Fei Hua
email: fei.hua@oca.eu

Member, IEEE Roula Nassif
email: roula.nassif@aub.edu.lb

Haiyan Wang
email: hywang@sust.edu.cn

Fellow, IEEE Ali H Sayed
email: ali.sayed@epfl.ch

C Richard
email: cedric.richard@unice.fr

Diffusion LMS with Communication Delays: Stability and Performance Analysis

Keywords: Communication delays, distributed estimation, diffusion LMS, performance analysis, stability. I

We study the problem of distributed estimation over adaptive networks where communication delays exist between nodes. In particular, we investigate the diffusion Least-Mean-Square (LMS) strategy where delayed intermediate estimates (due to the communication channels) are employed during the combination step. One important question is: Do the delays affect the stability condition and performance? To answer this question, we conduct a detailed performance analysis in the mean and in the mean-square-error sense of the diffusion LMS with delayed estimates. Stability conditions, transient and steady-state mean-square-deviation (MSD) expressions are provided. One of the main findings is that diffusion LMS with delays can still converge under the same step-sizes condition of the diffusion LMS without delays. Finally, simulation results illustrate the theoretical findings.

of consensus and gossip strategies is investigated in [START_REF] Tsitsiklis | Distributed asynchronous deterministic and stochastic gradient optimization algorithms[END_REF]- [START_REF] Kar | Distributed consensus algorithms in sensor networks: Quantized data and random link failures[END_REF] in the presence of link-failures, switching topology, and noisy links. Single-task and multi-task diffusion strategies over asynchronous networks are also examined in [START_REF] Zhao | Asynchronous adaptation and learning over networks-Part I: Modeling and stability analysis[END_REF]- [START_REF] Nassif | Multitask diffusion adaptation over asynchronous networks[END_REF], while diffusion LMS over multi-task networks is investigated in [START_REF] Nassif | Diffusion LMS over multitask networks with noisy links[END_REF] in the presence of noisy links.

The aforementioned works focus on asynchronous events such as link-failures, switching topology, noisy links, and agents turning on and off randomly. Another critical challenge in a distributed implementation is the presence of communication delays. For instance, in underwater acoustic sensor networks, one of the main characteristics of underwater communication channels is their long propagation delays due to the low speed of sound. It is therefore crucial to consider distributed solutions that take into account the communication delays. In [START_REF] Olfati-Saber | Consensus problems in networks of agents with switching topology and time-delays[END_REF], the authors studied a situation where the agents in the network seek average-consensus in the presence of time delays. It was shown that, in a deterministic setting, time delays should be smaller than a threshold to reach consensus. Recently, several other works have also derived interesting findings on consensus strategies in the presence of communication delays [START_REF] Tsianos | Distributed consensus and optimization under communication delays[END_REF]- [START_REF] Yi | Asynchronous distributed algorithms for seeking generalized nash equilibria under full and partial-decision information[END_REF]. In [START_REF] Wu | Decentralized consensus optimization with asynchrony and delays[END_REF], the authors propose and analyze the decentralized asynchronous primaldual algorithm using fixed step-sizes which converges to the exact solution. In [START_REF] Yang | Distributed optimization based on a multiagent system in the presence of communication delays[END_REF], a distributed consensus algorithm is proposed and analyzed for continuous-time multiagent systems in the presence of communication delays. Proportional-Integral (PI) consensus-based distributed optimization algorithm is presented to handle constant communication delays in [START_REF] Hatanaka | Passivity-based distributed optimization with communication delays using PI consensus algorithm[END_REF]. In most cases, these works are problematic for processing streaming data in an adaptive context. It has been shown that diffusion strategies outperform consensus or primal-dual strategies in the stochastic setting due to gradient noise [START_REF] Tu | Diffusion strategies outperform consensus strategies for distributed estimation over adaptive networks[END_REF], [START_REF] Towfic | Stability and performance limits of adaptive primal-dual networks[END_REF]. By using constant step-sizes, diffusion strategies ensure continuous learning and adaptation. In this letter, we carry out a detailed performance analysis of diffusion LMS strategy in the presence of delays and provide stability conditions in the mean and mean-square sense. One of the main findings of this work is that the communication delays do not affect the convergence condition of adaptive diffusion networks.

II. DATA MODEL AND DIFFUSION STRATEGIES

Consider a network of N agents, labeled with k = 1, . . . , N . At each time instant i ≥ 0, each agent k is assumed to have access to a zero-mean scalar measurement d k,i ∈ R and a realvalued regression vector

x k,i ∈ R M with positive-definite co- variance matrix R k = E{x k,i x k,i }. Let r dx,k E{d k,i x k,i }.
The data {d k,i , x k,i } are assumed to be related via the linear regression model:

d k,i = x k,i w * + v k,i (1)
where w * ∈ R M is an unknown parameter vector to be estimated, and v k,i ∈ R is a zero-mean spatially independent measurement noise with variance σ 2 v,k . In order to find an estimate for w * , the objective of the network is to minimize the cost function J(w) : R M → R given by:

min w J(w) = N k=1 E(d k,i -x k,i w) 2 .
(

) 2
It can be verified that the solution of the above problem is given by w * = (

N k=1 R k ) -1 ( N k=1 r dx,k
). This requires the signal statistical information R k and r dx,k , which are rarely available in practice. To solve the problem in a fullydistributed and adaptive manner, the following adapt-thencombine (ATC) diffusion LMS strategy can be employed [START_REF] Cattivelli | Diffusion LMS strategies for distributed estimation[END_REF]:

ψ k,i = w k,i-1 + µ k x k,i (d k,i -x k,i w k,i-1 ) (3a)
w k,i = ∈N k a k ψ ,i (3b)
where N k denotes the set of neighbors of agent k including itself and w k,i denotes the estimate of w * at agent k and iteration i. The first step (3a) is an adaptation step where agent k uses its own data available at time i to update the previous estimate w k,i-1 to an intermediate estimate ψ k,i . Then, in the combination step (3b), agent k convexly combines the intermediate estimates ψ ,i from its neighbors to obtain w k,i . The parameter µ k is a positive step-size, and the combination coefficients {a k } are non-negative, chosen to satisfy the following conditions:

N =1 a k = 1, and
a k > 0, if ∈ N k , a k = 0, otherwise. (4)
The above conditions imply that the matrix [A] k = a k collecting the parameters a k is left-stochastic. Notice that, in the combination step (3b), each node k at time i is assumed to have access to the estimates ψ ,i obtained by its neighbors at the same time instant i. This requires the network to be time synchronized and each node should transmit its estimate to its neighbors before the next iteration i + 1. In this work, we are interested in scenarios where there are time delays in information exchange. In this case, at time i, the estimates ψ ,i from neighbors are not available at node k. Instead, previous estimates may just have been received. In principle, each node k could pause the adaptation step until receiving the required timely estimates and delay the processing of data by the network. Alternatively, we can directly use the delayed information without any additional complexity. By doing so, we arrive at ATC diffusion LMS with delays:

ψ k,i = w k,i-1 + µ k x k,i (d k,i -x k,i w k,i-1 ) (5a)
w k,i = ∈N k a k ψ ,i-τ k (5b)
where the adaptation step (5a) is the same as (3a). However, instead of using the timely estimates ψ ,i , the combination step (5b) uses delayed estimates ψ ,i-τ k from neighbors, where τ k ≥ 0 is an integer denoting the communication delay from node to k.

III. PERFORMANCE ANALYSIS

We now analyze the stability and performance of diffusion LMS with delays [START_REF] Olfati-Saber | Consensus and cooperation in networked multi-agent systems[END_REF]. Before proceeding, we introduce the following independence assumption.

