Dr Thomas K Ellis

Dr Melodie Galerne

Joseph J Armao

Dr Artem Osypenko

Prof. Dr David Martel

Dr Mounir Maaloum

Dr Gad Fuks

Odile Gavat

Emilie Moulin

Nicolas Giuseppone
email: giuseppone@unistra.fr

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Keywords: Supramolecular polymers, Electrochemistry, Smart materials, Nanowires, Supramolecular electronics

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Supramolecular Electropolymerization

Thomas K. Ellis, § Melodie Galerne, § Joseph J. Armao IV, Artem Osypenko, David Martel, Mounir Maaloum, Gad Fuks, Emilie Moulin, Odile Gavat, and Nicolas Giuseppone* Abstract: Gaining control over supramolecular polymerization mechanisms is of high fundamental interest to understand selfassembly and self-organization processes at nanoscale. It is also expected to significantly impact the design and improve the efficiency of advanced materials and devices. Up to now, supramolecular polymerization has been shown to take place from unimers in solution, mainly by variations of temperature or of concentration. Here we show that supramolecular nucleation-growth of triarylamine monomers can be triggered by electrochemistry in various solvents. The involved mechanism offers new opportunities to precisely address in space and time the nucleation of supramolecular polymers at an electrode. To illustrate the potential of this methodology, we grow and orient supramolecular nanowires over several tens of micrometers in between different types of commercially available electrodes submitted to a single DC electric field, reaching a precision unprecedented in the literature.

Because supramolecular bonds are labile in essence, supramolecular polymers are highly dynamic chemical objects. [START_REF] Ciferri | Supramolecular Polymers[END_REF][START_REF] De Greef | [END_REF] The kinetics of formation and dissociation of such polymers can be partially controlled by intensive parameters (e.g. temperature, concentration, mechanical stress) or by modifying in situ their supramolecular recognition units. [3] This toolbox is of particular interest for the design of responsive materials. [4,5] However, one drawback of such an intrinsic dynamics is that gaining control over their polydispersity index is particularly difficult. To improve that aspect, recent seminal works have been developed to access living supramolecular polymerization processes. [6,7] Another very challenging aspect of supramolecular polymerization concerns its precise control in space (i.e. with a spatial resolution approaching the size of the polymer itself). In this direction, the possibility to precisely address the nucleation and growth of supramolecular polymers by electrochemistry attracted our attention. The rational design of the present study rests on the supramolecular polymerization of triarylamine molecules (TAA) substituted with (an) amide function(s), such as molecule 1 in Figure 1. [8][9][10] We have previously shown that, in chlorinated solvents, light-triggered oxidation of TAA to the corresponding radical cation (TAA •+ ) can promote supramolecular polymerization. The associated mechanism was thoroughly investigated by a number of experimental and theoretical means, [11,12] and it was proved to involve a nucleation/growth process composed of four elementary steps: (i) photoexcitation of a catalytic amount of triarylamines and subsequent oxidation to their radical cations, with concomitant reduction of the chlorinated solvent producing chloride counterions; (ii) formation of a nucleus of triarylammonium radicals in a double columnar arrangement involving hydrogen bonds; (iii) stacking of neutral triarylamines onto the nucleus and subsequent growth of the primary fibril; (iv) lateral secondary aggregations of fibrils by van der Waals forces to reach larger bundles of fibers. Based on this mechanism, we envisioned to suppress the lightirradiation step and to replace it by a direct electrochemical oxidation (Figure 1a). We first checked the possibility to oxidize 1 using light irradiation in the presence of a high concentration of commonly used electrolytes ([1] = 1 mM; [electrolyte] = 100 mM in 1,1,2,2-tetrachloroethane (TCE)). By probing the system with UV-Vis-NIR spectroscopy, we confirmed the effective photoinduced oxidation in the presence of tetrabutylammonium perchlorate (TBAClO4), hexafluorophosphate (TBAPF6), or chloride (TBACl) (section S5.1, Figure S5). Indeed, as already described extensively, [8,11] one of the typical signatures associated to radical cation 1 •+ , and further to the self-assembly process, is the appearance of a stable absorption band around 750 nm (Figure 1b, dashed line). We then performed cyclic voltammetry (CV) at a scan rate of 100 mV•s -1 for TAA 1 in TCE and in the presence of TBAPF6 (Figure 1c), showing a first oxidation associated with radical cation 1 •+ (E1/2 = -71 mV), and a second oxidation leading to the corresponding dication 1 2+ (E1/2 = 704 mV). [13] Spectroelectrochemistry was also performed on this solution, showing the expected UV-Vis spectrum of 1 •+ as it passes over the first oxidation potential (Figure 1d). The resulting spectrum perfectly matches with the one observed upon photoirradiation (Figure 1b, plain line). We further tested whether the wires can self-assemble electrochemically from 1 and 1 •+ in the presence of the electrolyte and in the dark. For that, we probed by [START_REF] Ciferri | Supramolecular Polymers[END_REF] H NMR spectroscopy the behavior of a solution of 1 in deuterated chloroform (5 mM) and in the presence of TBAPF6 (100 mM), while holding a potential of 0.2 V vs Fc/Fc + (Ferrocene / Ferrocenium reference redox couple) for 30 min. This time period of electro-oxidation involves the production of approximately 1% of 1 •+ in the solution as determined by coulometry (section S5.2. Figure S7). [START_REF] Ciferri | Supramolecular Polymers[END_REF] H NMR spectrum shows that, after this electrochemical treatment at constant concentration and temperature, the aromatic proton resonance signals of the triarylamine fully disappear. A similar behavior was observed when electro-oxidation of 1 was performed in the absence of electrolyte with a high potential of 25 V held for 30 min in TCE-d2 (Figure S8a). The disappearance of the NMR signals after the oxidation of a small percentage of the triarylamine molecules is a typical signature of a nucleation step involving radical cations and followed by the formation of large and rigid anisotropic stacks involving all the remaining neutral TAAs as elucidated in our previous works. [8,11] Drop cast samples from this solution were also imaged by both TEM and AFM, confirming the formation of bundled supramolecular triarylamine nanowires (Figure 1e andS8b). By analogy with our previous investigations involving a photogenerated radical cation [11] and accordingly to a series of control experiments (see sections S6-S8 in the supporting information), we can conclude that the electrochemically triggered self-assembly mechanism involves the following sequence (Figure 1f): (i) diffusion of neutral molecule 1 into the double layer, (ii) oxidation of the neutral species at the electrode surface producing 1 •+ , (iii) migration of the radical cation into the diffusion layer, (iv) nucleation of the radical species, and (v) growth of the supramolecular fibers by addition of neutral monomers onto the nuclei. Knowing the possibility to electrochemically trigger the nucleation/growth of supramolecular triarylamines in solution, we then investigated the possibility to directly interconnect commercially available microelectrodes in the presence of a sole Direct Current (DC) electric field, without electrolyte. However, when using chloroform or 1,1,2,2-tetrachloroethane to dissolve TAA molecules, we were unable to grow nanowires at low voltages (probably because of the difficulty to form 1 •+ in the absence of electrolyte), neither to align them at higher voltages due to short-circuits (probably because of the red-ox activity of the chlorinated solvent itself). Interestingly, we found that TAA 1 can dissolve at elevated temperatures in alkanes (such as heptane in the following study), and that it forms a thermo-responsive fibrillar organogel upon subsequent cooling (with an onset gelation concentration of 2.5 mg/mL at T ≈ 10 ˚C) (Figure 2a). A peculiar property of this thermal transition is its large hysteresis showing a good stability of the gel when heating up to T ≈ 30 °C (Figure S13). By making use of this hysteresis, we found that at 20°C the supramolecular fibers formation can be triggered by electroactivation from solution. Indeed, by introducing a 5 mg/mL solution of 1 between 5 μm path sealed ITO plates (which avoids evaporation), and after application of a DC field of 1 x 10 7 V•m -1 , we observed the formation of a dense array of fibers by polarized optical microscopy (POM) (Figure 2b). We subsequently questioned whether fibers formation within the electrical field requires an electrical contact at the electrode to start nucleation. A blocked ITO layer device coated with a photoresist did not show fibers formation, except at the edges that contain flaws in the photoresist (Figure S2c). The necessity of the electrical contact for complete wire formation indicates that, even without electrolyte, the immediate contact with the electrode in such high field is able to create an oxidized seeding surface from which neutral TAA can grow. In addition, the continuous growth of individual fibers from the electrode edges across the solution is also highlighting that the fibers formation is clearly not triggered by a sol-gel transition (Figure 2c, and section S10). We then turned our investigations towards gold interdigitated electrodes (IDEs) of lateral geometry, separated by parallel gaps of 10 m, and in the presence of a DC electric field. The substrates were analyzed by POM, SEM, and AFM, all showing the homogeneous alignment of mono-disperse fibers filling all gaps with an orientation parallel to the applied electric field (Figure 2d-f). Using this set-up configuration, we first determined unambiguously that the fibers formation is the result of an electrooxidation and does not come from thermal or concentration effects. Indeed, a first control experiment without an electric field revealed only the presence of amorphous and unlocalized material (Figure S14). Thus, in the time span needed for the solvent to evaporate, the nucleation-growth process does not take place, which also excludes the involvement of dielectrophoretic alignment processes. In a second control experiment, a solution that had been previously allowed to form a gel at low temperature was partly re-solubilized with a gentle heating at 30 °C. At this temperature, although being macroscopically in the sol state, fibers seeds remain in suspension. However, after drop-casting on the IDEs in absence of an electric field and evaporation of the solvent, only a disordered network of fibers was observed (Figure S15a). In addition, application of an electric field of 1.25 x 10 7 V•m - 1 for duration of 5 minutes after deposition and concentration to the gel state by solvent evaporation, did not show any further alignment (Figure S15b). This last observation indicates that the reorientation of the fibers after entanglement in a low mobility medium is not possible. Overall, these series of experiments in vertical sealed ITO cells and horizontal gold IDEs configurations reveal that (i) the successful polymerization procedure involves electro-oxidation of TAA 1 at the electrode surface as the sole cause for localized nucleation of fibers, and that (ii) the good alignment of these fibers requires their oriented supramolecular growth in the electric field before the formation of a kinetically trapped disordered entangled network. Our description of the self-organization process in the DC field is reinforced by the observation that the nucleation starts always from the anode (+ electrode) with the fibers growing towards the cathode (-electrode), as clearly shown by SEM and AFM on Figures 3a and3c. Nucleation/growth at anode was in addition visualized by a real time movie in the interdigitated configuration (Supplementary Movie 1, and section S11.5). Another striking feature of this process is illustrated by the precise topology of the bridging fibers observed by AFM (Figure 2g) and showing that the supramolecular polymers are strictly following the curvature of the electric field during their growth (Figure 2h). We then studied the effects of field strength and concentration on fibers polymerization and alignment. We determined that the minimum field strength required for the polymerization is Ef ≈ 1 × 10 6 V•m -1 . However, with such a field, the fibers are still highly disordered (Figure S17a,b). Alignment started to occur at 5 × 10 6 V•m -1 and was found to be optimal at 1.25 × 10 7 V•m -1 (Figure S17c,d). For the optimal voltage, we determined that a minimal 2 mg/mL concentration of TAA leads to isolated fibers spanning the entire gap (Figure S16). A higher concentration of 5 mg/mL resulted systematically in fully filled electrode gaps with dense bundles of fibers, with some isolated fibers passing through the cathode (Figure 2e). We quantified the degree of alignment by calculating the order parameter S (0 ≤ S ≤ 1) as a function of the electric field value (Figure 3b, and section S12). At a voltage of 1.25 × 10 7 V•m -1 , we found average order parameters of 0.94±0.08 for single fibers spanning the entire gap (2 mg/mL) (Figure 3d) and of 0.87±0.09 for single fibers spanning the negative electrodes (5 mg/mL) (Figure S23). This means that high TAA concentration increases the number of fibers spanning the gaps without modifying the very high quality of the alignment. In addition, by applying an electric field to a device with a more complex geometry, that is using circular concentric interdigitated electrodes, we unambiguously proved that the wires precisely follow in any point the field's lines (Figures S18 a-c). As a last implementation, using microarray electrodes, we were able to grow single supramolecular wires between two pores of less than 5 m in diameter and over a distance of 50 m in length (Figure 3d). The POM image confirms the nucleation from the anode and the fiber growth towards the cathode only when the pores align in the DC field direction. To the best of our knowledge, such a precise bridging configuration is unprecedented in the literature, either by using top-down techniques such as zonecasting [14] or by orienting organogelators [15] and liquid crystal polymers [16][17][18][19] in an AC field.

