case G = Sp 2n . Recall that x and ỹ are obtained from x and y by replacing pairs (s, s) of equal consecutive elements by (s -1/2, s + 1/2). Then for all 1 ≤ i ≤ t,

We get

Suppose S i (ỹ) -S i (x) < 0 for some i. Then we must have i / ∈]a, b], y i-1 = y i , and x i-1 = x i , which can only happen if i = a and y a-1 = y a + 1, or i = b + 1 and y b -1 = y b+1 . In both cases, we derive a contradiction for the entries of y weakly increase from left to right. Since S i (x) ≤ S i (ỹ) for any 1 ≤ i ≤ t, we conclude that x ≤ ỹ and we are done. Now assume G = SO 2n+1 . Then t = 2m + 1. According to §7.3, we can form disjoint pairs (y i-1 , y i ) of consecutive letters in y such that y i-1 = y i . For such a pair we have (ỹ i-1 , ỹi ) = (y i-1 , y i ) if y i-1 = y i and i is even, (y i-1 -1, y i + 1) if y i-1 = y i and i is odd.

Let K be the set of indices i such that y i belongs to one of these pairs. Now for any i /

Consider i ∈ {1, . . . , 2m + 1}. When i is even, (y i-1 , y i ) is not a distinguished pair. Moreover, the set U i = {j ∈ {i, . . . , 2m} | j / ∈ K} contains the same number of even and odd indices. Thus S i (ỹ) = S i (y). When i is odd, by computing the number of odd and even indices in U i in both cases (y i-1 , y i ) distinguished or not, we obtain

We have the same relation between S i (x) and S i (x). Moreover S i (x) = S i (y) -1 ]a,b] (i). This gives

When i is even, we have clearly S i (x) ≤ S i (ỹ). This is also true for i odd by using the same arguments as for Sp 2n .

For G = SO 2n , we proceed as in the case G = SO 2n+1 , switching the cases i even and i odd due to the fact that t = 2m is even.