Assumption 1 (Independent regressors). The regression vector x k,i arises from a stationary random process that is temporally white and spatially independent with covariance matrix

R k = E{x k,i x k,i } > 0.
A consequence of Assumption 1 is that we can consider the regressors {x k,i } independent of w ,j for all and j < i.

Although not true in general, this assumption is commonly employed for analyzing adaptive filters and networks since it simplifies the derivations without constraining the conclusions. Furthermore, there are extensive results in the adaptive filtering literature indicating that the performance results obtained using this assumption match well the actual performance for sufficiently small step-sizes [START_REF] Sayed | Adaptive Filters[END_REF].

A. Error Recursion

We introduce the error vectors at node k and time instant i:

ψ k,i w * -ψ k,i , w k,i w * -w k,i , (6)
and collect all error vectors into network block error vectors:

ψ i col{ ψ 1,i , . . . , ψ N,i }, w i col{ w 1,i , . . . , w N,i }. ( 7
)
Subtracting w * from both sides of the adaptation step (5a) and using the data model (1), it can be verified that

ψ i = (I M N -MR i ) w i-1 -Ms i (8)
where M and R i are N × N block diagonal matrices with each block of size M × M , s i is an N × 1 block vector whose entries are of size M × 1 each:

M bdiag{µ 1 I M , . . . , µ N I M }, (9)
R i bdiag{x 1,i x 1,i , . . . , x N,i x N,i }, (10)
s i col{x 1,i v 1,i , . . . , x N,i v N,i }. ( 11
)
It holds that R = ER i = bdiag{R 1 , . . . , R N } and Es i = 0. Due to the communication delays, the relation between w i and ψ i cannot be obtained by simply subtracting w * from both sides of the combination step (5b). Following the same line of reasoning as in [START_REF] Lee | Spatio-temporal diffusion strategies for estimation and detection over networks[END_REF], [START_REF] Hua | Penalty-based multitask estimation with non-local linear equality constraints[END_REF], we introduce the following N (Γ + 1) × 1 extended network block error vectors with each block of size M × 1:

ψ e i col{ ψ 1,i , . . . , ψ N,i , ψ 1,i-1 , . . . , ψ N,i-1 , . . . , ψ 1,i-Γ , . . . , ψ N,i-Γ }, (12)
w e i col{ w 1,i , . . . , w N,i , ψ 1,i , . . . , ψ N,i , . . . , ψ 1,i-Γ+1 , . . . , ψ N,i-Γ+1 }, (13)
where Γ = max{τ k }, 1 ≤ , k ≤ N . For simplicity, we let T Γ + 1. Using the fact that the matrix A is left-stochastic, and from (5b), we obtain

w e i = A e ψ e i (14)
with A e an M N T × M N T matrix given by

A e = A 0 A 1 . . . A Γ I M N Γ 0 (M N Γ)×(M N ) (15)
where A τ A τ ⊗ I M (τ = 0, 1, . . . , Γ) with the ( , k)-th entry of the N × N matrix A τ is given by

[A τ ] k = [A] k , if τ k = τ ; [A τ ] k = 0, otherwise. (16) Observe that A = Γ τ =0 A τ and A A ⊗ I M = Γ τ =0 A τ .
Likewise, from (8), we can derive the extended error vector recursive relation between ψ e i and w e i-1 :

ψ e i = (I M N T -R e i ) w e i-1 -s e i ( 17
)
with

R e i MR i 0 0 0 , s e i Ms i 0 (M N Γ)×1 . ( 18
)
By combining ( 14) and ( 17), we conclude that the extended network error w e i evolves according to the following recursion:

w e i = B i w e i-1 -A e s e i (19)
where

B i A e (I M N T -R e i ).

B. Mean-error Behavior

Taking the expectation of both sides of (19), using Assumption 1 and the fact that Es i = 0, we arrive at the mean-error recursion:

E w e i = A e (I M N T -R e )E w e i-1 = BE w e i-1 (20) R e = ER e i = bdiag{MR, 0, . . . , 0}, (21)
B = A e (I M N T -R e ). (22)
Lemma 1. Consider the block matrix B defined by [START_REF] Wu | Decentralized consensus optimization with asynchrony and delays[END_REF]. The matrix B is stable, i.e., its spectral radius ρ(B) is less than 1, when I M N -MR b,∞ < 1 where the notation • b,∞ denotes block maximum norm of its argument. 1

Proof: The proof can be found in the supplemental material. Theorem 1 (Convergence in the mean). Assume the linear data model (1) and Assumption 1 hold. Then, for any initial condition, algorithm (5) converges asymptotically in the mean toward the optimal vector w * if the step-sizes in M are chosen to satisfy:

0 < µ k < 2 λ max (R k ) , k = 1, . . . , N. (23)
Proof: The proof can be found in the supplemental material.

Theorem 1 allows us to conclude that diffusion LMS with communication delays will continue to converge in the mean 1 For more details and properties of the block maximum norm, refer to [31, Appendix D].

sense under the same step-sizes condition of the algorithm without delays. It is worth noting that this work focuses on single-task problems where all nodes seek to estimate the same parameter vector w * in (1). In this case, the resulting estimates w k,i will be unbiased according to Theorem 1. When heterogeneity in the model exists, diffusion multitask algorithms [START_REF] Nassif | Multitask learning over graphs[END_REF], [START_REF] Chen | Multitask diffusion adaptation over networks[END_REF] and exact diffusion algorithms [START_REF] Yuan | Exact diffusion for distributed optimization and learning-Part I: Algorithm development[END_REF] can be used in order to obtain unbiased estimates.

C. Variance Relation

We now study the mean-square-error behavior. We consider the mean-square error vector weighted by a positive semidefinite matrix Σ, i.e., E w e i 2 Σ

E( w e i ) Σ w e i . The freedom in selecting Σ allow us to derive different performance measures about the network and the nodes. Evaluating the weighted square measures on both sides of ( 19), we get:

w e i 2 Σ = B i w e i-1 -A e s e i 2 Σ
=( w e i-1 ) B i ΣB i w e i-1 + (A e s e i ) ΣA e s e i -(B i w e i-1 ) ΣA e s e i -(A e s e i ) ΣB i w e i-1 . (24) Taking the expectation of both sides of ( 24), using Assumption 1 and the fact that the expectations of the last two terms on the right-hand side (RHS) of ( 24) are zero, we obtain:

E w e i 2 Σ = E w e i-1 2 Σ + E{(A e s e i ) ΣA e s e i } (25)
where Σ E{B i ΣB i }. Let σ vec(Σ) denote the vector obtained by stacking the columns of the matrix Σ on top of each other. Note that, in the sequel, we will use the notation

• 2 σ and • 2 Σ interchangeably to refer to the same quantity. Considering the following properties of matrices:

vec(AΣB) =(B ⊗ A)σ, (26)
Tr(ΣB) =[vec(B )] σ, (27)
we find that σ vec(Σ ) = E{B i ⊗ B i }σ. Let F E{B i ⊗ B i }. Under sufficiently small step-sizes [START_REF] Sayed | Adaptive Filters[END_REF], the matrix F can be approximated by:

F ≈ B ⊗ B . (28)
The first item on the RHS of ( 25) can be rewritten as

E w e i-1 2
Σ = E w e i-1 2
F σ . Now we evaluate the second term:

E{(A e s e i ) ΣA e s e i } = Tr(ΣG) = [vec(G )] σ (29)
where

G A e S e (A e ) , (30)
S e E{s e i (s e i ) } = bdiag{S, 0, . . . , 0},

S M • bdiag{σ 2 v,k R k } N k=1 • M. (31)
Therefore, the variance relation ( 25) can be approximated as

E w e i 2 σ = E w e i-1 2
F σ + [vec(G )] σ. (33)
Theorem 2 (Mean-square stability). Consider the same settings as in Theorem 1. The diffusion LMS with delays algorithm (5) is mean-square stable if the matrix F is stable.