In addition, our previous method consisting in the photo-triggered supramolecular polymerization of triarylamine between nanoelectrodes [20] is not adapted to the various configurations of the present study. First, photo-oxidation appeared experimentally limited to path lengths below 100 nm, a distance almost three orders of magnitude smaller compared to the present devices. We postulate that this limitation partly comes from the necessary use of chlorinated solvents to trigger TAA oxidation with light. Indeed, the large electric field necessary to align the wires on long distances would be shielded by the high amount of photogenerated triarylamonium radical cations and their respective chloride counter ions. If this effect can indeed be overcome by using nanoelectrodes in which the Debye length is larger than the electrode gap, it precludes using photo-oxidation for higher path lengths (section S13). In addition, in our previous approach, [20] fibers were grown everywhere in solution, necessitating the removing of unattached fibers all over the device. Here, making

Anode +E v Cathode -E v Gap +E v +E v Gap -E v -E v
Gap use of a single electrical stimulus, the supramolecular electropolymerization mechanism reaches a much higher precision in the sense that fibers can only nucleate from the anode, opening new possibilities to grow selectively single wires between two-point electrodes (as in Figure 3d). Finally, from a functional point of view, the mobility of charge carriers within the supramolecular nanowires formed by this technique in interdigitated electrodes was derived from the current vs voltage measurement and by using the space charge limiting approximation [21] (which is justified by the low concentration of ions introduced in the system) (section S14). By assuming the number of wires in the electrodes, and by assuming all of them being connected, we can estimate a minimum value as high as 0.56 cm 2 •V -1 •s -1 , close to the theoretical prediction published in the literature. [22] In conclusion, we have described the first example of what can be defined as a "supramolecular electropolymerization" process. It represents a new tool of fundamental and practical interest for supramolecular chemistry in general. To this end, we have used triarylamines but one can imagine to extend the concept to other molecules with red-ox functionalities and capable of forming supramolecular polymers or self-assemblies involving charge transfer complexes. [23] In addition, we have shown that the mechanism involved can be used to gain control over the precise placement and orientation of supramolecular polymers in between electrodes with a precision that cannot be reached by other current means, up to the bridging of two electrodes of less than 5 m in diameter and over 50 m in length by a single nanowire. Such a precise control in every aspect of supramolecular polymerization, including localization and orientation on nano-and micrometer scales, is at the moment quite unique in the literature. Other potential implementations resulting from these experiments include for instance: (i) spatial control over self-assembly processes in various optoelectronic devices requiring efficient charge transport and ranging from sensors to solar cells, [24,25] (ii) use of waste-free electrochemistry tools to control (living) supramolecular polymerization or to create out-of-equilibrium redox environments for self-organized systems, [26] and (iii) design of 3D electrical wiring in organogels.

Figure 1 .

 1 Figure 1. (a) Electro-or photo-triggered oxidation of TAA 1 in its radical cation 1 •+ ; (b) Normalized absorption spectra of electro-and photo-oxidized solutions of 1 at a concentration of 1 mM in TCE and in the presence of 0.1 M TBAPF6; (c) CV data of compound 1 at a concentration of 1 mM in TCE with 100 mM TBAPF6 and at a scan rate of 0.1 V•s -1 ; (d) Spectroelectrochemical behavior of 1 for different potential scans over the first oxidation wave in the range -0.15 to 0.15 V vs Fc/Fc + in TCE with 100 mM TBAPF6; (e) (i) TEM and (ii) AFM height images of the supramolecular fibers formed from 1 by an electrochemically triggered process; (f) Simplified electrochemical mechanism of the (nucleation / growth) supramolecular polymerization process activated by electro-oxidation.

Figure 2 .

 2 Figure 2. (a) TEM image of the fibrillar organogel of TAA 1 in heptane; (b) Cross-polarized optical microscopy image of supramolecular polymers formed between ITO plates: a 5 mg/mL solution was let to cool down to room temperature before deposition into the ITO cell with a path of 5 μm, and the solution was subsequently subjected to a DC electric field to trigger the supramolecular polymerization; (c) Cross-polarized optical microscopy image of a cell reproducing the protocol described in (b), but using blocked ITO electrodes, avoiding the electrochemically triggered formation of the supramolecular polymers, except at the edge of the ITO domain; (d-f) POM (d), SEM (e) and AFM (f) images of IDEs after drop-casting a 5 mg/mL solution of 1 and under application of an electric field of 1.25 x 10 7 V•m -1 ; (g) Topological representation of the AFM image shown in (f); (h) 3D schematic depiction of the supramolecular electropolymerization process occurring between IDEs and following the electric field lines as observed in (g) (the green and red ovals represent 1 and 1 •+ , respectively).

Figure 3 .

 3 Figure 3. (a) SEM image of an IDE prepared with a 2 mg/mL solution of 1 in the presence of an applied field of 1.25 x 10 7 V•m -1 and highlighting the nucleation taking place from the anode only; (b) plot of the order parameter (S) as a function of the field strength (dots: [1] = 5 mg/mL; triangle: [1] = 2 mg/mL), the error bars are calculated from four different regions of the IDEs; (c) AFM image of an IDE prepared with a 2 mg/mL solution of 1 in the presence of an applied field of 1.25 x 10 7 V•m -1 and highlighting the formation of thin fibers with nucleation taking place from the anode only; (d) POM image of a microelectrode array showing a single wire grown from the anode to the cathode, parallel to the electric field (white arrow) and over a distance of 50 μm.

  oxidation of triarylamines at anodes triggers their nucleation and growth into supramolecular polymers. This mechanism can be defined as a supramolecular electropolymerization process. It is used to grow and precisely address single nanowires over a distance of 50 m between two electrodes in a DC electric field. Thomas K. Ellis, Melodie Galerne, Joseph J. Armao IV, Artem Osypenko, David Martel, Mounir Maaloum, Gad Fuks, Emilie Moulin, Odile Gavat, and Nicolas Giuseppone* Page No. -Page No.

Acknowledgements

The research leading to these results has received funding from the European Research Council under the European Community's Seventh Framework Program (FP7/2007-2013) / ERC Starting Grant agreement n°257099 (N.G.), the European Union (FP7/2007-2013, ITN Read), the Agence Nationale de la Recherche (STANWs ANR-11-EMMA-0009) and the international center for Frontier Research in Chemistry (icFRC). We also thank the Centre National de la Recherche Scientifique (CNRS), the COST action (CM 1304), the Laboratory of Excellence for Complex System Chemistry (LabEx CSC), the University of Strasbourg (UdS), and the Institut Universitaire de France (IUF). Part of this work was performed within the framework of the IRTG "Soft Matter Science: Concepts for the Design of Functional Materials" (M.G., E. M. and N.G.). We acknowledge the Electron Microscopy Facility of the Institut Charles Sadron. We finally thank André Schroeder for help with optical microscopy.

Conflict of interest

The authors declare no conflict of interest.


Gwyn Bellamy
email: gwyn.bellamy@glasgow.ac.uk

Cédric Bonnafé
email: cedric.bonnafe@umontpellier.fr

Baohua Fu
email: bhfu@math.ac.cn

Daniel Juteau
email: daniel.juteau@u-picardie.fr

Paul Levy
email: p.d.levy@lancaster.ac.uk

Eric Sommers
email: esommers@math.umass.edu

Lamfa

A NEW FAMILY OF ISOLATED SYMPLECTIC SINGULARITIES WITH TRIVIAL LOCAL FUNDAMENTAL GROUP by

We construct a new infinite family of 4-dimensional isolated symplectic singularities with trivial local fundamental group, answering a question of Beauville raised in 2000. Three constructions are presented for this family: (1) as singularities in blowups of the quotient of 4 by the dihedral group of order 2d , (2) as singular points of Calogero-Moser spaces associated with dihedral groups of order 2d at equal parameters, (3) as singularities of a certain Slodowy slice in the d -fold cover of the nilpotent cone in sl d .

Introduction

Symplectic singularities were introduced by Beauville [Bea]. Their introduction has led to numerous important developments in both algebraic geometry and geometric representation theory. Basic examples of symplectic singularities include symplectic quotient singularities and singularities in normalizations of nilpotent orbit closures in semisimple Lie algebras [START_REF] Beauville | Symplectic singularities[END_REF]Section 2]. Note that any two dimensional symplectic singularity is just a rational double point.

An isolated symplectic singularity of dimension 4 is a normal isolated singularity whose smooth locus admits a holomorphic symplectic 2-form [START_REF] Beauville | Symplectic singularities[END_REF](1.2)]. It follows that such a singularity is canonical Gorenstein, hence its local fundamental group is finite [Bra]. Symplectic quotient singularities give many examples of isolated symplectic singularities, but they all have non-trivial local fundamental groups.

Minimal nilpotent orbit closures g min in simple Lie algebras g give examples of isolated symplectic singularities with trivial local fundamental group. In this case, the projective tangent cone PT 0 ( g min ) of g min at 0 is isomorphic to P g min = g min / × , which is smooth and G -homogeneous (where G is the adjoint Lie group of g). It turns out that the smoothness of projective tangent cones characterizes g min , by the following result.

Theorem 1.1 (Beauville). -Let ( , x ) be an isolated symplectic singularity whose projective tangent cone at x is smooth. Then there exists a simple Lie algebra g such that ( , x ) is locally analytically isomorphic to ( g min , 0). Moreover, the singularity ( g min , 0) has trivial local fundamental group if and only if g sp 2n ( ) for any n 1.