Assuming further that the step-sizes are small enough to justify [START_REF] Sayed | Adaptive Filters[END_REF], condition (23) ensures mean-square stability.

Proof: The proof can be found in the supplemental material.

D. Network Transient and Steady-state MSD

Iterating [START_REF] Chen | Multitask diffusion adaptation over networks[END_REF] starting from i = 0, we obtain:

E w e i 2 σ = E w e -1 2 F i+1 σ + [vec(G )] i t=0 F t σ (34)
where w e -1 = 1 N T ⊗ w * is an initial condition by assuming w -1 = 0. Comparing relation [START_REF] Yuan | Exact diffusion for distributed optimization and learning-Part I: Algorithm development[END_REF] at time instants i and i-1, we can derive the weighted variance recursion

E w e i 2 σ = E w e i-1 2 σ + w e -1 2 (F -I)F i σ +[vec(G )] F i σ. (35) Let ζ i 1 N E w i
2 denote the network transient MSD averaged over all nodes at time i. By replacing σ with σ = vec( Σ), Σ = bdiag{I M N , 0, . . . , 0}, we find that the network transient MSD evolves according to

ζ i = ζ i-1 + 1 N w e -1 2
(F -I)F i σ + [vec(G )] F i σ (36) with ζ -1 = 1 N ( w e -1 ) Σ w e -1 .
Notice that the evaluation of (36) involves the manipulation of the (M N T ) 2 × (M N T ) 2 matrix, which will be prohibitive in computing. However, using property [START_REF] Towfic | Stability and performance limits of adaptive primal-dual networks[END_REF], expression (36) can be rewritten as

ζ i = ζ i-1 + 1 N Tr w e -1 ( w e -1 ) (B i+1 ) ΣB i+1 - (B i ) ΣB i + (B i ) ΣB i G (37)
where the matrix operations ease to the order of O(M N T ).

The network steady-state MSD is defined as

ζ * 1 N lim i→∞ E w i 2 .
Assuming that F is stable and that the algorithm converges in the mean-square sense, taking the limit on both sides of (34), we observe that the first term on the RHS converges to zero. Then, by setting σ = 1 N vec( Σ) and using properties [START_REF] Tu | Diffusion strategies outperform consensus strategies for distributed estimation over adaptive networks[END_REF] and [START_REF] Towfic | Stability and performance limits of adaptive primal-dual networks[END_REF], we get:

ζ * = 1 N ∞ t=0 Tr (B t ) ΣB t G . (38)
IV. SIMULATIONS

We consider a network of 30 nodes with the topology depicted in Fig. 1(a). The length of the parameter vector is set to M = 10 and the optimal vector is w * = [0.495, -0.134, 0.139, -0.328, 0.367, -0.049, -0.141, -1.858, -0.253, -0.602] . The regression vectors x k,i are generated from a zero-mean Gaussian distribution with covariance matrix R k = σ 2

x,k I 3 . The noises v k,i are zero-mean i.i.d. Gaussian random variables with variances σ 2 v,k . The variances σ 2

x,k and σ 2 v,k are shown in Fig. 1(b). The communication delay τ k between two connected nodes and k is proportional to their distance. The step-sizes are set to µ k = µ for all nodes. Combination coefficients for diffusion strategies are chosen according to the uniform rule, i.e., a k = 1/|N k | for ∈ N k . All simulated results are averaged over 500 independent trials.

In the first experiment, we set µ = 0.02 for non-cooperative LMS, diffusion LMS with delays, diffusion LMS with ideal communications where there is no delay, and synchronous diffusion LMS. Note that, for the synchronous diffusion LMS, all nodes need to wait for the longest delayed information to complete one adaptation and combination process. Observe from Fig. 2 that the diffusion LMS strategies perform better than non-cooperative LMS in terms of steady-state MSD, and that diffusion LMS with delays achieves better network steady-state MSD compared to synchronous diffusion LMS at a faster convergence rate and to diffusion LMS with ideal communications at a slower rate under the same stepsizes. In the second experiment, for comparison purposes, we set µ = 0.035 for diffusion LMS with delays in order to meet the same steady-state MSD. It is seen that diffusion LMS with delays converges much faster than the synchronous counterpart. This implies that one can adjust the step-sizes for diffusion LMS with delays to obtain faster convergence rate without additional asynchronous, computational and storage overheads. Moreover, the algorithm will be stable as long as the step-sizes are small enough, which is independent of delays. Finally, from Fig. 2, we observe that the simulated results match well the theoretical curves.

V. CONCLUSION

In this work, we considered the problem of distributed estimation over adaptive networks in the presence of communication delays. We derived the stability condition for diffusion LMS strategy with delays. Stochastic behaviors in the mean and mean-square sense were also provided. Simulation results confirmed the theoretical findings.

Fig. 1 .

 1 Fig. 1. Simulation settings. Variances are generated as σ 2x,k ∼ U (0.8, 1.2), σ 2 v,k ∼ U (0.18, 0.22).

  Theoretical steady-state MSD Synchronous diffusion LMS =0.02 Non-cooperative LMS =0.02 Diffusion LMS with delays =0.035 Diffusion LMS with delays =0.02 Diffusion LMS with ideal communications =0.02

Fig. 2 .

 2 Fig. 2. MSD comparison.

This work was supported in part by the NSFC under Grants 61901385, 61671386, 61771396, and in part by National Key R&D Program of China (2016YFC1400200). The work of C. Richard was supported in part by the ANR under Grant ANR-19-CE48-0002. The work of A. H. Sayed was supported in part by Grant 205121-184999 from the Swiss National Science Foundation. F. Hua is with the School of Marine


Tristan Bozec

Anton Mellit

IRREDUCIBLE COMPONENTS OF THE GLOBAL NILPOTENT CONE

Keywords: Moduli stacks of sheaves over curves of higher genus, Higgs sheaves, Lagrangian subvarieties, stability. Mathematics Subject Classification (2010): 14H60

This paper gives a combinatorial description of the set of irreducible components of the semistable locus of the global nilpotent cone, in genus ≥ 2.

INTRODUCTION

Given a smooth projective curve X of genus g, the moduli stack of Higgs sheaves of rank r and degree d is known to be of dimension 2(g -1)r 2 . It can be viewed as the cotangent stack of the stack of coherent sheaves of class (r, d) over X, and Laumon proved in [START_REF] Laumon | Un analogue global du cône nilpotent[END_REF] that the substack Λ r,d of nilpotent Higgs pairs is Lagrangian (see also [START_REF] Faltings | Stable G-bundles and projective connections[END_REF][START_REF] Ginzburg | The global nilpotent variety is Lagrangian[END_REF]). This substack, which is the 0-fiber of the Hitchin map, is a global analog of the nilpotent cone and a plays a critical role in the geometric Langlands program. The global nilpotent cone is highly singular, and one first interesting step toward its comprehension is the study of its set of irreducible components (see [START_REF] Beilinson | Quantization of Hitchin's integrable system and Hecke eigensheaves[END_REF]2.10.3] for rather implicit results in this direction).

The stack of stable Higgs pairs is known to be smooth, and several results have been proved recently regarding the counting of the number of stable irreducible components of the global nilpotent cone. Its is known from [START_REF] Hausel | Mirror symmetry and Langlands duality in the non-abelian Hodge theory of a curve[END_REF]Corollary 3.11] that the Poincaré polynomial of genus g twisted character varieties, and hence of the diffeomorphic moduli spaces of stable Higgs bundles, is independent from the degree d, provided that it is coprime to the rank r. In [START_REF] Hausel | Mixed Hodge polynomials of character varieties[END_REF], Hausel and Rodriguez-Villegas establish several conjectures dealing with the E-polynomial (a specialization of the mixed Hodge polynomial) of these character varieties. In particular, they conjecture a combinatorial relation with the Kac polynomial A g,r of the quiver with one vertex and g loops in dimension r, which counts absolutely indecomposable isoclasses of g-tuples of matrices over finite fields.

With a different perspective, Schiffmann establishes in [START_REF] Schiffmann | Indecomposable vector bundles and stable Higgs bundles over smooth projective curves[END_REF] that the number of absolutely indecomposable vector bundles of rank r and degree d over F q (still over a curve X of genus g) is given by an expression A g,r,d , polynomial in the Weil numbers of X. These polynomials are therein proved to be related to the moduli space of stable Higgs bundles, and, for instance, the number of stable irreducible components of Λ r,d is given by A g,r,d (0).

In a recent work [START_REF] Mellit | Poincaré polynomials of moduli spaces of Higgs bundles and character varieties (no punctures)[END_REF], Mellit relates the formulas obtained in [START_REF] Hausel | Mixed Hodge polynomials of character varieties[END_REF][START_REF] Schiffmann | Indecomposable vector bundles and stable Higgs bundles over smooth projective curves[END_REF], and proves as a consequence that the polynomials A g,r,d and the E-polynomials of the moduli spaces of stable Higgs pairs are both independent of the degree d coprime with r. A very particular consequence of this work is the equality A g,r,d (0) = A g,r (1).

The aim of the present paper is to give a combinatorial description of the set of irreducible components of Λ r,d , and explain which ones subsist in the subset of semistable components. It is motivated by the W = P conjecture claimed by de Cataldo, Hausel and Migliorini [START_REF] Andrea | Topology of Hitchin systems and Hodge theory of character varieties: the case A 1[END_REF]. In the light of the works above-mentioned, one can expect the polynomial A g,r to play a role in the understanding of the perverse filtration. Adding structure to the set of irreducible components could lead to an interpretation of each of the coefficients of A g,r , rather than just their sum (= A g,r (1)).

The first main result of this paper is Corollary 2.5, which states that the set of irreducible components of the global nilpotent cone is given by the very natural decomposition in twisted Jordan cells, which are smooth. It is based on a direct computation of the dimensions of these cells and previous works [START_REF] Schiffmann | Indecomposable vector bundles and stable Higgs bundles over smooth projective curves[END_REF][START_REF] Mozgovoy | Counting Higgs bundles and type A quiver bundles[END_REF]. Then we move on to the semistable locus and obtain Theorem 3.1 which gives purely combinatorial conditions on the twisted Jordan type to be semistable. The proof uses an analogous result from [START_REF] Bradlow | Irreducibility of moduli of semi-stable chains and applications to U(p, q)-Higgs bundles[END_REF] obtained in the context of moduli stacks of chains, and shows that semistability can be tested on the most 'simple' subsheaves -the ones built with iterated kernels and images. The proof is constructive and do not rely on the coprimality of r and d, in particular we get in Corollary 3.3 that the attracting cells are irreducible in any case.

The Corollary 4.5 describes this set of semistable irreducible components in terms of integral polytopes, which sheds a new light on the quantity A g,r (1), whose behaviour is still very poorly understood.

RECOLLECTION ON COHERENT AND HIGGS SHEAVES

1.1. Coherent sheaves over a curve. Let X be a smooth projective curve of genus g over a field k. We will denote by Coh the category of coherent sheaves over X, and by

[F] = (rank F, deg F) ∈ H = {(r, d) ∈ N × Z | d ≥ 0 if r = 0}
[F(D)] = [F ⊗ D] = [F](p).
We will use the usual slope defined on H by µ(r, d) = d/r ∈ Q ∪ {∞} and we set µ(F) = µ([F]). We say that F is semistable if

{0} ⊂ G ⊂ F ⇒ µ(G) ≤ µ(F),
stable if the right-hand side inequality is strict. Note that these notions coincide if deg F and rank F are coprime. We will use the following basic property.

Proposition 1.1. For any short exact sequence in Coh, one of the following is true

µ(E) < µ(F) < µ(G) µ(E) = µ(F) = µ(G) µ(E) > µ(F) > µ(G).
The category Coh being hereditary (i.e. of homological dimension one or less), the Euler form is defined by

F, G = dim Hom(F, G) -dim Ext 1 (F, G),
and we will denote by (-, -) its symmetrized version

(F, G) = F, G + G, F .
The Euler form only depends on the class of the sheaves, and satisfies (r, d), (r , d ) = (1 -g)rr + rd -r d thanks to the Riemann-Roch theorem.

1.2. Higgs sheaves. A Higgs sheaf is a pair (F, θ), where F ∈ Coh and θ ∈ Hom(F, F(Ω)), Ω being the canonical divisor of degree l = 2g -2. We will denote by M r,d the moduli stack of pairs (F, θ) satisfying [F] = (r, d), whose dimension is lr 2 . A Higgs sheaf (F, θ) is said to be semistable

{0} ⊂ G ⊂ F θ(G) ⊆ G(Ω) ⇒ µ(G) ≤ µ(F),
θ k = θ((k -1)Ω) • • • • • θ(Ω) • θ : F → F(kΩ).
A pair (F, θ) is said to be nilpotent if θ k = 0 for some k, and we denote by

Λ r,d = {(F, θ) ∈ M r,d | (F, θ) nilpotent} the global nilpotent cone.
It is nothing but the zero fiber of the Hitchin map M r,d → ⊕ 1≤i≤r H 0 (X, Ω i ), mapping (F, θ) to the coefficients of the characteristic polynomial of θ. It is known, thanks to Laumon [START_REF] Laumon | Un analogue global du cône nilpotent[END_REF], to be a Lagrangian substack of M r,d , but its irreducible components are still not well understood (see [Ibid., Remarque 3.9 (ii)]). The aim of this article is to give a precise combinatorial description of these components, as well as the ones of the semistable locus Λ sst r,d .

THE TWISTED JORDAN CELL DECOMPOSITION

2.1. The setting. In this section we will recall and make use of notations and results established in [START_REF] Mozgovoy | Counting Higgs bundles and type A quiver bundles[END_REF][START_REF] Schiffmann | Indecomposable vector bundles and stable Higgs bundles over smooth projective curves[END_REF]. Consider a nilpotent Higgs sheaf (F, θ) ∈ Λ r,d , with r > 0 but r and d not necessarily coprime. Set F k = (Im θ k )(-kΩ) and denote by s the nilpotency index of θ. We have a chain of epimorphisms

F 0 F 1 (Ω) • • • F s (sΩ) = {0}
that allows us to define

F k = ker{F k → F k+1 (Ω)}.
We also have a chain of inclusions

{0} = F s ⊂ F s-1 ⊂ • • • ⊂ F 1 ⊂ F 0 = F
whose successive quotients are denoted by F k = F k /F k+1 . These two chains induce the following ones

F 0 F 1 (Ω) • • • F s (sΩ) = {0} {0} = F s ⊂ F s-1 ⊂ • • • ⊂ F 1 ⊂ F 0 ,
and we define