If ( , x ) is an isolated symplectic singularity whose projective tangent cone at x is smooth, then the simple Lie algebra g such that ( , x ) is locally analytically isomorphic to ( g min , 0) is uniquely determined. It can be recovered as the Lie algebra structure on the cotangent space T *

x ( ) of at x induced by the Poisson bracket. In other words,

(1.2)

T * 0 ( g min ) g
as Lie algebras.

When g sp 2n ( ), the singularity ( g min , 0) is locally analytically isomorphic to ( 2n /µ 2 , 0), where µ 2 = {1, -1} acts by multiplication on 2n , so the local fundamental group is isomorphic to µ 2 .

In the same paper, Beauville asked [START_REF] Beauville | Symplectic singularities[END_REF](4.3)] whether there exist other isolated symplectic singularities with trivial local fundamental group. In the intervening two decades, no other examples have come to light.

Constructing examples of new isolated symplectic singularities is also motivated by the long-standing conjecture (attributed to LeBrun and Salamon) that any Fano contact manifold is isomorphic to P g min for some simple Lie algebra g. In fact, if Z is a Fano contact manifold and L is the contact line bundle, then there exists a map L * → X which contracts the zero section to one point o . The contact structure on Z induces a symplectic structure on the complement of the zero section in L * , which gives a symplectic form on X \ {o }. This implies that (X , o ) is an isolated symplectic singularity. By the same argument as in [START_REF] Beauville | Symplectic singularities[END_REF]Proposition 4.2], X has trivial local fundamental group except in the case where Z 2n -1 .

The aim of this paper is to describe a new infinite family of 4-dimensional isolated symplectic singularities with trivial local fundamental group. We will give three different constructions of this family: one from blowups of symplectic quotient singularities, one from Calogero-Moser spaces associated with dihedral groups, and the last one from slices in the d -fold cover of the nilpotent cone of sl d .

For this, let V be a complex vector space of dimension 2, let d 4, let W d ⊂ GL (V ) denote the dihedral group of order 2d and let (d ) denote the symplectic quotient singularity (V × V * )/W d . We define two varieties:

• Let ˜ (d ) denote the blowup of (d ) at its singular locus.

• Let (d ) denote the Calogero-Moser space associated with W d at non-zero equal parameters [START_REF] Bonnafé | On the Calogero-Moser space associated with dihedral groups[END_REF]: it is a Poisson deformation of (d ).

Then both ˜ (d ) and (d ) are normal varieties of dimension 4, which have a unique singular point (denoted by 0 for in both cases). Now ( (d ), 0) is a symplectic singularity by general results about Calogero-Moser spaces [Gor]. It is of course an isolated singularity.

As will be shown in the paper, ( ˜ (d ), 0) is also a symplectic singularity (Corollary 3.8), which is locally analytically isomorphic to ( (d ), 0) (Corollary 2.15). In the case ˜ (5), this was observed in [START_REF] Fu | Generic singularities of nilpotent orbit closures[END_REF]Remark 12.8]. Our first main result is the following: Of course, the analogous statements also hold for the singularity ( ˜ (d ), 0). The proof relies on the explicit computation of the equations for (d ) (see [START_REF] Bonnafé | On the Calogero-Moser space associated with dihedral groups II. The equal parameter case[END_REF]) and ˜ (d ). A key role is played by an action of SL 2 on both (d ) and ˜ (d ).

Symplectic singularities of Calogero-Moser spaces associated with the complex reflection groups denoted by G (d , 1, n ) in Shephard-Todd classification [ShTo] are relatively well-understood, as they are quiver varieties [EtGi]. However, we think it might be interesting to study the symplectic singularities of Calogero-Moser spaces associated with other complex reflection groups. For the infinite family G (d e , e , n ) (note that W d = G (d , d , 2)), and the 34 exceptional groups, it is appealing to expect to find more interesting examples of symplectic singularities.

The third construction is a certain cover of a type A Slodowy slice [Slo]. The proof of this theorem is rather indirect. The cyclic group µ d acts on the singularity ( (d ), 0) and the quotient can be identified with the Calogero-Moser space (at a nongeneric parameter) ( (d , 1, 2), x 0 ) associated to G (d , 1, 2). By [EtGi, Mar], the latter space is isomorphic to a certain quiver variety (M λ (v), x 0 ) associated to the framed affine quiver of type A, which can be shown to be isomorphic to (d ) by using results of Nakajima [Nak] (see also [Ma]). This implies that both (d ) and ˜ (d ) are d -fold coverings of (d ), which are shown to be locally holomorphically isomorphic by using the triviality of the local fundamental group of (d ).

It is interesting to remark that the singularity ( ˜ (5), x5 ) is exactly the generic singularity of the nilpotent orbit closure of A 4 + A 3 along the codimension 4 boundary A 4 + A 2 + A 1 in E 8 . More surprisingly, the whole of ˜ 5 is in fact isomorphic to the Slodowy slice of (A 4 + A 3 , A 4 ) in E 8 (see [START_REF] Fu | Generic singularities of nilpotent orbit closures[END_REF]Section 12.3]).

Acknowledgements. We wish to thank S. Baseilhac, A. Bruguières and C. Xu for useful discussions about fundamental groups. We also thank the group of A. Hanany, and particularly A. Bourget, for many conversations about physics and symplectic singularities; based on the Hilbert series in §6.B, Z. Zhong proposed a magnetic quiver whose Coulomb branch should provide yet another description of our singularities.

Convention, notation.

We work over the field of complex numbers (polynomials, Lie algebras, algebraic varieties, schemes are supposed to be complex). If is an affine scheme, we denote by [ ] its ring of regular functions.

We fix a natural number d 4, we denote by µ d the cyclic group of order d and we fix a primitive d -th root of unity ζ. Let V = 2 and denote by (x , y ) its canonical basis and by (X , Y ) its dual basis. Thus, X , Y ∈ [V ] and we may view x and y as elements of [V * ]. If j ∈ or /d , we set

s j = 0 ζ j ζ -j 0 ∈ GL 2 ( ) = GL (V ), s = s 0 , t = s 1 and W d = 〈s , t 〉 = 〈s 0 , s 1 , . . . , s d -1 〉: it is the dihedral group of order 2d .
Let U 2 denote the standard representation of sl 2 so that Sym d (U 2 ) is the unique simple sl 2 ( )-module of dimension d + 1. We denote by g d the Lie algebra sl 2 ⊕ Sym d (U 2 ), where the Lie structure is determined by the fact that Sym d (U 2 ) is a commutative ideal of g d on which sl 2 acts through its natural action.

Deformation and blowup of a symplectic quotient singularity

2.A. Symplectic quotient (V × V * )/W d . -The induced action of W d on V × V * is sym- plectic, making the quotient (d ) = (V × V * )/W d
an affine variety with only symplectic singularities. The singular locus S ⊂ (V × V * )/W d consists of W d -orbits with non-trivial stabilizer, namely the image of the points of the form (λ(xζ j y ), µ(Yζ j X )), with λ, µ ∈ and j ∈ /d . It is easy to check that S is of codimension 2 and the singularity of

(V × V * )/W d along a nonzero point of S is an A 1 -singularity.
We denote by (Ψ k ) k 0 the sequence of polynomials in three indeterminates q , Q , e , defined by

   Ψ 0 = 1, Ψ 1 = e , Ψ k = e Ψ k -1 -qQ Ψ k -2 , if k 2.
(see [START_REF] Bonnafé | On the Calogero-Moser space associated with dihedral groups II. The equal parameter case[END_REF](1.4)]). Note that Ψ k is homogeneous of degree k . Alternatively, Ψ k can be defined by the generating series k 0

Ψ k t k = 1 1 -e t + qQ t 2 in [q ,Q , e ][[t]].
In particular,

(2.1)

Ψ k (q , 0, e ) = Ψ k (0,Q , e ) = e k and Ψ k (q ,Q , 0) = 0 if k is odd, (-qQ ) k /2 if k is even. Consider the following W d -invariant polynomials in [V × V * ]. q = x y ,Q = X Y , e = x X + y Y , a i = x d -i Y i + y d -i X i , 0 ≤ i ≤ d .
By [START_REF] Bonnafé | On the Calogero-Moser space associated with dihedral groups II. The equal parameter case[END_REF]Theorem 1.6], we have (2.2)

(d ) = {(q ,Q , e , a 0 , a 1 , . . . , a d ) ∈ d +4 | ∀ 1 j k d -1, e a j = q a j +1 + Q a j -1 , a j -1 a k +1 -a j a k = (e 2 -4qQ )q d -k -1 Q j -1 Ψ k -j (q ,Q , e ) }.

2.B. The variety (d ).

-The Calogero-Moser spaces associated with complex reflection groups are extensively studied in [EtGi]. Using the notation of [START_REF] Bonnafé | On the Calogero-Moser space associated with dihedral groups II. The equal parameter case[END_REF], we denote by (d ) the Calogero-Moser space associated with the dihedral group W d of order 2d at equal parameter a = 1. By [EtGi], the variety (d ) is endowed with a Poisson structure. The case a = 0 is isomorphic to the case a = 1 and the only change is that the Poisson structure is multiplied by a scalar: this is irrelevant for our purpose. It turns out this Calogero-Moser space is a Poisson deformation of the quotient (d ), with equations given as follows [Bon2, Theorem 2.9]:

(d ) = {(q ,Q , e , a 0 , a 1 , . . . , a d ) ∈ d +4 | ∀ 1 j k d -1, e a j = q a j +1 + Q a j -1 , a j -1 a k +1 -a j a k = (e 2 -4qQ -d 2 )q d -k -1 Q j -1 Ψ k -j (q ,Q , e ) }.
Note that the point 0 = (0, 0, . . . , 0) ∈ d +4 belongs to (d ) and we denote by T * 0 ( (d )) the cotangent space of (d ) at 0.

Let us first recall some facts about the variety (d ) with d 4 (recall that the Lie algebra g d = sl 2 ⊕ Sym d (U 2 ) has been defined at the end of the introduction). Proof. -The fact that (d ) is irreducible, normal, of dimension 4 is a general fact about Calogero-Moser spaces [EtGi].

sl 3 ( ) if d = 4, g d if d 5, (
(a) was proved in [Bel] (see also [START_REF] Bonnafé | On the Calogero-Moser space associated with dihedral groups[END_REF]). (b) is proved in [START_REF] Bonnafé | On the Calogero-Moser space associated with dihedral groups[END_REF]Proposition 8.4] and [START_REF] Bonnafé | On the Calogero-Moser space associated with dihedral groups II. The equal parameter case[END_REF]Proposition 2.12].

(c) The fact that (d ) is a symplectic singularity is again a general fact about Calogero-Moser spaces [START_REF] Gordon | Symplectic reflection algebras[END_REF]Proposition 4.5].

(d) A computation with MAGMA [Mag] (and the CHAMP package built by Thiel [Thi]) based on the equations of (4) given in [START_REF] Bonnafé | On the Calogero-Moser space associated with dihedral groups[END_REF]Proposition 8.3] shows that the projective tangent cone PT 0 ( (4)) of (4) at 0 is smooth. So the result follows from (b) and Beauville's Theorem 1.1 [Bea, Introduction]. We will provide in Remark 3.4 a proof which does not rely on MAGMA computations.