α k = (r k , d k ) = ker{F k-1 ((k -1)Ω) → F k (kΩ)} .
The family α = (r, d) = (r 1 , . . . , r s , d 1 , . . . , d s ) is called the Jordan type of (F, θ) and is denoted by J(F, θ). We will call s the length of α or r. One good way to understand its definition is to fill the triangular Young tableau T s of size s in the following way (here with s=4)

α4 α4(-l) α3 α4(-2l) α3(-l) α2 α4(-3l)α3(-2l) α2(-l) α1
(2.1) and then notice that

[F k ] = i>k α i (-kl)
is the sum of the classes in the boxes of the k-th subdiagonal. Hence,

[F k ] = i≥k [F i ]
corresponds to the region below this subdiagonal. Denote by [R] the sum of the classes of the boxes in a region R. We have, for instance with s = 5,

[F 1 ] =       [F 2 ] =       [F 4 ] = [F 4 ] =      
where the classes are summed over the blackened regions. In particular the sum over all boxes is (r, d) := [F] and we write α (r, d) (and call (r, d) the size of α). This implies the following equality

k kd k = d + l k k(k -1) 2 r k . (2.2)
We also have

[ker θ k ] -[ker θ k-1 ] = [F 0 ] -[F k (kΩ)] -{[F 0 ] -[F k-1 ((k -1)Ω)]} = [F k-1 ((k -1)Ω)] -[F k (kΩ)] = j>i≥k-1 α j ((k -1 -i)l) - j>i≥k α j ((k -i)l) = - j>k-1 α j (l(j -k + 1)(j -k)/2) + j>k α j (l(j -k)(j -k -1)/2) = j≥k α j ((k -j)l)
which corresponds to the k-th (from the bottom) horizontal strip. Graphically, we have for instance

[ker θ] =       [ker θ 3 ] =      
We call canonical the subsheaves of F obtained by intersections and sums of the F k and the ker θ k . The corresponding regions in the Young tableau are the ones saturated in the west, south and south-east directions (note that there is a mistake in the corresponding statement in [Sch16, 3.1]). We denote by R this set of regions, which we will also call canonical.

The slope of the sheaf F R corresponding to a region R ∈ R is given by

µ α (R) = d R r R = ∈R deg ∈R rank with respect to the filling (2.1) by α = (r, d) = J(F, θ) (we will call µ α (R) the α-slope of R). For instance µ(ker θ 2 ∩ F 2 + ker θ ∩ F 1 ) = µ α       = 2d 5 + 2d 4 + d 3 + d 2 -7lr 5 -5lr 4 -2lr 3 -lr 2 2r 5 + 2r 4 + r 3 + r 2 .
The definition of the Jordan type yields a cell decomposition

Λ r,d = α (r,d) Λ α
where

Λ α = J -1 (α) ⊆ Λ r,

Irreducible components.

In this section, we will study the map of stacks

π α : Λ α -→ k Coh α k (F, θ) -→ ker{F k-1 ((k -1)Ω) → F k (kΩ)} k
for any Jordan type α. Denote by F α the stack of chains of epimorphisms

H 0 H 1 • • • H s = {0} satisfying α k = [ker{H k-1 H k }] and write π α = ρ α • χ α where χ α : Λ α -→ F α (F, θ) -→ (F 0 F 1 (Ω) • • • F s (sΩ))
and

ρ α : F α -→ k Coh α k (H 0 H 1 • • • H s ) -→ ker{H k-1 → H k } k .
The following equalities are obtained in [MS20, Proposition 5.2] and [Sch16, 3.1] respectively.

Proposition 2.4. The maps χ α and ρ α are iterations of vector bundle stacks and their respective relative dimensions are

d χα = - k F k , F k+1 d ρα = - i<j α j , α i . Corollary 2.5. The set of irreducible components of Λ r,d is Irr Λ r,d = Λ α | α (r, d) .
Proof. From Proposition 2.4, since all Coh α k are irreducible, we know that the Λ α are also irreducible (even smooth), thus we just have to prove that they have the same dimension. From Proposition 2.4 and the fact that dim Coh r,d = -(r, d), (r, d) , we get

dim Λ α = - k F k , F k+1 - i<j α j , α i - k α k , α k = - i>k α i (-kl), F k+1 - i≤j α j , α i . Now, since [ker θ k ] = j≤k-1 [F j (jΩ)], we have [F k ] = ([ker θ k+1 ] -[ker θ k ])(-kl) = j>k α j ((1 -j)l), thus dim Λ α = - i>k j>k+1 α i (-kl), α j ((1 -j)l) - i≤j α j , α i = - i>k j>k+1 ( α i , α j + l(k + 1 -j)r i r j ) - j≤i α i , α j = - i≥j j α i , α j -l j(j -1) 2 r i r j - i<j i α i , α j + l i(i + 1 -2j) 2 r i r j = - i<j i(α i , α j ) - i i α i , α i -l i(i -1) 2 r 2 i + l 2 i<j (i(i -1) -i(i + 1 -2j))r i r j = l i<j ir i r j + l 2 i (i + i(i -1))r 2 i + l i<j i(j -1)r i r j = (g -1)r 2
as expected.

THE SEMISTABLE LOCUS

As semistability is an open condition, we know that the irreducible components of Λ sst (r,d) form a subset of Irr Λ (r,d) which can be now identified, thanks to Corollary 2.5, with the set of all Jordan types of size (r, d). We say that a Jordan type is semistable if it appears in Irr Λ sst (r,d) . The aim of this section is to prove the following. Theorem 3.1. Assume that g ≥ 2. A Jordan type α (r, d) is semistable if and only if for every canonical strict subregion R ∈ R we have

µ α (R) ≤ d r . (3.2)
Note that this condition is strictly numerical and obviously necessary since a canonical subsheaf G satisfies by construction θ(G) ⊆ G(Ω). This result is optimal in the way that it says that it is sufficient to test (generic) semistability on the most trivial θ-stable subsheaves. To prove it, we will use the results of [START_REF] Bradlow | Irreducibility of moduli of semi-stable chains and applications to U(p, q)-Higgs bundles[END_REF], which deals with the moduli stack of chains E s → . . . → E 1 . In this article is obtained an analogous result in the way that it gives necessary and sufficient conditions on the numerical invariants (n

• , p • ) = [E • ]
of chains for these to be generically semistable (we will call semistable types of chains such invariants). The conditions obtained therein are somewhat unnatural, in the way that they do not correspond to proper subchains -see [Ibid., Remark 2.11]. However, in this section (c.f. Proposition 3.10), we will build a injection between Jordan types satisfying (3.2) and semistable types of chains, which are related to the semistable components of the nilpotent cone in the following way. We consider stability of chains with respect to the α Higgs -slope

µ α Higgs (E • ) = 1≤i≤s (deg(E i ) + α Higgs i rank(E i )) 1≤i≤s rank(E i ) where α Higgs = ((i -1)(2g -2)) 1≤i≤s . Then the direct sum ⊕ k E k ((k -1)Ω)
C - n•,p• = {(F, θ) | lim t→0 t.(F, θ) ∈ C n•,p• }.
It is known (see e.g. [START_REF] Bradlow | Irreducibility of moduli of semi-stable chains and applications to U(p, q)-Higgs bundles[END_REF]§6]) that the closures of these attracting varieties, for (n • , p • ) of semistable type, are unions of irreducible components of Λ sst r,d . Hence, building the injection announced will imply Theorem 3.1 as all inequalities in the following chain will have to be equalities: This was only known in the coprime case.

# Irr Λ sst r,d ≤ #{α (r, d) | (3.2)} ≤ #{semistable types (n • , p • ) | k (n k , p k )((k -1)l) = (r, d)} ≤ # Irr Λ sst r,
Fix for now a type α of length s satisfying (3.2), and (F, θ) ∈ Λ α . We introduce a couple of notions before rephrasing the main result of [START_REF] Bradlow | Irreducibility of moduli of semi-stable chains and applications to U(p, q)-Higgs bundles[END_REF] in our context (c.f. Proposition 3.7). Definition 3.4. We call 1-flags the flags R

• = (∅ = R 0 ⊂ R 1 ⊂ • • • ⊂ R t = T s ) of subregions of T s such that: (i) t = s; (ii) R k ∈ R;
(iii) if the number of boxes in a given column of T s is increased by 1 from R k to R k+1 , the same must be true for every column on its left.

We call strips the (noncanonical) subregions S k = R k \ R k-1 , and 1-chains the chains

E • = (E s → . . . → E 1 ) associated to a given 1-flag R • , where E k = F R k /F R k-1 (-(k - 1)Ω)
, and the morphisms are induced by θ.

Remark 3.5.

• A 1-chain E • depends on the data (R • , F, θ), but its type (n • , p • ) = [E • ] only depends on (R • , α).
• The 1notation comes from the fact that ht(S k ) = 1 (the height being the number of boxes in the higher column), which is why θ induces morphisms E k+1 → E k .

• Each strip has a box on the left border of T s .

Example 3.6.

• The flag of regions (s = 3)

⊂ ⊂ ⊂ is not a 1-flag, as condition (iii) is not satisfied by R 1 ⊂ R 2 . • The flag of regions (s = 3) ⊂ ⊂ ⊂
is not a 1-flag, as condition (ii) is not satisfied by R 2 (region not saturated in the south-east direction).

Consider a 1-flag R • and the associated 1-chain E • (again, the pair (F, θ) is fixed for now). We will denote by

|E k | = #S k the number of boxes in S k and set (n k , p k ) = [E k ].
Thanks to Definition 3.4, each 1-flag can be seen as the permutation σ on s elements given by (σ(1), . . . , σ(s)) = (|E 1 |, . . . , |E s |).

It will be convenient to represent 1-flags with horizontal strips, for instance write

R • =
for the 1-flag associated to σ = (3, 2, 5, 4, 1, 6).

Take two integers 1 ≤ k < j ≤ s. Note that

n j < n k implies |E j | < |E k |. When n j < min{n k , . . . , n j-1 }, the α Higgs -slope of the chain (see [BGPGH18, Definition 2.10]) E s → . . . → E j = • • • = E j → E k-1 → . . . → E 1
is the α-slope of the (noncanonical!) subregion obtained by only considering the #S j leftmost boxes in each S t , k ≤ t ≤ j. We denote by R j k the complementary of this region. For instance if s = 6 and σ = (3, 2, 5, 4, 1, 6), and if we represent the strips S t horizontally, we get

R • = ⇒ R 4 3 = , R 5 2 =
where the black boxes are the ones contained in the region. Note that in order to take the αslope, one has to project the boxes in the south direction, and then proceed to the previously mentioned filling (2.1) of T s . For instance in our example µ α (R 4 3 ) = µ(α 2 (-l)).

(ii) if ( Čjp kp ) (resp. (C jp kp )) fails for (ρ jp-1 kp-1 ) p-1 . . . (ρ j1 k1 ) 1 R • then p = 1 (resp -1) for all p = 1 . . . t; (iii) ρR • := (ρ jt kt ) t . . . (ρ j1 k1 ) 1 R • satisfies all conditions (C j k ) and ( Čj k )
. As explained at the beginning of the section, the following completes the proof of Theorem 3.1. Proposition 3.10. There is an injective map κ from the set of Jordan types α (r, d) satisfying (3.2) to the set of semistable types of chains

E • such that k [E k ((k -1)Ω)] = (r, d).
Proof. Thanks to 3.9, from any Jordan type α (r, d), we build a 1-flag satisfying all conditions (C j k ) and ( Čj k ). Thanks to 3.7, the associated 1-chain is of semistable type. Denote by (n • , p • ) this type, the map κ : α → (n • , p • ) hence built is injective because any type of 1-chain characterizes α, according to the following argument.

We prove more generally that for any s and any 1-flag R • of T s , the map ι : α → [E • ] from Jordan types of length s to types of chains of length s is injective, where -(k -1)Ω) for any choice of Higgs pair (F, θ) of type α (ι doesn't depend on this choice). We use induction on s, initialization being trivial at s = 0. Then take s > 0, α of length s, and E • as above. Consider t maximizing rk(E k ), as well as deg(E k ) if several pieces of the chain E • have same rank. Then (3.11)

E k = F R k /F R k-1 (
α 1 + α 2 (-l) + • • • + α s (-(s -1)l) = [F Rt ] -[F Rt-1 ](l) = [ker θ].
Consider the 1-flag R• of T s-1 obtained by quotienting by ker θ, then the associated map ῑ is injective by induction hypothesis. Hence, writing ᾱ = (α 2 , . . . , α s ), we get

ι(α) = ι(α ) ⇒ ῑ( ᾱ) = ῑ( ᾱ ) ⇒ ᾱ = ᾱ ⇒ α = α ,
thanks to 3.11 for the last implication.

For instance here with t = s = 6,

R • = × × × × × × ⇒ R• =
where ×'s correspond to ker θ.

One could wonder how geometric this bijection is. We state the following.

Theorem 3.12. For any semistable Jordan type α, we have Λ α = C - κ(α) . Proof. Denote by λ the bijection from the set of semistable Jordan types to the set of semistable types of chains given by

Λ α = C - λ(α)
. We want to prove that λ = κ. We will use the partial order ≤ on Jordan cells introduced in [SS20, (2.8)]:

β ≤ α ⇔ ∀k : [ker θ k ] ≤ [ker θ k ] ⇔ ∀k : [ker θ k ] -[ker θ k ] ∈ H.

Consider a semistable type α and set

β = λ -1 κ(α). Consider (F, θ) ∈ Λ β such that there exists a θ-stable flag (F 1 ⊂ • • • ⊂ F s = F) of F of graded type κ(α): if E k = F k /F k-1 (-(k -1)Ω), we have [E • ] = κ(α).
By definition, such a pair (F, θ) exists in Λ sst β , and

lim t→0 t.(F, θ) = (E • , φ • ) is a semistable chain, where φ • is induced by θ.
Assume that β = α (r, d), and since we know that for a fixed pair (r, d) ∈ H there are finitely many semistable Jordan types of size (r, d), pick α ≤-maximal of size (r, d) with that property. We conclude thanks to the following Lemma 3.13. Indeed by maximality, β > α would imply κ(β) = λ(β) = κ(α) hence again α = β by injectivity of κ. Lemma 3.13. We have β ≥ α.

Proof. Fix an arbitrary auxiliary pair (G, ψ) of type α, and denote by G • the flag constructed along the proof of 3.10, whose associated chain is of type κ(α). The pair (F, θ) ∈ Λ β being as above, we want to prove that [ker(ψ k )] ≤ [ker(θ k )] for every k. Note that

[F k ] = [G k ]
for every k and that [ker(ψ k )] only depends on α. We prove the following statement, which doesn't involve semistability.