(e) Assume that d 5. By (b), T * 0 ( (d )) g d is not a simple Lie algebra. So, by 1.2, the singularity ( (d ), 0) is not locally analytically isomorphic to ( g min , 0) for any simple Lie algebra g.

(f) follows from (b) and the fact that dim(g

d ) = d + 4 < d + 4 = dim(g d ). 2.C. Blowup. -A polynomial f ∈ [V ×V * ] is called W d -semi-invariant if w (f ) = det(w )f for all w ∈ W d . We denote by [V × V * ] W d -sem the set of W d -semi-invariant polynomials: it is a [V × V * ] W d -module. Let δ = x X -y Y and β j = x d -j Y j -y d -j X j for 0 j d . Then δ, β 0 , β 1 ,. . . , β d are W d -semi-invariants. Let Γ d denote the (normal) cyclic subgroup of W d generated by t s = diag(ζ, ζ -1 ). It is easily seen that [V × V * ] Γ d = [x y , X Y , x X , y Y , (x d -j Y j ) 0 j d , (y d -j X j ) 0 j d ].
Using the action of

W d /Γ d /2 on [V × V * ] Γ d , this can be rewritten (2.4) [V × V * ] Γ d = [V × V * ] W d [δ, β 0 , β 1 , . . . , β d ].
As the product of two elements of the list δ, β 0 , β 1 ,. . . ,

β d is a W d -invariant, we get that (2.5) The [V × V * ] W d -module [V × V * ] Γ d is generated by 1, δ, β 0 , β 1 ,. . . , β d . Since [V × V * ] Γ d = [V × V * ] W d ⊕ [V × V * ] W d -sem , we get that (2.6) The [V × V * ] W d -module [V × V * ] W d -sem is generated by δ, β 0 , β 1 ,. . . , β d .
Therefore, the ideal

I = (δ [V × V * ]) W d = δ [V × V * ] W d -sem of [V × V * ] W d is generated by δ 2 , δβ 0 , δβ 1 ,. . . , δβ d .
Write D = e 2 -4qQ = δ 2 and consider the following W d -invariant rational functions:

b j = β j δ = x d -j Y j -y d -j X j x X -y Y , 0 j d . Then D b j = δβ j is W d -invariant and I is the ideal of [V × V * ] W d generated by D , D b 0 , D b 1 ,. . . , D b d . Let ˜ (d )
be the blowup of (d ) along this ideal. The (contracting) ×action on V × V * by homothety induces a contracting action on the quotient (d ). For this action, D and the b j 's are homogeneous (of degree 2 and d -2 respectively) so the blowup ˜ (d ) inherits a × -action.

Recall that the blowup of an affine variety Spec(A) along an ideal a generated by elements f 1 ,. . . , f r admits an affine open covering indexed by the f j 's and such that the affine open subset associated with f j is isomorphic to Spec(A[a/ f j ]). In our case, the variety

˜ (d ) is covered by affine open subsets (d ), 0 , 1 ,. . . , d defined by (d ) = Spec( [V × V * ] W d [I /D ]) and j = Spec( [V × V * ] W d [I /(D b j )]), for j = 0, 1, • • • , d .
They are all × -stable. The following relations follow from straightforward computations:

(2.7)

a j = e b j -2q b j +1 if 0 j d -1, a j = 2Q b j -1 -e b j if 1 j d , (2.8)    D b 0 = e a 0 -2q a 1 , D b j = Q a j -1 -q a j +1 if 1 j d -1, D b d = 2Q a d -1 -e a d .
Let η be a primitive 4d -th root of unity. The action of diag(η, η -1 ) sends a j to -1β j , while e , q and Q remain invariant. Applying this transformation to the equations (2.2) describing (d ), and dividing the first line by δ and the second line by δ 2 , we obtain the following relations (note the sign change in the second line!):

(2.9)

e b j = q b j +1 + Q b j -1 b j b k -b j -1 b k +1 = q d -k -1 Q j -1 Ψ k -j (q ,Q , e ) if 1 j k d -1.
Lemma 2.10. -With the above notation, we have:

(a) 0 d 4 . (b) If 1 r d -1, then r is smooth. Proof. -(a) We claim that 0 = Spec( [q , a 0 , 1 b 0 , b 1 b 0 ]
), which will imply that 0 4 as 0 is of dimension 4. For this, we will express other variables in terms of q , a 0 ,

1 b 0 , b 1 b 0 . By (2.7), a 0 = e b 0 -2q b 1 , whence e = 1 b 0 a 0 + 2q b 1 b 0 , which gives the expression for e . Take j = k = 1 in (2.9), we have b 2 1 -b 0 b 2 = q d -2 as Ψ 0 = 1, hence b 2 b 0 = ( b 1 b 0 ) 2 -q d -2 ( 1 b 0 ) 2 , which gives the expression for b 2 b 0 . By (2.9), we have e b 1 = q b 2 + Q b 0 , which gives Q = e b 1 b 0 -q b 2 b 0 , the expression for Q , as b 2 b 0 is a polynomial in q , a 0 , 1 b 0 , b 1 b 0 . By (2.9), we have b 1 b k -b 0 b k +1 = q d -k -1 Ψ k -1 (q ,Q , e ), which gives inductively the ex- pressions for b k b 0 for all k ≥ 3. For 1 ≤ j ≤ d -1, we have a j = e b j -2q b j +1 . Replacing e = 1 b 0 a 0 + 2q b 1
b 0 and using (2.9), we have

a j = a 0 b j b 0 + 2q b 1 b j -b 0 b j +1 b 0 = a 0 b j b 0 + 2q d -j Ψ j -1 (q ,Q , e ) 1 b 0 .
It remains to express a d . Recall that a d = 2Q b d -1e b d and a 1 = 2Q b 0e b 1 by (2.7). This gives

a d = 2Q b 0 b d -1 b 0 -e b d = a 1 b d -1 b 0 + e b 1 b d -1 -b 0 b d b 0 = a 1 b d -1 b 0 + e Ψ d -2 (q ,Q , e ) 1 b 0 ,
as desired. This completes the proof for 0 (and the proof for d is similar).

(b) For ease of notation, let

B r = 1 b r , B j = b j b r for 0 j d , j = r . By (2.7) we have a r = e b r -2q b r +1 , hence e = a r B r +2q B r +1 . (We also have e = 2Q B r -1 -a r B r .) For j > r +1, the equation b r +1 b j -1 -b r b j = q d -j Q r Ψ j -r -2 (q ,Q , e
) allows us to iteratively express B j in terms of q ,Q , and e ; we can deal with B 0 , . . . , B r -2 similarly.

For

0 j < r we substitute e = a r • 1 b r + 2q b r +1
b r into (2.7) to obtain:

a j = a r 1 b r + 2q b r +1 b r b j -2q b j +1 = a r B j + 2q d -j +2 Q r Ψ j -r -3 (q ,Q , e )B r .
Similarly, we substitute e = 2Q B r -1 -a r B r into (2.7) to express a r +1 , . . . , a d in terms of a r , q , Q and e . Thus we have obtained

[q ,Q , e , a 0 , • • • , a d , B 0 , • • • , B d ] = [q ,Q , a r , B r , B r -1 , B r +1 ].
We have two relations:

(2.11)

q B r +1 -Q B r -1 + a r B r = 0 and B r -1 B r +1 + q d -r -1 Q r -1 B 2 r -1 = 0.
The first of these follows from the equality of the two expressions for e ; the second from (2.9). It is straightforward to check that the Jacobian has rank 2 (under the same conditions on q ,Q , a r , B r , B r -1 , B r ), which implies that r is smooth (of dimension 4). More specifically, it implies that the subvariety of 6 with equations (2.11) is smooth of dimension 4; by dimensions we know that r is isomorphic to one of its irreducible components.

2.D. Singularity

(d )] = [V × V * ] W d [b 0 , b 1 , . . . , b d ].
Using (2.7), we see that (2.12)

[ (d )] = [q ,Q , e , b 0 , b 1 , . . . , b d ].
This shows that (d ) is a closed subvariety of d +4 and the × -action on (d ) is contracting (to 0). Here is a presentation of (d ) as well as a first result on its singularities:

Proposition 2.13. -Equations for (d ) ⊂ d +4 are given by (2.9). Moreover, (d ) is normal and 0 is its only singular point.

Proof. -Let be the affine scheme defined by the equations (2.9). These equations being homogeneous (recall that e , q and Q have degree with the equations (2.9) shows that we have an isomorphism

(#) [ ] 0 [ (d )] 0
given by e → e , q → q , Q → Q and b j → a j / d 2 + 4qQ -e 2 for 0 j d .

By Theorem 2.3, the variety (d ) is normal so [ (d )] 0 is a normal domain by Zariski's main Theorem. This shows that [ ] 0 is a normal domain. As the natural map [ ] -→ [ ] 0 is injective (because [ ] is -graded and [ ] 0 is the completion with respect to the unique maximal homogeneous ideal), this shows that [ ] is a domain. Therefore,

Kdim [ ] 0 = Kdim [ (d )] 0 = Kdim [ (d )] = 4,
where Kdim denotes the Krull dimension. Hence is irreducible reduced of dimension 4. As (d ) is irreducible of dimension 4, this shows that = (d ). In other words, equations for (d ) ⊂ d +4 are given by (2.9). Moreover, (#) becomes (2.14)

[ (d )] 0 [ (d )] 0
In particular, the singularities ( (d ), 0) and ( (d ), 0) are locally analytically isomorphic, so 0 is an isolated singularity of (d ). Since the × -action on (d ) is contracting, any irreducible component of the singular locus of (d ) must contain 0. Thus, 0 is the only singularity of (d ).

Finally, (d ) is normal at 0 because its completion at 0 is normal, so (d ) is normal because all other points are smooth. The proof of the proposition is complete.

When viewed inside ˜ (d ), the point 0 of (d ) will still be denoted by 0, as in the Introduction. [START_REF] Haiman | t , q )-Catalan numbers and the Hilbert scheme[END_REF]Proposition 2.6]. This gives the symplectic resolution of ( n × n )/S n .

(2) The fact that ˜ (5) has an isolated singularity has been observed in [START_REF] Fu | Generic singularities of nilpotent orbit closures[END_REF]§12.3.1].

Other descriptions of the blowup

3.A. Orbit closures. -We denote by GSp(V × V * ) the general symplectic group of V × V * , which is the group of linear automorphisms of V × V * which preserve the symplectic form up to a scalar. Since W d is a Coxeter group, the finite group W d commutes with a subgroup of GSp(V ×V * ) isomorphic to GL 2 = GL 2 ( ) (see for instance [START_REF] Bonnafé | Cherednik algebras and Calogero-Moser cells[END_REF]§3.6]): to see this, note that V V * as a W d -module, so V × V * 2 ⊗ V with W d acting trivially on 2 and the action of GL 2 is on the left hand side of this tensor product decomposition. Note that the restriction of this action to × (identified with the center of GL 2 ) is by homothety, so is the action considered in §2.C.

Remark 3.1. -Note that the above action of GL 2 on V × V * does not coincide with the action of GL (V ) = GL 2 ( ) given by the natural action on V and the contragredient action on V * . To distinguish both actions, the natural module for GL 2 will be denoted by U 2 (as in the subsection about conventions/notation in the Introduction) and we use the notation GL 2 (instead of GL 2 ( )) when we talk about the action respecting the symplectic form on V × V * up to scalar. The canonical basis of U 2 will be denoted by ( 1 , 2 ).

Here, an explicit isomorphism of W d -modules V ∼ -→ V * is given for instance by x → Y , y → X . This leads to the following formulas:

Action of a b c d ∈ GL 2 on V × V *          x -→ a x + c Y , y -→ a y + c X , X -→ b y + d X , Y -→ b x + d Y .
Note that the action of SL 2 is symplectic while the action of × scales the symplectic form (and so scales the Poisson bracket).

Hence GL 2 acts on [q ,Q , e , a 0 , . . . , a d ]; by a straightforward calculation, q , Q , e span a copy of the adjoint representation sl 2 tensored with the determinant, and a 0 , a 1 ,. . . , a d span a copy of Sym d (U 2 ) where U 2 is the natural representation for GL 2 . We denote by sl

(1) 2 the above representation of GL 2 : its underlying space is sl 2 (and we will forget the exponent (1) whenever we forget the GL 2 -action) but the usual adjoint action of GL 2 on sl 2 is multiplied with the determinant. This discussion shows that (d ) may be viewed as a GL 2 -stable closed subvariety of sl

(1) 2 ⊕ Sym d (U 2 ).
This action is very useful for understanding the geometry of (d ). While there are infinitely many orbits for the action of SL 2 on its adjoint representation, there are only two non-zero orbits for the above action of GL 2 on sl (1) 2 , with representatives

H = 1 0 0 -1 and E = 0 1 0 0 .
In particular, the GL 2 -orbit of any non-zero semisimple element is dense in sl Proof. -It follows immediately from the generating set for the ring of invariants and the above remarks that

(V × V * )/W d is isomorphic (via a closed immersion) to a Zariski closed GL 2 -stable subset of sl (1) 2 ⊕ Sym d (U 2 ).
Inspecting the relations for the generators, we see that if q = Q = 0 and e = 1 (corresponding to a non-zero multiple of H in sl 2 ) then a j = 0 for 1 j d -1 and a 0 a d = 1. It is easy to see that the stabilizer of the point

(H , d 1 + d
2 ) = (0, 0, 1, 1, 0, . . . , 0, 1) in GL 2 is finite, which (after suitably scaling a basis of Sym d (U 2 ) if necessary) proves (a).

To establish (b), note that the singular locus has dimension 2, and all non-zero GL 2orbits have dimension 2 or 4; the orbit is dense if and only if the projection to sl

(1) 2 is semisimple. Thus, the singular locus is a union of orbits of elements of the form (E , u ) or (0, u ). Now we recall that the singular locus is the image in (V ×V * )/W d of the points with

x = ζ k y , Y = ζ k X , in which case q = ζ k y 2 , e = 2ζ k y X , Q = ζ k X 2 and a j = 2y d -j X j .
If q = e = Q = 0 then clearly a j = 0 for all j , so there are no non-zero orbits in the singular locus of the form (0, u); for the orbits of the form (E , u ), we assume q = e = 0 so that y = 0. Then Q = ζ k X 2 , a 0 = . . . = a d -1 = 0 and a d = 2X d . In particular, a 2 d = 4Q d , so there are at most two such orbits for each fixed non-zero value of Q ; if d is odd then the two different possibilities for a d are obviously conjugate via the µ 2 -action; if d is even then 2Q d /2 = ±a d accordingly as k is even or odd, so we do obtain two disjoint orbits.

An immediate computation shows that g (δ) = det(g )δ for g ∈ GL 2 (this will be better explained in Remark 3.4) and that the subspace

β 0 ⊕ β 1 ⊕ • • • β d is GL 2 -stable
and is isomorphic to the representation Sym d (U 2 ). In particular, δ is SL 2 -invariant (note that D = δ 2 corresponds to the determinant in sl 2 ). It follows that GL 2 acts on the blowup ˜ (d ) and stabilizes the affine open subset

(d ) = Spec [q ,Q , e , b 0 , . . . , b d ]. Now, the previous remark shows that b 0 ⊕ b 1 ⊕• • • b d is GL 2 stable and is isomorphic to the representation Sym d (U 2 ) (-1)
, where the exponent (-1) means that the action on Sym d (U 2 ) (-1) is the natural action on Sym d (U 2 ) tensored with the inverse of the determinant. In particular, for the

× -action, b 0 , b 1 ,. . . , b d have weight d -2.
This discussion shows that (d ) may be viewed as a GL 2 -stable closed subvariety of sl

(1) 2 ⊕ Sym d (U 2 ) (-1) . Lemma 3.3. -Through this embedding, (d ) is the closure of the GL 2 -orbit of (H , d 1 -d 2 ).
Proof. -Inspecting the relations (2.9), we see that if e = 1 and q = Q = 0 then b j = 0 for 1 j d -1 and b 0 b d = -1. We can now complete the proof with the same argument as in Lemma 3.2(a).

Remark 3.4. -Via the inclusion sl 2 so 3 ⊂ sl 3 , we have a GL 2 -equivariant isomorphism

sl (1) 2 ⊕ Sym 4 (U 2 ) (-1) -→ sl 3 E -→ 0 2 0 0 0 -2 0 0 0 H -→ 2 0 0 0 0 0 0 0 -2 F -→ 0 0 0 2 0 0 0 -2 0 4 1 -→ 0 0 2 0 0 0 0 0 0 3 1 2 -→ 0 2/2 0 0 0 2/2 0 0 0 2 1 2 2 -→ -1/3 0 0 0 2/3 0 0 0 -1/3 1 3 2 -→ 0 0 0 -2/2 0 0 0 -2/2 0 4 2 -→ 0 0 0 0 0 0 2 0 0 In particular, H + 4 1 -4 2 -→ 2 0 2 0 0 0 -2 0 -2
, which is of rank 1 and trace 0, i.e. an element of the minimal nilpotent orbit. Now, the GL 2 -orbit of H + 4 1 -4 2 has dimension 4 by Lemma 3.3 and is contained in the minimal nilpotent orbit which has also dimension 4. This proves that (4) is isomorphic to the closure of the minimal nilpotent orbit of sl 3 . This provides a MAGMA free proof of Theorem 2.3(b) and (d).

3.B. Singular locus of (d ).

-In the previous subsection, we have identified (d ) with a closed subvariety of sl

(1) 2 ⊕ Sym d (U 2 ). It is easily seen that I = (δ [V × V * ]) W d
is the ideal vanishing on elements of (d ) whose projection onto sl 2 is nilpotent. Now, let I sing denote the ideal of [V × V * ] W d vanishing on the reduced singular locus sing (d ) of (d ).

Lemma 3.5. -The ideal I sing is generated by I and (β j β k ) 0 j ,k d .

Proof. -Note that β j β k = D b j b k obviously vanishes at any singular point. We will use the action of SL 2 on the coordinate ring; clearly I sing and the singular locus are stable under this action.

Since

β j β k ≡ a j a k modulo [q ,Q , e ], it follows that [V × V * ] W d /I
sing is generated as a [q ,Q , e ]/(e 2 -4qQ )-module by 1, a 0 , . . . , a d . Now [q ,Q , e ]/(e 2 -4qQ ) is the coordinate ring of the nullcone of sl 2 ; by a well-known theorem of Kostant [Kos, Theorem 0.9], it contains one copy of (the SL 2 -module) Sym 2l (U 2 ) in each even degree 2l . Inspecting the relations (2.7) and using the fact that e 2 -4qQ ∈ I sing , we find that the following monomials form a basis of the submodule of [V × V * ] W d /I sing spanned by a 0 , . . . , a d : a 0 , . . . , a d ; q a 0 , e a 0 , e a 1 , . . . , e a d ,Q a d ; q 2 a 0 , q e a 0 , qQ a 0 , qQ a 1 , . . . , qQ a d ,Q e a d ,Q 2 a d ; . . . and in particular there is one copy of Sym d +2l U in each degree d + 2l . For d odd, this is especially useful: the quotient [V × V * ] W d /I sing decomposes over SL 2 as a sum of nonisomorphic irreducible submodules, each one appearing in its own degree; for d even, the submodules Sym 2l (U 2 ) with 2l ≤ d are doubled. By the SL 2 -action, it only remains to determine which highest weight vectors vanish on the singular locus. These highest weight vectors are: Q j and Q j a d for d odd; Q j and any linear span of Q j a d ,Q j +d /2 if d is even. Our result will be proved if we can establish that none of these highest weight vectors vanishes on sing (d ).

Recall that the singular locus is the set of orbits of points satisfying

x = ζ k y , Y = ζ k X for some k . Then Q = ζ k X 2 and a d = 2X d .
None of these vanish, so for d odd, we obtain our result immediately. If d = 2m is even then Q m = ±X d accordingly as k is even or odd; these two possibilities correspond to the two irreducible components of sing (d ). It follows that there do exist highest weight vectors in degree d + 2l vanishing on each of these irreducible components (specifically, Q l (a d ∓ 2Q m )). However, no nonzero linear span of Q l a d and Q m+l can vanish on the whole singular locus, so our proof is complete.

Corollary 3.6. -For d = 2m even, the ideals of the two irreducible components of the singular locus are:

J 1 = (D , a 0 -2q m , a 1 -q m -1 e , a 2 -2q m -1 Q , . . . , a d -2Q m ),
and

J 2 = (D , a 0 + 2q m , a 1 + q m -1 e , a 2 + 2q m -1 Q , . . . , a d + 2Q m ).
For ease of notation, let a ± 2 j := a 2 j ± 2q m-j Q j and a ± 2 j +1 := a 2 j +1 ± q m -j -1 Q j +1 e . Note that:

a ± 2 j = (x m -j Y j ± y m -j X j ) 2 and a ± 2 j +1 = (x m -j Y j ± y m -j X j )(x m -j -1 Y j +1 ± y m -j -1 X j +1 ),
so that the generators of J 1 (resp. J 2 ) can be expressed naturally in terms of the generating invariants (resp. semi-invariants) for the dihedral group of order 2m .

Proof.

-By the proof of Lemma 3.5, it will suffice to show that I sing ⊂ J 1 and

I sing ⊂ J 2 . It follows easily from (2.7) that D b j ∈ J 1 and D b j ∈ J 2 . Since D b j b k -D b j -1 b k +1 ∈ (D ), it will therefore suffice to show that each of D b 2 0 , D b 0 b 1 , D b 2 1 , • • • , D b 2 d belongs to J 1 ∩ J 2 . We recall that D b 2 2 j = (x 2m -2 j Y 2 j -y 2m -2 j X 2 j ) 2 = a - 2 j a + 2 j ∈ J 1 ∩ J 2 .
Similarly, [START_REF] Godbillon | Éléments de topologie algébrique[END_REF]Theorem 2.3]. Now, × ×

D b 2 2 j +1 = 1 4 a + 2 j a - 2 j +2 + 2a + 2 j +1 a - 2 j +1 + a - 2 j a + 2 j +2 and D b j b j +1 = 1 2 a + j a - j +1 + a - j a + j +1 . Proof. -Let p 0 = (1, 1, 0) ∈ . Let • denote the open subset {(x , y , z ) ∈ | y = 0}. Then p 0 ∈ • and the map π 1 ( • , p 0 ) -→ π 1 ( , p 0 ) is surjective
• through the variables (x , y ) and the map (x , y ) -→ (x , y , y -2 (x 2 -1)), so π 1 ( • , p 0 ) is generated by the loop γ defined by γ(t ) = (1, e 2 -1πt , 0). Then it remains to show that γ is homotopic, in , to the trivial loop. But γ is contained in {1} × × {0} ⊂ , so the result follows.

Coming back to our aim of proving Theorem 1.3, it remains to prove that the local fundamental group of the singularity ( (d ), 0) is trivial. Moreover, the existence of a contracting × -action implies that the local fundamental group of ( (d ), 0) is just the fundamental group of (d ) \ {0}. The above discussion shows that it only remains to check the following proposition, whose proof uses in an essential way the SL 2 -action:

Proposition 4.2. -The variety (d ) \ {0} is simply-connected.
Proof. -We fix a base point p = (0, 0, 1, 1, 0, . . . , 0, 1) ∈ (d ) \ {0} and we want to show that π 1 ( (d ) \ {0}, p ) = 1. We divide the proof into several steps.

• First step: projection to sl 2 . First, if (q ,Q , e ) ∈ 3 , we denote by M (q ,Q , e ) the matrix

M (q ,Q , e ) = e 2Q -2q -e ∈ sl 2 .
Viewing (d ) as a closed subvariety of sl 2 ⊕Sym d (U 2 ) as in Lemma 3.3 (we have forgotten the exponents (1) and (-1) in sl 2 ⊕Sym d (U 2 ) because we are only considering the SL 2 -action, and not the GL 2 -action), the map pr : (d ) → sl 2 obtained by projecting to the first component is given by

pr(q ,Q , e , b 0 , b 1 , . . . , b d ) = M (q ,Q , e ).
It is SL 2 -equivariant. Also, the matrices pr(p ) = M (0, 0, 1) = diag(1, -1) = H and M (1, -1/4, 0) are in the same SL 2 ( )-orbit because they have the same non-zero determinant. We denote by g 0 an element of SL 2 such that g 0 M (0, 0, 1)g -1

0 = M (1, -1/4, 0).
Another easy property of this action is the description of its restriction to the diagonal torus of SL 2 : let ∆ op : × -→ SL 2 , ξ → diag(ξ, ξ -1 ). Then (4.3)

∆ op (ξ) • (q ,Q , e , b 0 , b 1 , . . . , b d ) = (ξ -2 q , ξ 2 Q , e , ξ -d b 0 , ξ 2-d b 1 , . . . , ξ d b d ).
• Second step: fibration. Now, let denote the open subset (det •pr) -1 ( × ) of (d ) and let denote the open subset det -1 ( × ) of sl 2 ( ), so that = pr -1 ( ). Then is the open subset of regular semisimple elements. Moreover:

Fact 1. The map det •pr : → × is a principal fibration with fiber isomorphic to SL 2 ( )/∆ op (µ d ). Proof. -Let ζ ∈ × . Let ξ be a square root of -ζ and set τ = diag(ξ, -ξ), so that τ ∈ det -1 (ζ). More precisely, det -1 (ζ) is the SL 2 -orbit of τ,
and the stabilizer of τ in SL 2 is the diagonal torus ∆ op ( × ). So it is sufficient to prove that ∆ op ( × ) acts transitively on pr -1 (τ) and that the stabilizer of some (any) point in pr -1 (τ) is equal to ∆ op (µ d ).

Using (2.1) and the equations for (d ), it follows that

pr -1 (τ) = {(q ,Q , e , b 0 , b 1 , . . . , b d ) ∈ d +4 | q = Q = b 1 = • • • = b d -1 = 0, e = ξ and b 0 b d = -ξ d -2 }.
Now, the result follows from (4.3).

• Third step: long exact sequence in homotopy. Applying the long exact sequence in homotopy to Fact 1 yields a short exact sequence of groups (4.4)

1 -→ π 1 (SL 2 ( )/∆ op (µ d ), 1) -→ π 1 ( , p ) -→ π 1 ( × , -1) -→ 1. But π 1 (SL 2 ( )/∆ op (µ d ), 1) µ d , generated by the loop α 0 : t → ∆ op (e 2 -1πt /d )∆ op (µ d ) while π 1 ( × , -1)
, generated by the loop β 0 : t → e 2 -1πt . Now, let

α : [0, 1] -→ t -→ (0, 0, 1, e -2 -1πt , 0, . . . , 0, -e 2 -1πt )
and

β : [0, 1] -→ t -→ g -1 0 ϕ(1, 0, -e 2 -1πt /4)g 0 .
Then α is the image of α 0 in π 1 ( , p ) while β is a lift of β 0 in π 1 ( , p ). The short exact sequence of groups (4.4) implies that (4.5)

π 1 ( , p ) = 〈α, β 〉.
• Last step: conclusion. Recall the following classical fact:

Fact 2. The map π 1 ( , p ) -→ π 1 ( (d ) \ {0}, p ) is surjective. Proof.
-By Proposition 2.13, the variety (d ) \ {0} is smooth, and is a Zariski open subset of (d ) \ {0}, so the result follows from [START_REF] Godbillon | Éléments de topologie algébrique[END_REF]Theorem 2.3].

Fact 2 shows that it suffices to check that α and β are homotopy equivalent, in (d ) \ {0}, to the trivial loop. But the loops α and β are both contained in g -1 0 ϕ( )g 0 , which is simply-connected by Lemma 4.1. So the proof of the proposition is complete.

Together with Theorem 2.3, this concludes the proof of Theorem 1.3.

Quiver varieties and Slodowy slices

In this section we show that the isolated singularity ( (d ), 0) is locally analytically isomorphic to a certain singularity ( ˜ (d ), xd ) constructed as a cover of a type A Slodowy slice. The isomorphism passes through a Nakajima quiver variety.

We begin with a quiver of type A with d vertices, labelled ρ 0 , . . . , ρ d -1 and d arrows ρ i → ρ i +1 , where indices are taken modulo d . We extend this by adding a framing ρ ∞ → ρ 0 and denote by Q the resulting quiver. Let Q be the doubled quiver, which is independent of the choice of orientation of Q.

We let δ be the minimal imaginary root for the A root system, thought of as a dimension vector for Q, and set

v = ρ ∞ + 2δ. Let Q = d -1 i =1 ρ i be the root lattice of the root system Φ of type A d -1 , with set of simple roots {ρ 1 , . . . , ρ d -1 }. Then Q + := d -1 i =1 0 ρ i and the set of positive roots is Φ + = Q + ∩ Φ. Let α h = ρ 1 + • • • + ρ d -1 be the highest positive root. For 1 i < j d -1, we define α i , j = ρ i + • • • + ρ j , an element of Φ + .
The symbol ≺ will refer to the dominance ordering on Q . We have (5.1)

{α ∈ Q + | (α, α) = 2} = Φ + .
Lemma 5.2. -The following holds:

(5.3) {α ∈ Q + | (α, α) = 4, and α 2α h } = {α i , j + α k ,l | 1 i < k < l < j d -1}
(5.4)

∪ {α i , j + α k ,l | 1 i < j < k < l d -1}.

and (5.4).

There is only one maximal vector, which is α

1,d -1 + α 2,d -2 . This corresponds to α = ρ ∞ + 2ρ 0 + ρ 1 + ρ d -1 .
Lemma 5.6. -The quiver variety M λ (v) has three symplectic leaves, M λ (v) τ 0 , M λ (v) τ 2 and M λ (v) τ 4 , of dimension 0, 2 and 4 respectively.

Proof. -As shown in [BeSc], the sympectic leaves of M λ (v) are labelled by the representation types τ = (β (1) , n 1 ; . . . ; β (k ) , n k ) of (v, λ). Here

β (i ) ∈ Σ λ (v), v = n 1 β (1) + • • • + n k β (k )
and the real roots in Σ λ (v) occur at most once amongst the β (i ) . The leaf labelled by τ has dimension i 2p (β (i ) ). In our case, Lemma 5.5 implies that the possible representation types are

τ 0 = (ρ ∞ + 2ρ 0 + ρ 1 + ρ d -1 , 1; ρ 1 , 1; ρ 2 , 2; . . . ; ρ d -1 , 1) τ 2 = (ρ ∞ + 2ρ 0 + α h , 1; ρ 1 , 1; ρ 2 , 1; . . . ; ρ d -1 , 1) τ 4 = (v, 1).
Since the symplectic leaf M λ (v) τ 0 is zero-dimensional and connected, it equals {x 0 } for some point x 0 ∈ M λ (v).

Recall from the introduction that d -2,2 is a Slodowy slice associated with the nilpotent orbit d -2,2 , that (d ) = d -2,2 ∩ d , and that x d is the unique element of d -2,2 ∩ d -2,2 .

Theorem 5.7. -The singularities (M λ (v), x 0 ) and ( (d ), x d ) are locally analytically isomorphic.

Proof. -This is an application of Crawley-Boevey's étale local picture [C-B], together with the isomorphism of Nakajima [Nak] (see also [Ma]). The point x 0 ∈ M λ (v) τ 0 corresponds to a semi-simple representation of the deformed preprojective algebra Π λ (Q) of the form

M = M ∞ ⊕ M 1 ⊕ M ⊕2 2 ⊕ • • • ⊕ M ⊕2 d -2 ⊕ M d -1 ,
where all summands are simple, dim M ∞ = ρ ∞ +2ρ 0 +ρ 1 +ρ d -1 and dim M i = ρ i . Since the dimension vector of each simple summand is real, there is a unique (up to isomorphism) simple representation of that dimension. Corollary 4.10 of [C-B] says that (M λ (v), x 0 ) is (étale locally) isomorphic to 0 in another (framed) quiver variety; as shown in [START_REF] Bellamy | Symplectic resolutions of quiver varieties[END_REF]Theorem 3.3] this isomorphism is Poisson.

The vertices e i of the new quiver are in bijection with the M i , and M ∞ corresponds to the framing data. There are -(dim M i , dim M j ) = δ i , j -1 + δ i , j +1 arrows from e i to e j in the doubled quiver. That is, the new (undoubled) quiver has underlying graph Dynkin of type A d -1 . Since all vectors are real there are no loops at any vertices. Finally there are

-(dim M ∞ , dim M i ) = 1 i = 2, d -2, 0 otherwise,
arrows from the framing vertex e ∞ to e i . As a (framed) quiver variety,

w : 1 1 v : 1 2 2 2 2 1
Here w is given by w 2 = w d -2 = 1 and w i = 0 otherwise and v 1 = v d -1 = 1 and v 2 = 2 otherwise. Following the construction given in section 8 of [Nak], we see that µ = (d ) and λ = (d -2, 2). Therefore, the isomorphism of [START_REF] Nakajima | Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras[END_REF]Remark 8.5(2)] says that the above framed quiver variety is isomorphic to (d ). It follows from [START_REF] Losev | Isomorphisms of quantizations via quantization of resolutions[END_REF]Lemma 4.6.4] that this isomorphism is as Poisson varieties.

Remark 5.8. -If M is a representation of Π λ lying on the 2-dimensional symplectic leaf M λ (v)

τ 2 then M = M ∞ ⊕ M 1 ⊕ M 2 ⊕ • • • ⊕ M d -1 ⊕ M d -1 ,
with dim M i = ρ i . The simple summands correspond to vertices e ∞ , e 1 , . . . , e d -1 . The associated doubled quiver is the affine type A quiver, except now p (ρ ∞ + 2ρ 0 + α h ) = 1, which implies that there are two loops at e ∞ . The dimension vector is the minimal imaginary root δ = (1, . . . , 1). This implies that (M λ (v), [M ]) is isomorphic to ( 2 ×( 2 /µ d ), 0).

In the notation of [START_REF] Martino | Stratifications of Marsden-Weinstein reductions for representations of quivers and deformations of symplectic quotient singularities[END_REF]Section 7.3], we choose c 1 generic, c = 0, and let (d , 1, 2) denote the spectrum of the centre of the rational Cherednik algebra associated to the wreath product group G (d , 1, 2) = µ d S 2 at t = 0 and (c 1 , c ). By [START_REF] Martino | Stratifications of Marsden-Weinstein reductions for representations of quivers and deformations of symplectic quotient singularities[END_REF]Theorem 1.4], we have an isomorphism (d , 1, 2) ∼ = M λ (v), which is Poisson up to a scalar factor. Therefore, Theorem 5.7 can be reinterpreted as saying that there is an isomorphism of symplectic singularities ( (d , 1, 2), x 0 ) ∼ = ( (d ),

Complements

6.A. Further deformation of (d ).

-If d 4 is even, then the Poisson variety (d ) admits a further deformation [EtGi], which is still a Calogero-Moser space associated with the dihedral group W d (see [START_REF] Bonnafé | On the Calogero-Moser space associated with dihedral groups[END_REF]). Let # (d ) be a generic such deformation. The variety # (4) is smooth so is uninteresting for our purpose but, if d 6 then # (d ) admits a single singular point that we still denote by 0 (see [Bel] or [START_REF] Bonnafé | On the Calogero-Moser space associated with dihedral groups[END_REF]).

So assume from now on that d 6 is even. Again, general facts about Calogero-Moser spaces say that ( # (d ), 0) is a symplectic singularity [START_REF] Gordon | Symplectic reflection algebras[END_REF]Proposition 4.5 sp 4 and computations with MAGMA [Mag], [Thi], based on the equations of # (6) given in [START_REF] Bonnafé | On the Calogero-Moser space associated with dihedral groups[END_REF]Table 5] show that the projective tangent cone of # (6) at 0 is smooth. So the result follows from Beauville's Theorem 1.1.

In other words, the singularity ( # (6), 0) is locally analytically isomorphic to the singularity ( 4 /µ 2 , 0). This shows in particular that the local fundamental group of ( # (6), 0) is isomorphic to µ 2 .

In the general even case, the action of µ 2 on (d ) given by (q ,Q , e , a 0 , . . . , a m ) -→ (q ,Q , e , -a 0 , . . . , -a d )

is free on the smooth part of an open neighbourhood of 0, hence (d )/µ 2 has an isolated singularity at 0. By the argument in the proof of Prop. 2.13, this is locally analytically isomorphic to ( (d )/µ 2 , 0) (with the same action on d +4 ). It is not difficult to show that the Calogero-Moser space associated with W 2d and (non-generic) parameters (c 1 , c 2 ) = (0, 1) is isomorphic to (d )/µ 2 .

Question: Assume d ≥ 4, is the singularity ( # (2d ), 0) locally analytically isomorphic to ( (d )/µ 2 , 0)?

We note here a further description of (d )/µ 2 . Recall by Corollary 3.6 that the singular locus of (2d ) has two irreducible components C 1 and C 2 , with ideals J 1 and J 2 . We can cover the blowup of (2d ) at C 1 by affine open subsets as as we did in §2.C. It is easy to see that the first affine open subset Spec(

[V × V * ] W 2d [J 1 /D ]) is isomorphic to (d )/µ 2 .
In fact, the singular locus of this blowup is a disjoint union of a 2-dimensional subset (the pre-image of C 2 , which can be resolved by blowing up once more) and the singular point For this, let E k ,l denote the elementary (2d -2) × (2d -2)-matrix whose only non-zero entry is is the (k , l )-entry, which is equal to 1. We set

0 ∈ (d )/µ 2 . Similarly, Spec( [V × V * ] W 2d [J 2 /D ] is isomorphic to (d )/µ 2 .
Ē = E 2,1 + E 3,2 + • • • + E d -1,d -2 and Bj = E d -1+ j ,1 -E d -1,2d -j -1
for 1 j d -1. Then the relations (#) show that there is a unique morphism of algebras θ : [ (d )]/〈m〉 -→ Mat 2d -2 ( ) sending ē to Ē and bj to Bj . As θ (1), θ ( ē ),. . . , θ ( ē d -2 ), θ ( b1 ),. . . , θ ( bd -1 ) are linearly independent, this proves the result. By (6.2), [ (d )]/〈m〉 is finite-dimensional, so the graded Nakayama Lemma implies that [ (d )] is a finitely generated R -module. This shows that (q ,Q , b 0 , b d ) is a system of parameters: since [ (d )] is Cohen-Macaulay, this shows that (6.3)

[ (d )] = R ⊕ R e ⊕ • • • ⊕ R e d -2 ⊕ R b 1 ⊕ R b 2 ⊕ • • • ⊕ R b d -1 .
The Hilbert series H (d ) (t) of [ (d )] is then easily computed: (6.4)

H (d ) (t) = 1 + t 2 + • • • + t 2d -4 + (d -1)t d -2
(1t 2 ) 2 (1t d -2 ) 2 .

6.C. Higher dimension. -

We explain here why the SL 2 -equivariant morphism (d ) → sl 2 is a particular case of a more general situation. For this, let (O , (., .)) be a finite dimensional orthogonal vector space and let (S , ω) be a finite dimensional symplectic vector space. Example 6.5. -Assume here that S = U 2 , the natural module of GL 2 endowed with the symplectic form given by the matrix 0 -1 1 0

, and that O = V endowed with the symmetric bilinear form given by the matrix 0 1 1 0

. Note that the quadratic form associated with (., .) is To respect the notation of the paper, if f ∈ Mat 2 ( ) has matrix

Q ∈ [V ]. Then V × V * V × V U 2 ⊗ V U * 2 ⊗ V
x Y y X

, then f * has matrix

0 -1 1 0 x Y y X 0 1 1 0 = X Y -y -x
. Computing π sp (f ) = f * f and π so (f ) = f f * , we get the following commutative diagrams

V × V * Mat 2 ( ) π sp { { π so " " sp(U 2 ) sl 2 det # # so(V ) det { { x Y y X S π sp z z © π so $ $ x X +y Y 2X Y -2x y -x X -y Y ¨det # # x X -y Y 0 0 -x X +y Y T det { { -(x X -y Y ) 2
The map π sp factorizes through the quotient by O(V ) but, since W d ⊂ O(V ), it factorizes through a map (d ) -→ sl 2 which is nothing but the composition of the first projection with the closed immersion (d ) → sl 2 ⊕ Sym d (U 2 ). On the other hand, π so identifies with the map δ (once we identify so(V ) with ). The relation det f * f = det f f * gives δ 2 = e 2 -4qQ (which can also be checked directly, of course).

It is a natural question whether the construction of Example 6.5 can be extended to higher dimensional settings to produce other interesting symplectic singularities.

Theorem 1. 3 .

 3 -The following statements hold: (a) The symplectic singularity ( (d ), 0) has trivial local fundamental group. (b) The symplectic singularity ( (4), 0) is locally analytically isomorphic to ( If d 5, then the singularity ( (d ), 0) is not locally analytically isomorphic to ( g min , 0) for any simple Lie algebra g. (d) If d > d 4, then the singularities ( (d ), 0) and ( (d ), 0) are not locally analytically isomorphic.

  If µ is a partition of d , let µ denote the associated nilpotent orbit in sl d ( ). Let d denote the nilpotent cone of sl d ( ). We denote by d -2,2 a Slodowy slice associated with d -2,2 and set (d ) = d -2,2 ∩ d . The nilpotent cone d admits a µ d -covering π d : ˜ d -→ d which is unramified above the regular nilpotent orbit d and bijective above the branch locus. Define ˜ (d ) = π -1 d ( (d )), denote by x d the unique element of d -2,2 ∩ d -2,2 and let xd denote its unique preimage in ˜ (d ). Then xd is the unique singular point of ˜ (d ) and: Theorem 1.4. -If d 4, the symplectic singularities ( (d ), 0) and ( ˜ (d ), xd ) are locally analytically isomorphic.

  Theorem 2.3. -The affine variety (d ) is irreducible, normal, of dimension 4. Moreover: (a) The point 0 is an isolated singularity of (d ) and its associated maximal ideal of [ (d )] is a Poisson ideal. This endows T * 0 ( (d )) with a Lie algebra structure. (b) We have an isomorphism of Lie algebras T * 0 ( (d ))

  c) If d 4, then (d ) is a symplectic singularity in the sense of Beauville. (d) The symplectic singularity ( (4), 0) is locally analytically isomorphic to ( If d 5, then the symplectic singularity ( (d ), 0) is not locally analytically isomorphic to any ( g min , 0) for a simple Lie algebra g. (f) If d > d 4, then the singularities ( (d ), 0) and ( (d ), 0) are not locally analytically isomorphic.

  of the blowup. -By Lemma 2.10, the singularities of the blowup ˜ (d ) are contained in the affine open subset (d ). First, by definition, [

  2 and b 0 , b 1 ,. . . , b d have degree d -2), [ ] is -graded, with 0-component equal to . Equations (2.9) show that we have a surjective morphism [ ] -→ [ (d )], hence a closed embedding (d ) → . Let [ ] 0 denote the completion of [ ] at the point 0 and, similarly, let [ (d )] 0 denote the completion of [ (d )] at the point 0. The comparison of the definition of (d )

  Corollary 2.15. -The blowup ˜ (d ) is normal and admits a unique singular point, namely the point 0 ∈ (d ). The singularities ( ˜ (d ), 0) and ( (d ), 0) are locally analytically isomorphic. Remark 2.16. -(1) If V is replaced by n and W d is replaced by the symmetric group S n , then the analogue of ˜ (d ) is the Hilbert scheme of n points in the plane by results of Haiman

2 ⊕

 2 -Viewing (d ) as a GL 2 -stable closed subvariety of sl (1) 2 ⊕ Sym d (U 2 ) as above, we have: (a) The variety (d ) is the closure of the GL 2 -orbit of (H , d Sym d (U 2 ). (b) If d is odd (resp. even), then the singular locus of (d ) is the closure of the GL 2 -orbit of (E , d 1 ) (resp. of the union of the GL 2 -orbits of (E , d 1 ) and (E , -d 1 )).

  x d ). Recall that the nilpotent cone d of sl d ( ) admits a µ d -covering π d : ˜ d -→ d which is unramified above the regular nilpotent orbit d and bijective above the branch locus. We have set ˜ (d ) = π -1 d ( (d )) and we denote by xd the unique element in π -1 d (x d ). We are now ready to prove our second main result: Theorem 5.9. -If d 4, the symplectic singularities ( (d ), 0) and ( ˜ (d ), x d ) are locally analytically isomorphic. Proof. -The group W d is a normal subgroup of G (d , 1, 2) with quotient µ d . By [BeTh, Proposition 4.17], the group µ d acts on (d ) such that (d )/µ d ∼ = (d , 1, 2). From now on, we consider all spaces as complex analytic spaces. Theorem 5.7 says that, in the analytic topology, there exist (analytic) open balls B and D around x 0 and x d in (d , 1, 2) and (d ) respectively such that ϕ : D ∼ -→ B , sending x d to x 0 . Let B and D denote the preimages of B and D in (d ) and ˜ (d ) respectively. By [Kuh, Theorem 4], B and D are normal complex spaces. The preimage of the smooth locus of B and D in B and D are denoted by B • and D • respectively. The complement to these open sets has (complex) codimension two.Thus Proposition 4.2 says that π 1 ( B • ) = 1. Hence π 1 (B sm ) = µ d since the map B • → B sm is unramified and Galois with group µ d . Therefore, there exists a (necessarily holomorphic; see[AnSt]) covering map ψ • : B • → D • making the diagram the degree of both vertical maps is d , we deduce that ψ • is an isomorphism. Composing ψ • with the embedding D• → ˜ (d ) gives a map ψ: B • → ˜ (d ).By[START_REF] Hartshorne | Ample subvarieties of algebraic varieties[END_REF] VI, Proposition 3.1], the complex space ˜ (d ) is Stein. Therefore,[START_REF] Lamfa | Extension of holomorphic maps[END_REF] says that ψ extends to a holomorphic map ψ: B → ˜ (d ).

  It is commutative when restricted to the dense open set B • . Therefore, it is everywhere commutative. Since the composition B → B → D → (d ) has image equal to D , we deduce that the image of π d • ψ equals D . The map ψ is µ d -equivariant (since it is so generically). Therefore, the fact that D = π -1 d (D ) means that the image of ψ equals D and ψ is bijective. Finally, since both B and D are normal, the analytic version of Zariski's main theorem implies that ψ is biholomophic. In particular, the singularities ( (d ), 0) and ( ˜ (d ), xd ) are locally analytically isomorphic. Corollary 5.10. -The isolated symplectic singularity ( ˜ (d ), xd ) has trivial local fundamental group.

  ] and we denote by T * 0 ( # (d )) the cotangent space of # (d ) at 0, endowed with its Lie algebra structure induced by the Poisson bracket on # (d ). The smallest singular case d = 6 is somewhat particular: Proposition 6.1. -The singularities ( # (6), 0) and ( sp 4 min , 0) are locally analytically isomorphic. Proof. -By [Bon1, Proposition 8.8], we have an isomorphism of Lie algebras T * 0 ( # (6))

  series. -We compute here the Hilbert series of the graded algebra[ (d )]. For this, recall that any symplectic singularity is Cohen-Macaulay (and even Gorenstein[START_REF] Beauville | Symplectic singularities[END_REF] Proposition 1.3]). Therefore, [ (d )] is a graded Cohen-Macaulay ring. Let R = [q ,Q , b 0 , b d ] and let m denote the unique graded maximal ideal of R . If f ∈ [ (d )],we denote by f its image in [ (d )]/〈m〉. Then (6.2)[ (d )]/〈m〉 = ⊕ ē ⊕ • • • ⊕ ē d -2 ⊕ b1 ⊕ b2 ⊕ • • • ⊕ bd -1Proof of (6.2). -By definition, [ (d )]/〈m〉 is the commutative -algebra whose presentation is given by: -1, ē bj = 0∀1 j k d -1, bj bk = 0 if j + k = d , -ē d -2 if j + k = d . Then + ē + • • • + ē d -2 + b1 + b2 + • • • + bd -1 is a subalgebra of [ (d )]/〈m〉 (because ē d -1 = ē b1 bd -1 = 0),which contains all the generators. So it is equal to [ (d )]/〈m〉. It only remains to show that dim [ (d )]/〈m〉 2d -2.

  For f ∈ Hom(S , O ) (resp. g ∈ Hom(O ,S )), we denote by f * (resp. g * ) the unique element of Hom(O ,S ) (resp. Hom(S , O )) such that (f (s ), o ) = ω(s , f * (o )) (resp. ω(g (o ), s ) = (o , g * (s ))) for all o ∈ O and s ∈ S . Note that f * * =f and g * * = -g . so(O ) where π sp (f ) = f * f and π so (f ) = f f * are the quotient maps by O(O ) and Sp(S ) respectively, which are equivariant with respect to Sp(S ) and O(O ) respectively [KP82].

  Hom(U 2 , V ) Mat 2 ( ) (see Remark 3.1 for the first two isomorphisms). Through this sequence of isomorphisms, the action of O(V ) (resp. GL 2 ) is by left (resp. right) multiplication. Note that sp(S ) = sl 2 and so(V ) = {diag(z , -z ) | z ∈ }.

C. B. is partially supported by the ANR projects GeRepMod (ANR-16-CE40-0010-01) and CATORE (ANR-18-CE40-0024-02). B. F. is supported by the NSFC grant No. 11688101. D. J. is grateful for the support of the ANR project GeRepMod (ANR-16-CE40-0010-01) and of the Charles Simonyi Endowment at the Institute for Advanced Study.

Proposition 3.7. -The blowup ˜ (d ) of (d ) at I is isomorphic to the blowup of (d ) at its reduced singular locus. Moreover, the symplectic form on V × V * induces a symplectic structure on ˜ (d ) \ {0}.

Proof. -The first assertion follows from the description of the blowup at I sing as the closure of the set of elements of d +4 × (d +1)(d +2)/2 of the form ((q ,Q , e , a 0 , . . . Let us now prove the second statement. We can consider (d ) as the µ 2 -quotient of (V × V * )/Γ d . Since there are no non-zero fixed points for the action of Γ d , the quotient

is the blowup of the singular locus, it restricts to the minimal resolution to (d ) \ {0}. This implies that ˜ (d ) \ π -1 (0) admits a symplectic structure (say ω) coming from that on V × V * . As π -1 (0) has dimension 2, ω extends to a symplectic structure on the smooth locus of ˜ (d ), which is just ˜ (d ) \ {0} by Corollary 2.15. By Corollary 2.15, the singularities ( ˜ (d ), 0) and ( (d ), 0) are analytically isomorphic but this does not ensure that the variety ˜ (d ) inherits a symplectic form on its smooth locus. Thanks to the previous proposition, we can now deduce: Corollary 3.8. -The variety ˜ (d ) has an isolated symplectic singularity.

The singularity ˜ (d ) does not admit a projective symplectic resolution. Since the blowup morphism is crepant (being Poisson) it would imply by composition that (V × V * )/W d admits a projective symplectic resolution, which is not the case. In fact, this also follows from the general fact that the only 4-dimensional isolated symplectic singularity admitting a symplectic resolution is analytically isomorphic to the minimal nilpotent orbit closure in sl 3 (see [START_REF] Wierzba | Small contractions of symplectic 4-folds[END_REF]Theorem 1.1]).

Note that the symplectic singularity ( (d ), 0) does not admit a contracting × -action. Therefore, the fact that symplectic singularity ( (d ), 0) is locally analytically isomorphic to ( (d ), 0) and admits a contracting × -action can be viewed as confirmation of a general conjecture of Kaledin [START_REF] Kaledin | Geometry and topology of symplectic resolutions[END_REF]Conjecture 1.8] in this particular case.

Local fundamental group

In order to compute the local fundamental group of (d ) around 0, we will consider the following smooth irreducible surface

It's easy to see that ϕ is a closed immersion, whose image is equal to Proof. -When 1 i < k < l < j d -1 (where the "shorter" root α k ,l sits on top of the "higher" root α i , j ) or when 1 i < j < k < l d -1 (where we have a disjoint union of two roots) we have (α i , j , α k ,l ) = 0 and hence

Therefore, the right hand sides of (5.3) and (5.4) are contained in the left hand side. Assume now we have α ∈ Q + with (α, α) = 4. If the support of α is disconnected, we can write α = α (1) + α (2) with the α (i ) = 0 having orthogonal supports. Then 4 = (α, α) = (α (1) , α (1) ) + (α (2) , α (2) ). Since (β , β ) 2 for all non-zero β ∈ Q + , we deduce from (5.1) that α (i ) ∈ Φ + and hence α belongs to the set in (5.4).

Thus, we assume that the support of α is connected. As a (connected) subgraph of a type A Dynkin diagram is again a type A Dynkin diagram, we may assume without loss of generality that the support of α is the whole Dynkin diagram. Thus, α h α 2α h and we can write

where

If this equals 4 then we must have k = 2 and i

belongs to the right hand side of (5.3). Now we consider the parameter λ, where λ ∞ = -2, λ 0 = 1 and λ i = 0 otherwise, for the quiver variety associated to Q.

Therefore we can associate to it the affine quiver variety M λ (v) as defined in [START_REF] Bellamy | Symplectic resolutions of quiver varieties[END_REF]Section 1.1]. This space parameterises semi-simple representations of the deformed preprojective algebra Π λ (Q) of dimension v. We define the set Σ λ (v) to be all positive roots α for Q such that (A) α v and λ • α = 0, (B) p (α) > p (β (1) ) + • • • + p (β (k ) ) for all proper decompositions α = β (1) + • • • + β (k ) with β (i ) a positive root, λ • β (i ) = 0. Here p (α) := 1 -(1/2)(α, α). A vector α belongs to Σ λ (v) if and only if there is a simple

Therefore, we assume that α ∞ = 1. As noted in [START_REF] Bellamy | Birational geometry of symplectic quotient singularities[END_REF]Lemma 4.3], this implies that α v, v ). Thus, we require:

First, if v = 0 then (v, v ) = 0 so the condition is vacuous. Next, we assume that (v, v ) = 2. Then v ∈ Φ + by (5.1) and the condition says v is maximal with respect to on Φ + . In other words, v = α h is the highest root. This corresponds to α = ρ ∞ + 2ρ 0 + α h . Finally, if (v, v ) = 4 then the condition says that v should be a maximal vector in the union of (5.3)