Consider (F, θ) ∈ Λ β and (G, ψ) ∈ Λ α , a 1-flag G • and a flag F • such that θ induces a chain (E • , φ • ) of same type as gr(G • ). Then [ker(ψ k )] ≤ [ker(θ k )] for every k.
We prove it by induction on the common length s of E • , gr(G • ). There is nothing to prove for s = 0. Assume that it is known for all lengths < s. Consider for any k the smallest index t k such that ker(ψ k ) ⊆ G t k .

We prove by induction on k ≥ 0 that [ker(θ k )] ≥ [ker(ψ k )] together with the following refinement

t k+1 > t k ⇒ [ker(ψ k )] ≤ η [F t k+1 -1 ∩ ker(θ k )] ≤ η [F t k+1 ∩ ker(θ k )] ≤ η [ker(θ k )] t k+1 = t k ⇒ [ker(ψ k )] ≤ η [F t k+1 ∩ ker(θ k )] ≤ η [ker(θ k )]
where we mean

[B] -[A] = γ ∈ H by [A] ≤ γ [B].
There is nothing to prove for k = 0.

Assume that the property is true for some k, and that t k+1 > t k . We have, using the notation

f p = [F p ] = [G p ], ker(θ k+1 ) ker(θ k ) ≥ F t k+1 ∩ ker(θ k+1 ) F t k+1 ∩ ker(θ k ) = ker F t k+1 F t k+1 ∩ ker(θ k ) → F t k+1 -1 F t k+1 -1 ∩ ker(θ k ) (Ω) ≥ f t k+1 -([ker(ψ k )] + η + η ) -f t k+1 -1 (l) + ([ker(ψ k )] + η)(l) ⇒ [ker(θ k+1 )] ≥ η + η(l) + f t k+1 -(f t k+1 -1 -[ker(ψ k )])(l) but t k+1 > t k ensures that f t k+1 -(f t k+1 -1 -[ker(ψ k )])(l) = [ker(ψ k+1 )] so we get [ker(θ k+1 )] ≥ [ker(ψ k+1 )]. Assume now that t k+1 = t k and set η = [ker(ψ k )] -[F t k+1 -1 ∩ ker(θ k )] ∈ Z 2 . We get this time [ker(θ k+1 )] ≥ η -η(l) + f t k+1 -(f t k+1 -1 -[ker(ψ k )])(l).
There exists some r ≥ 0 such that

[ker(ψ k )] -[G t k+1 -1 ∩ ker(ψ k )] = α k + • • • + α k-r which implies f t k+1 -(f t k+1 -1 -[ker(ψ k )])(l) = [ker(ψ k+1 )] + α k (l) + • • • + α k-r (l).
We use our induction hypothesis on the chains induced on F t k+1 -1 , G t k+1 -1 to get

α k + • • • + α k-r -η = α k + • • • + α k-r -[ker(ψ k )] + [F t k+1 -1 ∩ ker(θ k )] ≥ α k + • • • + α k-r -[ker(ψ k )] + [G t k+1 -1 ∩ ker(ψ k )] = 0
and thus [ker(θ k+1 )] ≥ [ker(ψ k+1 )] again.

Assume t k+2 > t k+1 . If t k+1 > t k , we have from the above chains of inequalities

[ker(ψ k+1 )] ≤ [F t k+1 ∩ ker(θ k+1 )] -η(l) ≤ [F t k+1 ∩ ker(θ k+1 )] ≤ [F t k+2 -1 ∩ ker(θ k+1 )].
If t k+1 = t k we have

[F t k+2 -1 ∩ ker(θ k+1 )] ≥ [F t k+1 ∩ ker(θ k+1 )] ≥ ker(ψ k+1 ) -η(l) + α k (l) + • • • + α k-r (l)
≥ ker(ψ k+1 ). Now assume t k+2 = t k+1 . We have [ker(ψ k+1 )] ≤ [F t k+1 ∩ ker(θ k+1 )] for the same reasons as above, which concludes the proof.

POLYTOPAL DESCRIPTION

We first reformulate Theorem 3.1, in order to use a slightly different set of inequalities to characterize the semistability. For any region R ∈ R we denote by R k the height of its k-th column, counted from the right. For instance

R = ⇒ (R 5 , R 4 , R 3 , R 2 , R 1 ) = (2, 2, 1, 1, 0). We set R p = {R ∈ R | R 1 = . . . = R p-1 = 0 = R p } and R >1 = ∪ p>1 R p . Note that from the definition of R, there is a bijective map R → R sending R = (R k ) to R = (k -R k ). For instance R = → R =
and we see that this map swaps R 1 and R >1 . For any Jordan type (r, d) (r, d) and any region R ∈ R, we finally define

p R (r) = r R r = ∈R rank r = k R k r k r . Proposition 4.1. A Jordan type α = (r, d) (r, d) is semistable if and only if for any region R ∈ R >1 we have p R (r)d + l k R k (R k -1) 2 r k ≤ k R k d k ≤ p R (r)d + l k R k ( Rk + k -1) 2 r k . (4.2)
Remark 4.3. Note that the bounds do not depend on d and that d 1 never appears in the central term. Hence, because of (2.2), it is only subject to

d 1 = d - k≥2 kd k + l k k(k -1) 2 r k . (4.4) Proof. First note that d R = ∈R deg = k R k d k -l k R k ( Rk + k -1) 2 r k so that (3.2) is equivalent to k R k d k ≤ p R (r)d + l k R k ( Rk + k -1) 2 r k . Also, d R = d -d R -l k R k Rk r k so that (3.2) with respect to R is equivalent to d -d R -l k R k Rk r k ≤ p R(r)d = (1 -p R (r))d ⇔ p R (r)d + l k R k (R k -1) 2 r k ≤ k R k d k .
This concludes the proof since R = R 1 R >1 .

Assume that r > 0. Then if (F, θ) ∈ Λ sst r,d has type α of length s, we necessarily have r s > 0 since r s = rank F s-1 and F s-1 ⊂ F. Proof. The polytope P r,d is defined by the set of linear inequalities (4.2) together with the hypothetical extra conditions d k ≥ 0 every time we have r k = 0. It is (s -1)-dimensional because of (4.4). Also because of this equation, note that a facet may be given by the equation

k≥2 kd k = d + l k k(k -1) 2 r k (4.6) if r 1 = 0.
We get the following correspondence between polytopes associated to different degrees.

- R (r) = l k R k (R k -1) 2 r k , b + R (r) = l k R k ( Rk + k -1) 2 r k , r • R = k R k r k , d • R = k R k d k .
Note that since we consider R ∈ R >1 we can replace d by d * = (0, d 2 , . . . , d s ) everywhere.

Finally, also note that if r 1 = 0 the equation (4.6) fits in this study since it can be written

d * - d r r • T s = l k k(k -1) 2 r k
where T s = (k) corresponds to the full tableau.

It is not clear how this bijection restricts to integral points, but we know, as explained in the introduction, that thanks to [START_REF] Mellit | Poincaré polynomials of moduli spaces of Higgs bundles and character varieties (no punctures)[END_REF] we have the following. Note that r S k > 0, otherwise by placing S k at the first position we would obtain a flag with µ(R 1 ) = +∞ which contradicts semistability. So the overall sign coincides with the sign of the expression in parenthesis. The latter expression can be simplified as follows:

µ j i=k+1 (r Si -r S k ) - j i=k+1 (d Si -d S k -(i -k)lr S k ) = µr Řj k -d Řj k .
So we have established

(A.3) f (ρ j k R • ) -f (R • ) = 2lr S k (µr Řj k -d Řj k ).
If condition ( Čj k ) fails for R • , then we have µ α ( Řj k ) > µ and so f (ρ

j k R • ) -f (R • ) < 0. If on the other hand condition (C j k ) fails for R • = ρ j k R • , then we have µ(R j k ) = µ( Řj k ) < µ, which implies f (R • ) -f ((ρ j k ) -1 R • ) = f (ρ j k R • ) -f (R • ) > 0.
Strictly speaking, we do not need the case l = 0 because it corresponds to g = 1, but we include it here for completeness. We construct f using a trick. Let us introduce l as a parameter, and make all d Q of all regions Q a function of l. Then (A.2) produces a family of values f l (R • ) depending on l for any 1-flag R • . Let

f (R • ) = ∂ ∂l f l (R • ) l=0 .
Differentiating the final formula (A.3) by l we obtain

f (ρ j k R • ) -f (R • ) = 2r S k (µr Řj k -d Řj k ),
and the proof continues in the same way as for l > 0.

  the class of F ∈ Coh. We will denote by Coh r,d ⊂ Coh the subcategory of coherent sheaves of class (r, d). If α = (r, d), we write r = rank α and d = deg α. For any α = (r, d) ∈ H and p ∈ Z, we set α(p) = (r, d + pr) so that if F ∈ Coh and D is a divisor of degree p over X we have

  stable if the right-hand side inequality is strict. Semistability defines an open substack M sst r,d ⊂ M r,d . 1.3. The global nilpotent cone. Definition 1.2. For any (F, θ) ∈ M r,d and k ≥ 1, set

  of a semistable chain yields a Higgs pair which is a semistable fixed point under the action t.(F, θ) = (F, tθ) of C * on M r,d , where k [E k ((k -1)Ω)] = (r, d). Denote by C n•,p• the substack of fixed points associated to chains of type (n • , p • ), and by C - n•,p• the corresponding attracting variety

  d . The last equality can be stated as Corollary 3.3. The closures of the semistable attracting cells C - n•,p• are irreducible regardless of the coprimality of r and d.

  Corollary 4.5. Fix a partition r r of length s. The set of degree vectors d = (d 1 , . . . , d s ) such that (r, d) (r, d) is semistable is the intersection P Z r,d of the integral lattice Z s with a convex (s -1)-polytope P r,d .

Proposition 4. 7 .

 7 Consider d, d ∈ Z and fix a partition r r of length s. The translation τ = (d -d)r/r of R s induces a bijection P r,d ∼ → P r,d . Proof. First rewrite (4.2) in the following way b - R (r) ≤ dd r r • R ≤ b + R (r) (4.8) where b

  Set δ = f (ρ j k R • ) -f (R • ). We have δ = (d S k + (j -k)lr S k -µ r S k ) 2 + j i=k+1 (d Si -lr S k -µ r Si ) 2 -j i=k (d Si -µ r Si ) 2 = 2(j -k)lr S k (d S k -µ r S k ) + ((j -k)lr S k ) 2 -j i=k+1 2lr S k (d Si -µ r Si ) + l 2 r 2 S k = 2lr S k (j -k)(d S k -µ r S k ) + j -k + 1 2 lr S k -j i=k+1(d Si -µ r Si ) .

This work was started during the author's postdoctoral appointment at MIT, before being supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). The author would like to thank Olivier Schiffmann for numerous extremely useful discussions.

Similarly, when n k < min{n k+1 , . . . , n j }, the α Higgs -slope of the chain

is the α-slope of the subregion obtained by only considering the #S k leftmost boxes in each S t , k ≤ t ≤ j. We denote by Řj k the complementary of this region, and with the same example σ = (3, 2, 5, 4, 1, 6), we have

where µ α ( Ř6 5 ) = 1≤t≤5 α t . The article [START_REF] Bradlow | Irreducibility of moduli of semi-stable chains and applications to U(p, q)-Higgs bundles[END_REF] gives necessary and sufficient conditions (C0,C1,C2,C3) for a type (n • , p • ) to be semistable, meaning that generically, chains of this type are semistable with respect to the α Higgs -slope. In our context, it yields the following.

and if for any

Remark 3.8.

• Note first that (3.2) ⇒ (C k ) for any 1-flag R • .

• Also, if σ is the permutation associated to a 1-chain E • of type satisfying n i-1 = n i but p i-1 < p i , one can always multiply on the left σ by the transposition (i -1, i) in order to satisfy the condition (C0) -without impacting any of the other conditions.

We are going to construct a 1-chain satisfying this set of conditions, under the assumption (3.2). Consider a 1-flag R • , and denote by E • and σ the chain and permutation associated to R • , and set

where means that we remove the underneath entry (this is just the multiplication on the right by the cycle (k, . . . , j)). Denote by ρ j k E • and ρ j k R • the corresponding chain and flag.

The following is true thanks to A.1. Proposition 3.9. For any R • , there exists a finite sequence ((k 1 , j 1 , 1 ), . . . , (k t , j t , t )) of integers such that (i) 1 ≤ k p < j p and p = ±1 for all p = 1 . . . t;

Theorem 4.9. The quantity r r #P Z r,d does not depend on degrees d coprime with r, and is equal to A g,r (1).

The Propostion 4.7 together with the rather explicit description (4.8) however shed an interesting light on Mellit's independence result, given our direct geometric approach of the global nilpotent cone, and its combinatorial flavor.

In fact, Rodriguez Villegas defined in [START_REF] Villegas | A refinement of the A-polynomial of quivers[END_REF] a refinement A g,r of A g,r for each partition r of r, satisfying A g,r (q) = r r A g,r (q), and establishes closed formulas for the quantities A g,r (1). Based on computations for small values of l(r) and the case r = (1 r ) established by Reineke [Rei12, §7], the following is expected.

Conjecture 4.10. We have #P Z r,d = A g,r (1).

APPENDIX A. DESCENT ARGUMENT, BY ANTON MELLIT

In this appendix we show that the procedure used to construct a flag satisfying conditions of Proposition 3.7 eventually stops. We briefly recall the setup. Fix l ≥ 0 and suppose r 1 , . . . , r s , d 1 , . . . , d s is a semistable Jordan type. Recall that 1-flags R • correspond to permutations (σ(1), . . . , σ(s)) where σ

is visually represented by a collection of horizontal strips of lengths σ(1), • • • , σ(s). The i-th box in S k has rank r i and degree

For every subset of boxes Q the rank r Q and degree d Q are defined by summing the ranks resp. degrees of its boxes. The slope is the ratio µ(Q) = d Q r Q , which by definition equals +∞ in the case r Q = 0. The slope of the Jordan type is the slope of the set of all boxes and denoted by µ. This does not depend on the flag. Semistability means that µ(R k ) ≤ µ for any 1 ≤ k ≤ s and any flag R • .

In the procedure described in Section 3 we consider regions R j k resp. Řj k where 1 ≤ k < j ≤ s and n j < min{n k , . . . , n j-1 } resp. n k < min{n k+1 , . . . , n j } and check the conditions C j k resp. Čj k , see Proposition 3.7. If Čj k fails, we apply permutation ρ j k , which places k-th strip at position j and shifts all the strips in between down. The resulting flag is denoted ρ j k R • . If on the other hand C j k fails, we apply the inverse permutation by placing the j-th strip at position k and shifting all the strips in between up. The resulting flag is denoted (ρ j k ) -1 R • .

Lemma A.1. In the above setup, there is a function f from the set of 1-flags to the set of real numbers such that if the condition ( Čj k ) fails for some k and j, then f (ρ j k R • ) < f (R • ), and if the condition (C j k ) fails for some k and j, then f ((ρ j k ) -1 R • ) < f (R • ). Therefore the procedure eventually stops.

Proof. Suppose l > 0 first. To any 1-flag R • we associate the following real number: