Alban Puech
email: alban.puech@se.com

Tristan Rigaut
email: tristan.rigaut@se.com

Adrien Le Franc

William Templier

Jean-Christophe Alais

Maud Tournoud
email: maud.tournoud@se.com

Victor Bossard

Alejandro Yousef

Elena Stolyarova

Controlling Microgrids Without External Data: A Benchmark of Stochastic Programming Methods

Keywords: microgrid, model predictive control, stochastic programming, stochastic optimization

. Our most effective algorithm produces notable cost reductions compared to an MPC algorithm that utilizes the same uncertainty model to generate predictions, and it demonstrates similar performance levels to an ideal MPC that relies on perfect forecasts.

I. INTRODUCTION

The US Department of Energy defines a microgrid as a "group of interconnected loads and distributed energy resources with clearly defined boundaries that act as a single, controllable entity and can connect and disconnect from the grid to operate in both grid-connected and island mode" [START_REF] Ton | The u.s. department of energy's microgrid initiative[END_REF]. Microgrids may play a prominent role in increasing climate resilience, with the opportunity to deploy more renewable energy and help to reduce energy costs.

Energy Management Systems (EMS) automatically control microgrids, optimizing an objective function (usually minimizing the cost), while respecting constraints under uncertainties in the energy production/demand/price. Model Predictive Control (MPC) is the most common method to implement an EMS [START_REF] Nor | Model predictive control based energy management system literature assessment for res integration[END_REF], and simply computes controls (i.e. actions to take) for a given time interval, using forecasts over a rolling horizon as proxies of future uncertainties. Since MPC is intuitive and scales well w.r.t the number of state variables, it has a wide history of industrial applications. Moreover, it relies on mature mathematical programming techniques and software. However, it is particularly sensitive to the quality of forecasts, so implementing a robust MPC for an industrial application can require paying third-party services to obtain them. These services are often expensive and increase the complexity of EMS development and maintenance.

Instead of being addressed by forecasts, uncertainties can be modelled as stochastic processes conditioned by past observations. This approach to EMS design has proved successful for various microgrid settings, e.g. in [START_REF] Rigaut | Stochastic optimization of braking energy storage and ventilation in a subway station[END_REF], [START_REF] Le Franc | EMSx: a numerical benchmark for energy management systems[END_REF], [START_REF] Pacaud | Optimization of a domestic microgrid equipped with solar panel and battery: Model predictive control and stochastic dual dynamic programming approaches[END_REF]. In all three references, stochastic programming methods [START_REF] Bertsekas | Dynamic programming and optimal control: Volume I[END_REF], [START_REF] Shapiro | Lectures on stochastic programming: modeling and theory[END_REF] outperform MPC controllers in most test scenarios. We find these results interesting from the perspective of removing thirdparty forecasts from the EMS components.

In this paper, we benchmark several EMS solutions that are only allowed to make decisions based on past microgrid data observations. We use a probabilistic modeling of future uncertainties fitted on these data, which relies on the Darts toolbox [START_REF] Herzen | Darts: User-friendly modern machine learning for time series[END_REF]. We compare four different algorithms, namely stochastic programming (SP), two-stage stochastic programming (2S-SP), two-stage stochastic programming with scenario clustering (2S-SP-C) and MPC. We further benchmark them against a heuristic (HEU) algorithm and an MPC algorithm relying on perfect forecasts (P-MPC). We assess them on a simple grid-connected microgrid consisting of a single energy storage, a local renewable production and a local load. Our main contributions are:

• A comprehensive mathematical formulation of four different control methods that do not use any external data, under the shared framework of Multistage Stochastic Programming. • A large-scale benchmark of the algorithms for the control of 61 anonymized industrial sites from the EMSx dataset made publicly available by Schneider Electric1 [START_REF] Le Franc | EMSx: a numerical benchmark for energy management systems[END_REF]. • A quantitative analysis and discussion of the performance of the algorithms on the different sites of the dataset.

II. MICROGRID MODEL AND SIMULATOR

In this paper, we design an EMS to control a simple grid-connected microgrid. The microgrid includes a building consuming an uncertain load; solar panels producing an uncertain amount of energy; and a battery with an effective capacity S ∈ R * + , a maximum charge power B ∈ R * + and a minimum discharge power B ∈ R * -.

A. Microgrid model

We model the microgrid as a discrete-time dynamical system. Decisions are made at every time-step of the sequence T = {0, . . . , T } of length T ∈ N * and time increment ∆t. We introduce the system dynamics during the time interval [t, t + 1) of length ∆t.

a) Battery operation: let s t denote the amount of energy stored in the battery at the beginning of the time interval; which is bounded by 0 ≤ s t ≤ S. The EMS can charge/discharge an amount p b t of energy in the battery, as expressed in the discrete-time dynamic equation:

s t+1 = s t + ρ c max{0, p b t } + ρ -1 d min{0, p b t } . (1)
p b t may be zero, positive (charge), or negative (discharge) and is bounded by B∆t ≤ p b t ≤ B∆t. Each microgrid in the benchmark specific charge/discharge efficiency coefficients ρ c ∈ (0, 1] and ρ d ∈ (0, 1], and an initial energy stock s 0 .

b) Energy production and demand: we denote by p t the energy produced by solar panels and by d t the energy demand.

c) Energy exchanges with the grid: the EMS can import an energy amount

e t = max{0, d t -p t + p b t } (2)
from the power grid. These imports occur when the net load d t -p t plus energy in/outflows p b t from the battery are nonnegative.

d) Grid import fee: electricity imports induce a cost that equates to the price of the energy volume e t , paid at a cost c t in e/kWh, plus a fixed penalty of C in e if this volume exceeds the subscribed power E of the installation. It is expressed as:

c t × e t + C × δ ≥E (e t ) , (3)
where δ ≥E is the indicator function equal to 1 when the import exceeds the maximum power. We assume that when the microgrid produces more energy than it consumes, the exported energy is not remunerated by the grid operator.

B. Optimal control framework a) Standard control notations: we adopt standard control notations as used in [START_REF] Bertsekas | Dynamic programming and optimal control: Volume I[END_REF]. State, control, and exogenous random variables are respectively noted as:

x t = s t , u t = p b t , w t = d t -p t . (4)
The aforementioned dynamics (1) and the constraints on (s t , p b t ) are summarized as follows:

x t+1 = f (x t , u t , w t ) , g t (x t , u t , w t ) ≤ 0 , (5)
where f t : R 3 → R + and g t : R 3 → R nc denote respectively the dynamics function and the constraints. The stage cost described in ( 3) is written as a function L t : R 3 → R + of (x t , u t , w t ).

b) Controller definition: At any time-step t ∈ T , the decision u t is made based on the current state x t , and on the history of the microgrid up to time t, defined as:

h t = {w s } s≤t . (6)
Then, a microgrid controller is a collection ϕ = {ϕ t } t∈T of mappings

ϕ t : (x t , h t ) → u t , ∀t ∈ T . (7)
By design, this generic definition of a microgrid controller is agnostic to the method employed to control the system. This opens our benchmark to any other method from all fields of discrete-time control theory. The core distinction with the microgrid controller from the EMSx benchmark [START_REF] Le Franc | EMSx: a numerical benchmark for energy management systems[END_REF] is that the definition of history in equation ( 6) does not include external forecasting data. c) Evaluating the controllers: We evaluate the performance of a controller by measuring the costs under its policy ϕ along the scenario h = {h t } t∈T , starting from an initial given state x 0 :

J(ϕ, h) = t∈T L t (x t , u t , w t ) , (8a)
where

     x t+1 = f (x t , u t , w t ) , g t (x t , u t , w t ) ≤ 0 , u t = ϕ t (x t , h t ) , ∀t ∈ T . (8b)

III. PRESENTATION OF THE CHALLENGERS

In this section, we introduce our challengers: the different control methods that we have implemented to benchmark against a perfect forecast MPC and a heuristic controller. We frame them all as scenario tree-based Multistage Stochastic Programming (MSSP) policies that differ by the scenario tree that they use.

A. Multistage Stochastic Programming policies

At every time-step t 0 an MSSP policy observes the state x t0 and its history h t0 and searches a control ϕ t0 (x t0 , h t0 ) in the argmin of the following stochastic optimization problem:

arg min ut 0 min (Ut 0 +i)1≤i≤R E t0+R t=t0 L t (X t , U t , W t ) , (9a)
s.t. X t+1 = f t (X t , U t , W t ) , (9b) g t (X t , U t , W t ) ≤ 0 , (9c) X t0 = x t0 , H t0 = h t0 , (9d) σ(U t ) ⊂ σ(H t0 , W t0+1 , . . . , W t ) . ( 9e
)
where R is a rolling horizon and the expectation integrates over the conditioned distribution P(W t0+1 , . . . , W t0+R |h t0 ).

The policy only returns a control u t0 for time-step t 0 , that is an element of the arg min in (9a). The min over the subsequent decisions in (9a) measures the expected impact of our current control on the future costs incurred by these decisions. The measurability constraint (9e) states these future decisions are functions of future uncertainties. If the distribution P(W t0+1 , . . . , W t0+R |h t0 ) has infinite support, the problem is hard to solve because the solution space is infinite-dimensional. We introduce the concept of scenario tree and how it is used to approximate (9) by a tractable optimization problem.

1) Scenario tree: A scenario tree G = (N , P) is a tree graph representing the distribution of a discrete stochastic process (Ξ 0 , . . . , Ξ R ) with finite support. N is a finite set of nodes, a node n at a depth t is a realization (ξ 1 , . . . , ξ t ) of the process up to time t. Ξ 0 has only 1 realization ξ 0 with probability 1 so there is a single node, the root n 0 , at depth 0. Elements (m, n) ∈ P ⊂ N × N are edges of the tree. We call d n the depth of a given node n. We define the probability π(n) of a node n as P(Ξ 0 = ξ 0 , . . . , Ξ dn = ξ dn ), or equivalently, the product of probability transitions from the root n 0 to n.

2) Construction of a scenario tree: Many methods exist to generate a scenario tree modeling the distribution of a given stochastic process (W 0 , . . . , W R ) from a collection of samples. We refer to [START_REF] Kammammettu | Scenario reduction and scenario tree generation for stochastic programming using sinkhorn distance[END_REF] and [START_REF] Kipngeno | New algorithms and fast implementations to approximate stochastic processes[END_REF] for recent reviews of the methods.

In this paper, we generate scenario trees to approximate the distribution of the netload stochastic process (W t+1 , . . . , W t+R ) conditioned by the netload history h t = (w 0 , . . . , w t ). To do so, we train a machine learning model to sample from P(W t+1 , . . . , W t+R |h t ). Our model uses exclusively past realizations to model future uncertainties: this is an autoregressive model.

3) Scenario tree approximation: Approximating the stochastic process P(W t0+1 , . . . , W t0+R |h t0 ) by a scenario tree G in (9) turns the solution space into a finite dimensional one. Indeed constraint (9e) implies that, for each node n of G, there is one decision u n , and due to constraint (9b) one state x n . We call w n = w dn the last netload realization of the node n. We rewrite (9) into its extensive formulation [START_REF] Kipngeno | New algorithms and fast implementations to approximate stochastic processes[END_REF]. arg min

un 0 min u n∈N \{n 0 } n∈N π(n) × L dn+t0 (x n , u n , w n ) , (10a) s.t. x n = f dm+t0 (x m , u m , w m ), ∀(m, n) ∈ P , (10b) g dn+t0 (x n , u n , w n ) ≤ 0, ∀n ∈ N , (10c) x n0 = x t0 . (10d)
In our case, problem ( 10) is a so-called Mixed-Integer Linear Program (MILP) that we model with Pyomo [START_REF] Michael | Pyomo-optimization modeling in python[END_REF] and solve using HiGHS [START_REF] Huangfu | Parallelizing the dual revised simplex method[END_REF]. The number of variables (resp. constraints) in [START_REF] Kipngeno | New algorithms and fast implementations to approximate stochastic processes[END_REF] grows linearly with the number of nodes (resp. edges) in the tree, while the latter grows exponentially with the number of time steps in the rolling horizon. This significantly deteriorates the performance of a MILP solver. We present hereunder four methods to compute controls using scenario trees that are small enough to solve problem [START_REF] Kipngeno | New algorithms and fast implementations to approximate stochastic processes[END_REF] in a reasonable amount of time.

B. Challengers

Without loss of generality, we fix R = 23 (control horizon of 24 hours). We formulate each of our challengers as an MSSP method and use a LightGBM model implemented using Darts [START_REF] Herzen | Darts: User-friendly modern machine learning for time series[END_REF] to draw K samples ( w0 , wi 1 , wi 2 , ..., wi 23 ), 1 ≤ i ≤ K from P(W t+0 , . . . , W t+R |h t ), that are later used to generate a

••• ••• ••• ••• ••• ••• ••• 2S-SP 2S-SP-C ••• SP ••• ••• MPC Fig. 1.
Diagram of the construction of the trees for each algorithm. We use R = 23 (control horizon of 24h) scenario tree. A challenger is identified by the kind of scenario tree it generates, as shown in Fig. 1: 1) SP: SP (short for Stochastic Programming) takes as input ( w0 , wi 1 , wi 2 , ..., wi 23 ), 1 ≤ i ≤ K and builds a scenario tree with K 23 leaves (w 0 , w 1≤i1≤K1 1 , w 1≤i2<K2 1 , ..., w 1≤i23<K23 23 ) using scenred [START_REF] Heitsch | Scenario reduction algorithms in stochastic programming[END_REF]. The reduction parameter defines the number of nodes K 1 , K 2 , ..., K 23 at each depth of the tree.

2) 2S-SP and 2S-SP-C: Two-stage stochastic programming, denoted by 2S-SP directly turns each sample into a branch of the tree connected to the root so that ∀1 ≤ i ≤ K : (w i 0 , w i 1 , ..., w i 23 ) = ( wi 0 , wi 1 , ..., wi 23 ). This type of tree is commonly referred to as a scenario fan. 2S-SP-C is a slightly different version of 2S-SP that uses K-means to separate the K samples into K' clusters, whose centres are used instead of the original samples to generate the tree.

3) MPC: Finally, Model predictive Control MPC is framed as an MSSP method, where the unique scenario (unique branch of the tree) is obtained by averaging all the samples: (w t ) t≤23 = (1/K) 1≤i≤K wi t t≤23

C. Two baselines

The challengers are compared with two baseline controllers.

1) P-MPC: An ideal MPC with perfect forecasts that permits us to assess the value of perfect forecasts compared to our autoregressive models of uncertainties.

2) HEU: A myopic heuristic that stores electricity when the netload is negative and discharges the battery when the netload is positive. This baseline is called HEU and represents the traditional optimization-free industrial method.

IV. RESULTS AND DISCUSSIONS

A. Simulation Setup

Our challengers (MPC, SP, 2S-SP and 2S-SP-C) are tested against the two baselines (HEU and P-MPC) through 61 simulations. Each simulation is associated with a time series of solar production and energy demand from a given site2 , with a 15-minute resolution, resampled into series with 1-hourlong time-steps. The energy demand time series corresponds to our exogenous uncertainty. Sites each have a microgrid setup, and we further set the subscribed power limit as

E = max t (d t -p t ) -B.
The penalty for exceeding the power subscription is fixed at 14.31e/h, as per the "Tarif Jaune" rates from France's main electric utility company EDF [START_REF]relatif aux tarifs réglementés de vente de l'électricité[END_REF].

The electricity prices (also displayed in Fig. 3) are set to 0.102e during off-peak hours (i.e. from 00:00 to 06:00, 09:00 to 11:00, 13:00 to 17:00, and 21:00 to 00:00) and to 0.153e during peak hours. We use 60 % of the dataset for training, and the rest for testing. Table I shows the hyperparameter values and other implementation specifics used for the different algorithms across the 61 simulations. Our control horizon is 24h (R = 23).

All experiments were run on a computer with an Intel i7-12800H CPU and 32 GB RAM.

B. Results on the 61 sites

We introduce the average performances, both in terms of savings and computation times, of the challengers and baselines across the whole dataset of 61 sites. Fig. 2. Relative cost savings from using each of the challenger (SP, 2S-SP, 2S-SP-C) and P-MPC over the heuristic algorithm HEU 1) Average savings: We show in Fig. 2 the distribution of the relative cost savings obtained using a given microgrid controller vs HEU. Using SP, 2S-SP, and 2S-SP-C over HEU always yields cost savings, and all three algorithms show similar performances. The savings obtained using MPC are smaller on average, sometimes negative, and show a much larger variance. Note that, in contrast to the other challengers, MPC is performing worse HEU on almost 25 % of 61 different simulations. We show in Table II the average results obtained by the different challengers across the 61 simulations. 2S-SP and 2S-SP-C together outperform SP in the majority of the scenarios (71 %), while SP yields higher cost savings on average (4.98 % against 4.94 and 4.93 % for 2S-SP and 2S-SP-C respectively). MPC sometimes performs worse than HEU, and the difference in performance between P-MPC and MPC highlights the importance of having accurate forecasts. Finally, the difference between SP and P-MPC was quite limited (1.51%): using a probabilistic model only with local historical data alleviates the impact of an imperfect forecast.

2) Computing times: We additionally report in Table II the mean processing time per control. It corresponds to an average over a 7700-time-step-long test dataset. The training is done on 60 % of the whole dataset and took 2.8s for each challenger.HEU computes a control in the smallest amount of time. P-MPC requires a bit more time as it creates and solves a MILP problem. MPC additionally uses the LightGBM model to compute forecasts. 2S-SP also draws samples from the LightGBM model but creates a larger MILP problem than MPC, there are more nodes in the scenario tree. 2S-SP-C adds a clustering step to 2S-SP to solve a smaller MILP but still requires more time than 2S-SP. Finally, SP stands between 2S-SP and 2S-SP-C, it adds the scenario tree generation to 2S-SP, which takes less time than the clustering of 2S-SP-C.

C. Comparison of the control trajectories on a simulation sample

We show in Fig. 3 a sample of the battery state of charge and grid import trajectories under the policies of the different algorithms3 2S-SP and 2S-SP-C have the closest trajectories to P-MPC, displayed in light gray. The gains obtained using the challengers over HEU are achieved in two different ways: a) Limiting the number of subscribed power limit overruns in high load conditions: Most of the savings are obtained in the very first (0-25) and last (>200) hours of the simulation slice shown in Fig. 3. The high net load forces all algorithms to exceed E to meet the energy demand, even when not charging the battery. This penalty being fixed, charging the battery on top of importing the energy necessary to satisfy the demand that already exceeds the subscribed power limit does not generate any additional cost (apart from the cost due to a larger volume being bought). The grid import trajectory under the policy of HEU constantly stays over E during those two time-frames, while it is more often below when following the policies of the challengers. On one hand, HEU constantly pays the penalty for exceeding E and imports the electricity necessary to meet the demand without charging the battery. On the other hand, the challengers repeatedly charge and discharge the battery, alternating between time-steps when the battery is charging and when the penalty is paid, and time-steps when discharging the battery allows the grid import to stay below E. In particular, notice how 2S-SP-C succeeds in keeping the grid import below E in the 205-210 time frame.

b) Arbitrage on the electricity prices and high net load anticipation: On the rest of the slice, additional savings are obtained thanks to the arbitrage done on the electricity prices. Indeed, they typically charge the battery during off-peak hours, when the price is low, and when the demand is low enough for the grid import (increased by the energy charged into the battery) to stay below E. The battery is then discharged when the electricity prices are higher, or when it allows the grid import that would otherwise exceed E power to stay below, by satisfying a bit of this demand using the energy stored in the battery. In the first case, this saves a cost equal to the amount of energy discharged, multiplied by the price spread. In the second case, this spares the penalty cost associated with the overrun of the subscribed power limit. V. CONCLUSION We introduced four different multistage stochastic control algorithms, namely, model predictive control, stochastic programming, two-stage stochastic programming, and two-stage stochastic programming with scenario clustering. We proposed a generic mathematical formulation of those algorithms under the shared framework of Multistage Stochastic Programming. We presented a simple microgrid model and benchmarked our four challengers against two baselines: a simple heuristic, and a model predictive control using perfect forecasts. The benchmark is done through 61 simulations, each having a different microgrid setup and exogenous uncertainty distribution. Our two-stage stochastic programming algorithm shows similar performances to our stochastic programming one, while at the same time being much easier to implement (as it doesn't require any scenario tree reduction step) and less computationally heavy. Both algorithms outperform the model predictive control approach, reducing the electricity cost by 2.2 % (which amounts to 7413e of average yearly cost savings) when used over model predictive control, and only generate an extra cost of 1.5 % compared to the perfect baseline. We believe this is an important finding, which questions the financial interest in relying on third-parties services, such as weather/irradiance forecasts, to improve the net load forecast accuracy.

Fig. 3 .

 3 Fig.3. Sample the control sequences/ trajectories of the challengers and the heuristic algorithm (HEU) running on site 19 of the EMSx dataset. "Import" refers to the amount of electricity imported from the grid, expressed in KWh.

https://zenodo.org/record/5510400#.YUizGls69hE

Nine sites from[START_REF] Le Franc | EMSx: a numerical benchmark for energy management systems[END_REF] were not used for our simulations, as they contain extended periods of missing data

Note that only a small portion of the simulation on site 19 of the EMSx dataset is displayed and that the full simulation on this site covers 6576 hours.


Bucciarelli Eva
email: eva.bucciarelli@univ-brest.fr

Stiger-Pouvreau Valerie

Connan Solène

Eva Bucciarelli

A new protocol using acidification for preserving DMSP in macroalgae and comparison with existing protocols

Keywords: acidification, DMSP, drying, freezing, macroalgae, storage protocol DMS, dimethylsulfide, DMSP, dimethylsulfoniopropionate, FW, fresh weight, GC, gas chromatography

Dimethylsulfoniopropionate (DMSP) plays many important physiological and ecological roles in macroalgae. The most common method to measure DMSP is by gas chromatography analysis of the dimethylsulfide (DMS) produced after NaOH hydrolysis (pH > 12). Storage of DMS, however, is not recommended for more than a week. We investigated if acidification can be a suitable method to preserve DMSP in macroalgal samples over three months of storage, compared to widely used protocols such as drying and freezing at -20°C. The DMSP content of green (Ulva sp. and Ulva compressa), red (Chondrus crispus) and brown (Bifurcaria bifurcata) macroalgae were analyzed 24 h after NaOH addition (control values); and after acidification (0.2 mol • L HCl-1) for 24 h of fresh material, followed by NaOH addition for 24 h. These values were compared to measurements after 3-month storage of samples that had been either dried in a heater (60°C for a night, and storage at room temperature), or frozen at -20°C, or kept in 0.2 mol • L HCl-1. There was no significant difference between DMSP measurements on freshly collected material and after acidification of the samples, whether 24 h later or after 3 months of storage. This was in contrast with 3-month storage protocols involving overnight drying at 60°C (75-98% DMSP loss), and to a lesser degree freezing at -20°C (37-80% DMSP loss). We thus advise to acidify macroalgal samples for preservation over long periods of time rather than drying or freezing, when assaying DMSP content.

Dimethylsulfoniopropionate (DMSP) is a tertiary sulfonium compound produced intracellularly mainly by micro-and macroalgae [START_REF] Malin | Minireview -Algal production of dimethylsulfide and its atmospheric role[END_REF]Kirst 1997, Bullock et al. 2017). It plays a wide range of physiological and ecological roles in macroalgae. Within the cells, it serves as an osmolyte, and as an antioxidant [START_REF] Stefels | Physiological aspects of the production and conversion of DMSP in marine algae and higher plants[END_REF][START_REF] Van Alstyne | Ecological and physiological roles of dimethylsulfoniopropionate and its products in marine macroalgae[END_REF][START_REF] Rix | Irradiance-mediated dimethylsulphoniopropionate (DMSP) responses of red coralline algae[END_REF]. DMSP and its enzymatic breakdown product dimethylsulfide (DMS) are also known to play a key role in chemical signalling, from bacteria [START_REF] Barak-Gavish | Bacterial virulence against an oceanic bloom-forming phytoplankter is mediated by algal DMSP[END_REF] to top predators [START_REF] Nevitt | The neuroecology of dimethyl sulfide: A global-climate regulator turned marine infochemical[END_REF].

Recent studies showed that DMSP from macroalgae can either inhibit [START_REF] Saha | Dimethylsulphopropionate (DMSP) and proline from the surface of the brown alga Fucus vesiculosus inhibit bacterial attachment[END_REF] or promote [START_REF] Kessler | Macroalgal-bacterial interactions: Role of dimethylsulfoniopropionate in microbial gardening by Ulva (Chlorophyta)[END_REF] bacterial activity. This production is thought to be part of an activated defense system (Van Alstyne et al. 2001, Wiesemeier et al. 2007), and to induce a foodsource chemical signal for bacteria, that in turn allow the morphogenesis and development of the DMSP-producer macroalga [START_REF] Kessler | Macroalgal-bacterial interactions: Role of dimethylsulfoniopropionate in microbial gardening by Ulva (Chlorophyta)[END_REF]Beemelmanns 2018). High DMSP concentrations are generally found in green macroalgae and a few red algal species, while brown macroalgae only contain small amounts (Reed 1983, Van Alstyne and[START_REF] Van Alstyne | DMSP in marine macroalgae and macroinvertebrates: Distribution, function, and ecological impacts[END_REF].

The most common method to measure DMSP is by GC analysis of the DMS released after 24 h of NaOH hydrolysis (pH> 12; White 1982). However, DMS is a volatile compound, prone to oxidation, that has been shown to decrease significantly in sealed vials after a week of storage [START_REF] Curran | Spatial distribution of dimethylsulfide and dimethylsulfoniopropionate in the Australasian sector of the Southern Ocean[END_REF]. Besides, other dimethylsulfonium compounds than DMSP can yield DMS after prolonged NaOH hydrolysis [START_REF] Howard | An alternative approach to the measurements of βdimethylsulfoniopropionate (DMSP) and other precursors of dimethylsulfide[END_REF]. Protocols to preserve the samples of macroalgae and plants have been proposed, including drying or freezing [START_REF] Karsten | A method for complete determination of dimethylsulfoniopropionate (DMSP) in marine macroalgae for different geographical regions[END_REF], Russell and Howard 1996[START_REF] Mcfarlin | Foliar DMSO:DMSP ratio and metal content as indicators of stress in Spartina alterniflora[END_REF][START_REF] Borges | Preservation protocol for dimethylsulfoniopropionate and dimethylsulfoxide analysis in plant material of the Mediterranean seagrass Posidonia oceanica, and re-evaluation of dimethylsulfoniopropionate leaf content[END_REF]. These methods, however, can strongly affect DMSP contents, compared to analyses on fresh samples [START_REF] Bischoff | Preliminary assessment of the β-dimethylsulfoniopropionate (DMSP) content of macroalgae from the tropical Island of Hainan (People's Republic of China)[END_REF], Karsten et al. 1994, Russell and Howard 1996). Still, storing samples over more than a week can be unavoidable in case of a cruise or field sampling, or when needing to analyze more than a few samples at a time. Acidification is an alternative method, which has been successfully used to preserve DMSP in seawater and phytoplankton culture samples over months of storage [START_REF] Curran | Spatial distribution of dimethylsulfide and dimethylsulfoniopropionate in the Australasian sector of the Southern Ocean[END_REF][START_REF] Del Valle | Effect of acidification on preservation of DMSP in seawater and phytoplankton cultures: Evidence for rapid loss and cleavage of DMSP in samples containing Phaeocystis sp[END_REF]. DMSP is highly stable in low pH solutions [START_REF] Dacey | Hydroxide decomposition of dimethylsulfoniopropionate to form dimethylsulfide[END_REF]. Furthermore, acidification lyses membranes of macroalgal cells and increases their porosity (Zemke-White et al. 1999[START_REF] Stefels | Physiological aspects of the production and conversion of DMSP in marine algae and higher plants[END_REF], thereby increasing the potential recovery and stabilisation of intracellular DMSP in macroalgae.

In this study, we investigated if acidification can be a suitable method to preserve DMSP over 3 months in samples of macroalgae, compared to drying and freezing.

Four species of macroalgae were collected by hand at low tide on the west coast of Brittany (Pointe du Diable, 48°28.871' N, 4°46.142' W), a field site nearby the laboratory. These macroalgae include two green, a brown and a red species, to test for the validity of our protocols

Accepted Article

This article is protected by copyright. All rights reserved on macroalgae with high (green species) and low (red and brown species) DMSP contents (Reed 1983, Van Alstyne and[START_REF] Van Alstyne | DMSP in marine macroalgae and macroinvertebrates: Distribution, function, and ecological impacts[END_REF]. Our green species include the foliose Ulva sp. Linnaeus (collected on Oct. 8 th 2018; Chlorophyta, Ulvophyceae, Ulvales, Ulvaceae) and the tubular Ulva compressa Linnaeus (collected on April 8 th 2019; Chlorophyta, Ulvophyceae, Ulvales, Ulvaceae).

The brown and red species were Bifurcaria bifurcata (collected on Oct. 8 th 2018; Ochrophyta, Phaeophyceae, Fucales, Sargassaceae), and Chondrus crispus (collected on Nov. 7 th 2018; Rhodophyta, Florideophyceae, Gigartinales, Gigartinaceae), respectively. Ten different thalli (i.e., individuals) per species were transported back to the laboratory in plastic bags. Shortly after returning to the laboratory, each individual was carefully dried with a tissue and most of the visible epiphytes (e.g., Ceramiales and Ulvales) were removed by hand. According to the DMSP values expected from the literature, and to ensure that measurements using GC would be detectable, a few micrograms (green macroalgae) to a few milligrams (red and brown macroalgae) from each individual was sampled using a clean scissor. This was repeated 5 times on each individual, so as to have 5 subsets of 10 replicates for the five different protocols. For the first protocol, the samples were placed in 10 mL glass vials, 5 mL of 5 mol • L NaOH -1 were added and the vials were sealed. DMS resulting from DMSP degradation in these basic conditions (pH > 12) was measured on the following day (Protocol P1-NaOH-T0) and this value was considered as the reference value. Samples from the second subset were wrapped in aluminium foil and dried in a heater at 60°C for a night (Protocol P2-Drying), and further kept at room temperature for 3 months. Samples from the third subset were placed in 10 mL glass vials that were sealed and stored at -20°C for 3 months (Protocol P3-Freezing). Samples from the fourth and the fifth subsets were placed in 10 mL glass vials with 4 mL of 0.2 mol • L HCl -1 (pH < 2) and the vials were sealed and kept at room temperature. A day later, the vials of the fourth subset were opened for addition of 1 mL of 5 mol • L NaOH -1 (pH > 12) and the vials were sealed again; DMS analyses were run on the following day (Protocol P4-HCl-T0). Vials of the fifth subset were stored for 3 months (Protocol P5-HCl-Tf). After 3 months, dried samples from Protocol P2-Drying were placed in 10 mL glass vials, and 5 mol • L NaOH -1 was added to all samples from P2-Drying (5 mL), P3-Freezing (5 mL) and P5-HCl-Tf (1 mL) so as to achieve a final volume of 5 mL in the vials. Analyses for DMS were run on the following day.

DMSP was analyzed as DMS using a Shimadzu 2010-Plus Gas Chromatograph (GC) equipped with a sulfur-selective flame photometric detector (air / H 2 : 70 mL • min -1 / 60 mL • min -1 ) and an Equity 1 capillary column (3.2 mm i.d., 30 m long, Supelco, T = 180°C, carrier gas: He, 6.7 mL • min -1 ). Detector and injection port temperatures were set at 250°C. Depending on the macroalgal DMSP content, 10 to 250 µL were collected from the headspace of the vials using a gas tight

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This article is protected by copyright. All rights reserved syringe and were directly injected into the GC injector port. The same headspace volume was sampled from the DMSP standards and the samples. Calibration curves for protocols P1-NaOH-T0, P2-Drying and P3-Freezing consisted in increasing additions of 10 -3 mol • L DMSP -1 into 5 mL of 5 mol • L NaOH -1 in 10 mL glass vials, which were sealed for a day before analyses.

Calibration curves for protocols P4-HCl-T0 and P5-HCl-Tf consisted in increasing additions of 10 -3 mol • L DMSP -1 into 4 mL of 0.2 mol • L HCl -1 in 10 mL glass vials, which were sealed for a day (P4-HCl-T0) or for 3 months (P5-HCl-Tf). One day before the analyses were run, the vials were opened for addition of 1 mL of 5 mol • L NaOH -1 and re-sealed. Five to ten standards were run for calibration. The correlation coefficients R 2 of the calibration curves ranged between 0.984 and 0.999 (0.998 ± 0.004, mean ± STD, n = 17).

Statistical analyses were performed using R software (version 3.6.3; R Core Team 2020) through the integrated development environment Rstudio (version 1.2.5033; Rstudio Team 2020). DMSP levels measured with Protocol P1-NaOH-T0 were compared among species using a Kruskal Wallis test with a confidence level of 95% as data did not meet the requirements for parametric tests. Multiple comparisons were performed using the pairwise non-parametric Wilcoxon test. Because the data did not follow a normal distribution (Shapiro-Wilks test, p-value < 0.05 for each algal species), the differences in DMSP content among protocols for each macroalgal species were tested using the non-parametric Friedman rank test for data in repeated measurement designs at a confidence level of 95%. When a statistical difference among protocols was detected, post-hoc pairwise comparisons using a Nemenyi-Wilcoxon test were applied. The effect of epiphytes occurrence on DMSP content was tested using a Kruskal Wallis test. The impact of hexane on the removal of epiphytes and therefore on DMSP content was tested using a Paired t test.

Figure 1 displays the mean DMSP concentrations for each macroalgal species according to the different protocols (in µmol • g -1 algal fresh weight (FW) for the two green Ulva spp. and in nmol • g FW -1 for the brown and red species). To our knowledge, values of DMSP for Bifurcaria bifurcata are not reported in the literature. For Chondrus crispus and U. compressa, our values of DMSP measured following Protocol P1-NaOH-T0 were similar to values reported previously following a similar protocol (respectively, in nmol • g FW -1 : < 50, Reed 1983; 47 ± 21, Russell and Howard 1996; and in µmol • g FW -1 : 25.4-39.5, Reed 1983; 20.2 ± 7.3, Bischoff et al. 1994; ~ 20, Steinke et al. 1996, see also Table S1 for comparison with available data from the literature for the same species, including other analytical methods than GC). Acidification gave the best results as a preservation method for the four studied species. Values measured after 24 h acidification (P4-HCl-T0) were not significantly different from values measured according

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This article is protected by copyright. All rights reserved to P1-NaOH-T0, including for U. compressa (Fig. 1). This last species exhibits the same, high, DMSP-lyase activity as the microalgal Phaeocystis sp. (~ 10 nmol • min -1 • mg -1 of protein; [START_REF] Steinke | DMSP lyase in marine macro-and microalgae. Intraspecific differences in cleavage activity[END_REF][START_REF] Del Valle | Effect of acidification on preservation of DMSP in seawater and phytoplankton cultures: Evidence for rapid loss and cleavage of DMSP in samples containing Phaeocystis sp[END_REF] showed that acidification was suitable to preserve DMSP for months in microalgal samples, including samples of single cell Phaeocystis spp., but that a substantial loss of DMSP occurs within a few minutes after acid addition when Phaeocystis spp. are present in the colonial stage. This has been attributed to possible differences in the physico-chemical characteristics of the enzymes or to differences in the cellular location of DMSP and the enzymes between single cell and colonial stages of Phaeocystis life cycle [START_REF] Del Valle | Effect of acidification on preservation of DMSP in seawater and phytoplankton cultures: Evidence for rapid loss and cleavage of DMSP in samples containing Phaeocystis sp[END_REF]). In our study there was no measurable DMSP loss in U. compressa after acidification. We cannot rule out, however, that there might be an effect of acidification in other species that have also been shown to express DMSP-lyase activity (e.g., [START_REF] Nishiguchi | Isolation, purification, and characterization of DMSP lyase (dirnethylpropiothetin dethiomethylase (4.4.1.3)) from the red alga Polysiphonia paniculata[END_REF][START_REF] Steinke | DMSP lyase in marine macro-and microalgae. Intraspecific differences in cleavage activity[END_REF][START_REF] Van Alstyne | Dimethylsulfide release during macroinvertebrate grazing and its role as an activated chemical defense[END_REF][START_REF] Garcia-Jimenez | Production of volatiles by the red seaweed Gelidium arbuscula (Rhodophyta): emission of ethylene and dimethyl sulfide[END_REF]. The nature of DMSP-lyases in macroalgae has begun to be unveiled only very recently, and their locations are still unknown (De Clerck et al. 2018). Until their characteristics and functioning are better constrained, we recommend to conduct preliminary tests to assess the validity of the acidification protocol before storage, e.g., by comparing values from P1-NaOH-T0 and P4-HCl-T0.

There was no statistical difference in DMSP content between acidified samples analyzed 24 h after sampling (P4-HCl-T0) and after 3 months of acidified storage (P5-HCl-Tf) for any of the four species (Fig. 1). On the contrary, only 2-25% of DMSP remained in all four macroalgae after overnight drying at 60°C and 3 months of storage (P2-Drying; Fig. 1). [START_REF] Bischoff | Preliminary assessment of the β-dimethylsulfoniopropionate (DMSP) content of macroalgae from the tropical Island of Hainan (People's Republic of China)[END_REF], who oven-dried Ulva compressa for 12 h at 80°C, also reported that only ~ 16% of DMSP remained after 3-4 months of storage. Freezing at -20°C for 3 months (P3-Freezing) also decreased the initial DMSP content of the macroalgae, but to a lesser extent than P2-Drying (20-63%; Fig. 1). We did not test the effects of freeze-drying of the tissues (e.g., [START_REF] Karsten | A method for complete determination of dimethylsulfoniopropionate (DMSP) in marine macroalgae for different geographical regions[END_REF], or freezing at -80°C, which may improve the preservation of DMSP compared to freezing at -20°C.

A high variability could however be noted with the acidification protocol, especially for Bifurcaria bifurcata and Chondrus crispus. Removing obvious outliers for these species (Fig. 1) did not change the statistical results: DMSP levels for protocols P4-HCl-T0, P5-HCl-Tf and P1-NaOH-T0 remained higher than for P2-Drying and P3-Freezing (Friedman rank tests; pvalue = 2.286.10 -5 and 1.247.10 -5 for B. bifurcata and C. crispus, respectively). This high variability may be due to intrathallus variability, and to the presence of epiphytes. In Ulva lactuca (Van Alstyne et al. 2007) and in the brown seaweed Taonia atomaria [START_REF] Paix | A multi-omics analysis suggests links between the differentiated surface metabolome and epiphytic microbiota along the thallus of a mediterranean seaweed holobiont[END_REF], DMSP concentrations are higher at the base of the thallus than in the middle of the thallus or at the

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This article is protected by copyright. All rights reserved edges. For B. bifurcata and C. crispus, we had to mix several pieces of the middle part of the thallus to ensure that there would be enough material to analyze (a few milligrams). This may have increased DMSP variability. Epiphytes can also increase DMSP content in some plant species [START_REF] Dacey | Herbivory by reef fishes and the production of dimethylsulfide and acrylic acid[END_REF]. We carefully removed epiphytes such as other algae from our samples, but other epiphytes like bryozoans were still evident and difficult to remove on some samples of C. crispus. Epiphytic microalgae and bacteria could also affect DMSP measurements [START_REF] Kessler | Macroalgal-bacterial interactions: Role of dimethylsulfoniopropionate in microbial gardening by Ulva (Chlorophyta)[END_REF][START_REF] Paix | A multi-omics analysis suggests links between the differentiated surface metabolome and epiphytic microbiota along the thallus of a mediterranean seaweed holobiont[END_REF]. In a supplementary experiment, we collected 20 individuals of C. crispus (same site of collection, Feb. 7 th 2019): 10 were colonised by epiphytes (including bryozoans), and 10 were devoid of epiphytes. Each one of the 20 individuals was divided in 2 subsamples. One of them was dipped in hexane for 30 s to remove microfouling organisms [START_REF] De Nys | A new method for determining surface concentrations of marine natural products on seaweeds[END_REF]) before being processed for DMSP measurements. As a result, we could compare DMSP contents (following Protocol P1-NaOH-T0) between populations with and without visible epiphytes (32 ± 42 nmol • g FW -1 , mean ± STD, n=10, and 7 ± 6 nmol • g FW -1 , mean ± STD, n=10, respectively), and when populations with and without visible epiphytes were dipped in hexane (53 ± 94 nmol • g FW -1 , mean ± STD, n=10, and 9 ± 11 nmol • g FW -1 , mean ± STD, n=10, respectively). The removal of fouling organisms by hexane dipping did not change the DMSP measurements, neither for the epiphyte colonised samples (Paired-t-test, t 9 = -1.221, p = 0.253) nor for the others (Paired-t-test, t 9 = -0.684, p = 0.511). DMSP contents on the contrary were significantly higher in epiphyte colonised samples than in the others (Kruskal-Wallis, Chi-Square 1 = 5.491, p = 0.019). The presence of epiphytes such as bryozoans may thus explain at least part of the observed DMSP variability in C. crispus.

In conclusion, there was no significant difference between DMSP measurements on freshly collected material (P1-NaOH-T0) and after acidification of the samples, whether 24 h later (P4-HCl-T0) or after 3 months of storage (P5-HCl-Tf), for any of the species that we considered. This is in contrast with 3-month storage protocols involving overnight drying at 60°C (P2-Drying), and to a lesser degree freezing at -20°C (P3-Freezing). We thus advise to use acidification (0.2 mol • L HCl -1 , pH < 2) for preservation of macroalgal samples over long periods of time rather than drying or freezing at -20°C, when assaying DMSP content.

ACKNOWLEGEMENTS

We acknowledge Flavia Bolloro and Lisa Morin for their help during preliminary experiments.

We are grateful to two reviewers for their comments that helped to improve the manuscript.

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This article is protected by copyright. All rights reserved Van Alstyne, K.L., Wolfe, G.V., Freidenburg, T.L., Neill, A. & Hicken, C. 2001

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This article is protected by copyright. All rights reserved Figure 1: DMSP concentrations in the four macroalgal species according to the five different protocols (in µmol or nmol • g -1 algal fresh weight depending on the algal species). Significant differences among protocols for each macroalgal species are marked with different letters (n = 10; Friedman rank test for data in repeated measurement designs with post-hoc Nemenyi-Wilcoxon tests; p-value < 0.05).

Table S1: DMSP content (in µmol or nmol DMSP • g of algal FW -1 depending on the algal species) for the four species of macroalgae according to the five different protocols of preservation in our study (see text for detail), and comparison with available data from the literature for the same species (or Ulva lactuca as a foliose Ulva sp.). For this study: mean ± standard deviation; n = 10; for literature data, depending on the available data: range, mean or mean ± standard deviation. # = Ulva lactuca as a foliose Ulva sp. * = Ulva mutabilis conspecific with Ulva compressa [START_REF] Steinhagen | Conspecificity of the model organism Ulva mutabilis and Ulva compressa (Ulvophyceae, Chlorophyta)[END_REF]).


Christian Miebach

ON QUOTIENTS OF BOUNDED HOMOGENEOUS DOMAINS BY UNIPOTENT DISCRETE GROUPS

We show that the quotient of any bounded homogeneous domain by a unipotent discrete group of automorphisms is holomorphically separable. Then we give a necessary condition for such a quotient to be Stein and prove that in some cases this condition is also sufficient.

INTRODUCTION

Given a Lie group of holomorphic transformations of a Stein space X, one would like to have a complex quotient space whose holomorphic functions are the invariant holomorphic functions on X and which is again Stein. In the case that the Lie group is compact, it is possible to average holomorphic functions over this group and to construct in this way an invariant-theoretic Stein quotient space, see [START_REF] Heinzner | Geometric invariant theory on Stein spaces[END_REF]. On the other hand, when an infinite discrete group acts properly by holomorphic transformations on a Stein space, then the orbit space is again a complex space. However, in general there is no averaging method in order to construct holomorphic functions on this quotient space. In fact, it is not hard to find examples where the quotient space is compact, hence where all invariant holomorphic functions are constant.

There are numerous results in the literature where certain quotients of Stein manifolds by proper actions of infinite discrete groups are shown to be holomorphically separable or Stein. In [START_REF] De Fabritiis | Quotients of the unit ball of C n for a free action of Z[END_REF] the authors showed that quotients of the unit ball B n ⊂ C n by proper Z-actions are Stein, which was then generalized to simply-connected bounded domains of holomorphy in C 2 ( [START_REF] Miebach | On proper R-actions on hyperbolic Stein surfaces[END_REF]), to arbitrary bounded homogeneous domains in C n ( [START_REF] Miebach | Quotients of bounded homogeneous domains by cyclic groups[END_REF]), and to Akhiezer-Gindikin domains ( [START_REF] Vitali | Quotients of the crown domain by a proper action of a cyclic group[END_REF]). Quotients of the unit ball and Akhiezer-Gindikin domains by discrete groups that act cocompactly on a real form of these domains are shown to be Stein in [4, Proposition 6.4] and [START_REF] Burns | The geometry of Grauert tubes and complexification of symmetric spaces[END_REF]Corollary 7], respectively. An analog result is shown in [START_REF] Loeb | Action d'une forme réelle d'un groupe de Lie complexe sur les fonctions plurisousharmoniques[END_REF] for quotients of complex solvable Lie groups by discrete subgroups that act cocompactly on a real form having purely imaginary spectrum. Quotients of complex Olshanski semigroups by certain discrete groups were studied in [START_REF] Achab | Discrete group actions on Stein domains in complex Lie groups[END_REF] and [START_REF] Miebach | Sur les quotients discrets de semi-groupes complexes[END_REF]. In [START_REF] Chen | Discrete groups and holomorphic functions[END_REF] actions of discrete groups on Kähler-Hadamard manifolds and their quotients are investigated. Recently, quotients of the unit ball by certain convex-cocompact discrete groups were studied from the viewpoint of complex-hyperbolic geometry in [START_REF] Dey | A note on complex-hyperbolic Kleinian groups[END_REF]. In [START_REF] Miebach | Schottky group actions in complex geometry, Geometric complex analysis[END_REF] Schottky group actions on the unit ball having Stein quotients are constructed. Most of these results strongly rely on Lie theory.

In this paper, we are concerned with the action of a unipotent discrete group Γ of holomorphic automorphisms of a bounded homogeneous domain. In order to state the main results, let D ⊂ C n be a bounded domain and recall that its automorphism group Aut(D) is a real Lie group that acts properly on D, see [START_REF] Cartan | Sur les groupes de transformations analytiques[END_REF]. Let G be the connected component of Aut(D) that contains the identity. The domain D is homogeneous if Aut(D) and hence G act transitively on it. Let us fix a base point p 0 ∈ D. Its isotropy group K := G p 0 is a maximal compact subgroup of G and there exists a decomposition G = KR where R is a simply-connected 2010 Mathematics Subject Classification. 32M10, 32E10. split solvable Lie group, see [START_REF] Kaneyuki | Homogeneuous bounded domains and Siegel domains[END_REF] and [START_REF] Vinberg | The Morozov-Borel theorem for real Lie groups[END_REF]. Consequently, R is isomorphic to a semi-direct product A N where N is the nilradical of R and A ∼ = (R >0 ) r .

Let D ⊂ C n be a bounded homogeneous domain and let G = KAN be the decomposition of G = Aut 0 (D) introduced above. A discrete subgroup of G will be called unipotent if it is conjugate to a subgroup of N. The research presented in this paper was motivated by the following result.

Theorem 1.1. Let D ⊂ C n be a bounded homogeneous domain and let Γ be a unipotent discrete group of automorphisms of D. Then the complex manifold D/Γ is holomorphically separable.

It is therefore natural to ask under which additional conditions on Γ the quotient manifold D/Γ is Stein. Suppose from now on that Γ is a discrete subgroup of N. It is well known that the simply-connected nilpotent group N admits a unique structure as a real-algebraic group such that its Zariski closed subgroups are precisely its connected Lie subgroups, see [START_REF]Lie groups and Lie algebras[END_REF]Chapter 2.4.2]. Hence, consider the Zariski closure N Γ of Γ in N. Then N Γ is a simplyconnected nilpotent Lie group such that N Γ /Γ is compact.

We have the following necessary condition for D/Γ to be Stein. Note that to the best of my knowledge it is not known whether this condition is also sufficient. Proposition 1.2. Let Γ ⊂ N be a discrete subgroup and consider its Zariski closure N Γ . If D/Γ is Stein, then all N Γ -orbits in D are totally real.

Since R ∼ = A N acts simply transitively on D, we can identify its Lie algebra r = a ⊕ n with T p 0 D = C n and thus obtain a complex structure j ∈ End(r). Let N Γ be the Zariski closure of a discrete subgroup Γ ⊂ N and let n Γ be its Lie algebra. The orbit N Γ • p 0 is totally real if and only if n Γ ∩ j(n Γ ) = {0}, in which case we call n Γ a totally real subalgebra of n. If n Γ is totally real and if the real dimension of n Γ coincides with the complex dimension of D, then we say that n Γ is a maximal totally real subalgebra of n. We have the following sufficient criterion for D/Γ to be Stein. Proposition 1.3. Let Γ ⊂ N be a discrete subgroup and consider its Zariski closure N Γ . If n Γ is contained in a maximal totally real subalgebra of n, then D/Γ is Stein.

The methods and results described above allow us to prove the main result of this paper, which in particular answers the question raised in [START_REF] Chen | Discrete groups and holomorphic functions[END_REF]Remark 7.6] in the negative.

Theorem 1.4. Let D be the unit ball or the Lie ball and let Γ be a unipotent discrete group of automorphisms of D having Zariski closure N Γ ⊂ N. Then D/Γ is Stein if and only if n Γ is totally real.

The proof of Theorem 1.4 relies in part on the fact that for the unit ball and the Lie ball the necessary condition given in Proposition 1.2 is indeed sufficient as well.

Let us outline the structure of this paper. In Section 2 we review some parts of the structure theory of bounded homogeneous domains and prove Theorem 1.1 as well as Propositions 1.2 and 1.3. Sections 3 and 4 contain the proves of Theorem 1.4 for the unit ball and the Lie ball, respectively. In the last section 5 we present an example that shows that Theorem 1.4 does not hold true for arbitrary bounded homogeneous domains.

Acknowledgments. I would like to thank Karl Oeljeklaus for helpful discussions. Financial support by the ANR-project QuaSiDy (ANR-21-CE40-0016-01) is gratefully acknowledged.

RESULTS FOR ARBITRARY BOUNDED HOMOGENEOUS DOMAINS

We begin by reviewing the structure of the automorphism group of a bounded homogeneous domain, its unbounded realization as a Siegel domain, and its associated normal j-algebra. As main references we refer the reader to [START_REF] Kaneyuki | Homogeneuous bounded domains and Siegel domains[END_REF] and [START_REF] Pyateskii-Shapiro | Automorphic functions and the geometry of classical domains, Translated from the Russian[END_REF]. We then show that the quotient of a bounded homogeneous domain by a unipotent discrete group Γ of automorphisms is holomorphically separable, see Theorem 1.1, and prove Propositions 1.2 and 1.3.

2.1.

Bounded homogeneous domains and normal j-algebras. Let D ⊂ C n be a bounded homogeneous domain with base point p 0 ∈ D and consider the decomposition G = KR where G = Aut 0 (D) and K = G p 0 . The elements of the Lie algebra g will be viewed as complete holomorphic vector fields on D. Since the group R is split solvable, we have R ∼ = A N where N is the nilradical of R. Moreover, the adjoint representation of the Abelian group A on n is diagonalizable over R, compare [START_REF] Kaneyuki | Homogeneuous bounded domains and Siegel domains[END_REF]Proposition 2.8].

It is well known that every bounded homogeneous domain D ⊂ C n is biholomorphic to a Siegel domain of the second kind D ⊂ C n , see [START_REF] Vinberg | Classification and canonical realization of complex homogeneous bounded domains[END_REF].

Example. The unit ball B n ⊂ C n is biholomorphically equivalent to

B n = (z, w) ∈ C × C n-1 ; Im(z) -w 2 > 0 , the Lie ball L n ⊂ C n is biholomorphic to L n = z ∈ C n ; Im(z n ) 2 -Im(z 1 ) 2 + • • • + Im(z n-1 ) 2 > 0, Im(z n ) > 0 ,
and the Siegel disk S n ⊂ Sym(n, C) may be realized as Siegel's upper half-plane

S n = Z ∈ Sym(n, C); Im(Z) is positive definite .
In Helgason's notation, see [16, Table V, Chapter X], the unit ball, the Lie ball and the Siegel disk correspond to the Hermitian symmetric spaces of types AI I I(p = 1, q = n), BDI(p = 2, q = n) and CI, respectively.

The split-solvable group R ∼ = A N acts by affine transformations on D. In particular, this R-action extends to the whole of C n . It follows from the explicit realization of its affine automorphism group, see e.g. [20, Chapters 2 and 3], that N and hence N C act algebraically by affine transformations on C n and that every transformation in N C has Jacobi determinant equal to 1.

Since R acts simply transitively on D, we may identify its Lie algebra r = a ⊕ n with T p 0 D = C n and thus obtain an integrable complex structure j ∈ End(r).

In order to show that (r, j) is a normal j-algebra we need to prove the existence of a linear form λ ∈ r * such that for all x, y ∈ r we have λ j(x), j(y) = λ[x, y] and λ j(x), x > 0 if x = 0, see part (III) of the definition in [30, p. 51].

In [START_REF] Kaneyuki | Homogeneuous bounded domains and Siegel domains[END_REF] and [START_REF] Pyateskii-Shapiro | Automorphic functions and the geometry of classical domains, Translated from the Russian[END_REF] existence of this linear form λ ∈ r * is deduced from [START_REF] Koszul | Sur la forme hermitienne canonique des espaces homogènes complexes[END_REF]. Let us explain here how λ can be constructed from the Bergman metric of D via a moment map. (Recall that the Bergman metric of a bounded domain D ⊂ C n is an Aut(D)-invariant Kähler metric on D. For its definition we refer the reader to [START_REF] Kobayashi | Hyperbolic complex spaces[END_REF]Chapter 4.10].) It is shown in [START_REF] Ishi | On symplectic representations of normal j-algebras and their application to Xu's realizations of Siegel domains[END_REF] that there exists an R-equivariant holomorphic embedding of D into Siegel's upper half-plane S n . Since the domain S n is symmetric, the group Aut 0 (S n ) is semisimple and thus there exists an equivariant moment map for the action of Aut 0 (S n ) on S n . Pulling back this moment map to D we obtain an R-equivariant moment map µ : D → r * with respect to the Bergman metric ω of D. Note that µ is a diffeomorphism onto its image, an open coadjoint orbit. Moreover, µ is a symplectomorphism with respect to the Kostant-Kirillov form on this coadjoint orbit, i.e., we have (2.1)

ω p 0 x(p 0 ), y(p 0 ) = µ(p 0 )[x, y]
for all x, y ∈ r, see [START_REF] Guillemin | Symplectic techniques in physics[END_REF]Equations (26.2) and (26.8)]. As a consequence, we can take λ := µ(p 0 ) ∈ r * in the definition of the normal j-algebra associated with D. For the structure theory of normal j-algebras we refer the reader to [START_REF] Pyateskii-Shapiro | Automorphic functions and the geometry of classical domains, Translated from the Russian[END_REF]Chapter 2].

Remark. For a connected closed subgroup R of R the orbit R • p 0 is totally real if and only if r ∩ j(r ) = {0}. In this case, we refer to r as a totally real subalgebra of r. Note that every isotropic orbit (with respect to the Bergman metric) is totally real. Moreover, every orbit of an Abelian subgroup of R is totally real since µ(p 0 ) j(x), x > 0 holds for x = 0. This last statement is also a consequence of the fact that D is Kobayashi-hyperbolic, since an Abelian group of automorphisms that is not totally real would yield a non-constant holomorphic map from C to D.

Conversely, every abstract normal j-algebra is the normal j-algebra associated with some bounded homogeneous domain, see [START_REF] Pyateskii-Shapiro | Automorphic functions and the geometry of classical domains, Translated from the Russian[END_REF]Appendix].

2.2.

Quotients by unipotent discrete groups are holomorphically separable. Let D ⊂ C n be a bounded homogeneous domain and let Γ be a unipotent subgroup of G = Aut 0 (D).

Proof of Theorem 1.1. Any unipotent discrete subgroup Γ ⊂ G acts freely and properly on D × C by γ • (z, t) := γ(z), t det dγ(z) -1 , so that we have the quotient manifold

L := D × Γ C := (D × C)/Γ. Moreover, the projection onto the first factor D × C → D is Γ-equivariant and the induced map L = D × Γ C → D/Γ defines a holomorphic line bundle on D/Γ. A holomorphic section in L corresponds to a Γ-equivariant holomorphic function from D to D × C.
Holomorphic sections in the k-fold tensor product L ⊗k can be constructed via Poincaré series for k 2 and separate the points of D/Γ in the following sense. For every pair of elements p, q ∈ D/Γ with p = q there exist k 2 (depending on p and q) and a holomorphic section s in L ⊗k such that s(p) = 0 and s(q) = 0, see [START_REF] Pyateskii-Shapiro | Automorphic functions and the geometry of classical domains, Translated from the Russian[END_REF]Lemma 3.4.1]. We will finish the proof by showing that for a unipotent discrete subgroup Γ the line bundle L admits a non-vanishing holomorphic section. It then follows that L, and thus all of its powers, are holomorphically trivial.

In order to do so, let ϕ : D → D be a Γ-equivariant biholomorphic map to a Siegel domain of the second kind on which Γ acts by affine transformations having Jacobi determinant 1. In other words, we have ϕ • γ = γ • ϕ where det d γ(z) = 1 for all z ∈ D. Then, the chain rule implies

det dϕ γ(z) = det dγ(z) -1 • det d(ϕ • γ)(z) = det dγ(z) -1 • det d( γ • ϕ)(z) = det dγ(z) -1 • det dϕ(z).
Consequently, the holomorphic map s : D → D × C * given by s(z, t) = z, det dϕ(z) is Γ-equivariant and thus defines a non-vanishing holomorphic section in L, as desired.

A necessary condition for Steinness.

In this subsection we prove Proposition 1.2.

An important ingredient for the proof is the following result of Loeb. Let Γ ⊂ G be a discrete group of unipotent automorphisms of D. After conjugation we may suppose that Γ is contained in N. Let N Γ be the real Zariski closure of Γ in N and let N C Γ be its universal complexification. Then the complex homogeneous space N C Γ /Γ is Stein, see [START_REF] Gilligan | On non-compact complex nil-manifolds[END_REF] and [23, Théorème 1].

Recall that the complexification N C is a unipotent complex algebraic group and its action on the Siegel domain D extends to an algebraic action on C n . In particular, the orbits of any algebraic subgroup of

N C are closed in C n , see [2, Proposition 4.10]. Lemma 2.1. The orbit N Γ • z is totally real in D if and only if the isotropy group (N C Γ ) z is trivial. Proof. For this, let x + iy ∈ n Γ ⊕ in Γ = n C
Γ be a holomorphic vector field on D and consider x(z) + j z y(z) ∈ T z D where j z is the complex structure of T z D, for some z ∈ D. Suppose that N Γ • z is totally real. Then x(z) + j z y(z) = 0 implies x(z) = y(z) = 0, and since N Γ acts freely on D, we obtain x = y = 0. Consequently, the isotropy (N C Γ ) z is discrete. Since N C Γ acts algebraically and has no finite subgroups, we conclude that (N C Γ ) z is trivial. Conversely, if N Γ • z is not totally real for some z ∈ D, then there are x, y ∈ n Γ such that x(z) = j z y(z). Thus the vector field xiy ∈ n C Γ vanishes at z, i.e., N C Γ does not act freely. We are now in position to prove the Proposition 1.2.

Remark. Under the hypotheses of Proposition 1.3, N

C Γ ∼ = N C Γ • D = C n is the universal glo- balization of the induced local N C
Γ -action on D in the sense of [START_REF] Palais | A global formulation of the Lie theory of transformation groups[END_REF], see the proof of Proposition 1.3.

THE CASE OF THE UNIT BALL

In this section we consider the unit ball B n ⊂ C n . Firstly, we illustrate Theorem 1.4 by two examples that can be analyzed by ad hoc methods. Then we review the structure of the normal j-algebra b n of B n , which will be used to show that every totally real subalgebra of b n is contained in a maximal totally real one.

Two examples.

We identify the unit ball B n with its unbounded realization

B n = (z, w) ∈ C × C n-1 ; Im(z) -w 2 > 0 .
For an explicit description of the vector fields belonging to its normal j-algebra b n as well as of the corresponding one-parameter groups we refer the reader to [24, First, let us present an example of a nonabelian discrete group Γ such that B 3 /Γ is Stein, thus answering the question raised in [6, Remark 7.6] in the negative. Note that 3 is the smallest dimension so that a similar example can be constructed.

Example. Let us consider the complete holomorphic vector fields

x 1 := 2iw 1 ∂ ∂z + ∂ ∂w 1 , x 2 := 2(w 1 + w 2 ) ∂ ∂z + i ∂ ∂w 1 + i ∂ ∂w 2
, and x 3 := ∂ ∂z on B 3 . Since their only non-vanishing Lie bracket is [x 1 , x 2 ] = 4x 3 , they generate a three-dimensional subalgebra in the nilradical of b 3 , isomorphic to the three-dimensional Heisenberg algebra. Moreover, this algebra is defined over Q and therefore the corresponding connected subgroup admits a cocompact discrete subgroup Γ, which justifies the notation N Γ = exp(Rx 1 ⊕ Rx 2 ⊕ Rx 3 ). One verifies directly that every N Γ -orbit in B 3 is maximally totally real. This implies that the universal globalization of the local N C Γ -action on B 3 is isomorphic to N C Γ and the N Γ -action on B 3 corresponds to left multiplication on N C Γ . Since N C Γ /Γ is a Stein manifold which contains B 3 /Γ as a domain, we see that B 3 /Γ is Stein due to [START_REF] Docquier | Levisches Problem und Rungescher Satz f ür Teilgebiete Steinscher Mannigfaltigkeiten[END_REF] while Γ is not Abelian.

Secondly, we present an explicit example of a unipotent discrete group Γ where B 2 /Γ is non-Stein but admits a (singular) Stein envelope.

Example. Let Γ be the discrete group consisting of the automorphisms

(z, w) → z + 2(n + im)w + i(m 2 + n 2 ) + 2k, w + m + in =: ϕ 2k,m+in (z, w),
where m, n, k ∈ Z and (z, w) ∈ B 2 . Although we do not need this fact, let us remark that the Lie algebra n Γ of the Zariski closure of Γ coincides with the nilradical of b 2 .

In order to determine the quotient B 2 /Γ, we first consider the action of the normal subgroup Γ 0 := {ϕ 2k,0 ; k ∈ Z} Γ. The map p : B 2 → C * × C given by p(z, w) = (e iπz , w) is Γ 0 -invariant and yields

B 2 /Γ 0 ∼ = p( B 2 ) = (z, w) ∈ C * × C; |z| < e -π|w| 2 . The induced action of Z ⊕ iZ ∼ = Γ/Γ 0 on p( B 2 ) is given by (m + in) • (z, w) = e iπ(2(n+im)w+i(m 2 +n 2 )) z, w + m + in .
It follows that the action of Z ⊕ iZ on p( B 2 ) extends to a proper action on the whole of C 2 and that the equivariant map C × C → C, (z, w) → w, induces a holomorphic line bundle

L := C 2 /(Z ⊕ iZ) → E = C/(Z ⊕ iZ). We see that B 2 /Γ ∼ = p( B 2 )/(Z ⊕ iZ) embeds into L
as an open neighborhood of the zero section minus this zero section. Note that B 2 /Γ is Kobayashi-hyperbolic since B 2 is so, see [START_REF] Kobayashi | Hyperbolic complex spaces[END_REF]Theorem 3.2.8(2)]. Since ∂ B 2 is strictly pseudoconvex, the zero-section in L has a strictly pseudoconvex neighborhood and hence is negative in the sense of Grauert, see [START_REF] Grauert | Über Modifikationen und exzeptionelle analytische Mengen[END_REF]Satz 1]. It follows that the zero section of L can be blown down to yield a Stein space Y with an isolated singularity containing B 2 /Γ as a neighborhood of this singularity minus the singularity. Consequently, B 2 /Γ is holomorphically separable but not Stein.

3.2. The normal j-algebra of the unit ball. The automorphism group of the unit ball B n is G = Aut(B n ) ∼ = PSU(n, 1). Let G = KAN be an Iwasawa decomposition with maximal compact subgroup K = G p 0 ∼ = U(n) for p 0 = 0. It is well known that G is of real rank 1, i.e., that dim A = 1. Moreover, under the identification T p 0 B n ∼ = a ⊕ n we can rewrite the moment map condition (2.1) as

(3.1) [x, y](p 0 ) = ω p 0 x(p 0 ), y(p 0 ) ζ(p 0 )
for all x, y ∈ n, compare [30, p. 52]. This means that n is the Heisenberg algebra of dimension 2n -1 with center Rζ defined by the symplectic form ω p 0 induced by the Bergman metric of B n . As we shall see in the following subsection, it is this close relation between the geometry of (B n , ω) and the structure of N that enables us to prove Theorem 1.4 for the unit ball.

Recall from [24, Section 4.2] that the normal j-algebra b n = a ⊕ n can be written as

a = Rα and n = n-1 k=1 Rξ k ⊕ n-1 k=1 Rξ k ⊕ Rζ,
where the only non-zero Lie brackets are Proposition 3.1. Every totally real subalgebra n of n is contained in a maximal totally real subalgebra n of n. Moreover, if n is Abelian, then n can also be chosen to be Abelian.

[ξ k , ξ k ] = ζ, [α, ξ k ] = -ξ k , [α, ξ k ] = -ξ k ,
Proof. Let Φ : n → n • p 0 be the linear isomorphism given by Φ(x) := x(p 0 ). Let V ⊂ n • p 0 be a real vector subspace. The following is a direct consequence of equation (3.1).

(a) The preimage Φ -1 (V) is an Abelian subalgebra of n if and only if V is isotropic with respect to ω p 0 . (b) The preimage Φ -1 (V) is a nonabelian subalgebra of n if and only if V is not isotropic and contains ζ(p 0 ). Now firstly suppose that n is Abelian. Then n • p 0 is not only totally real but isotropic, and by basic symplectic linear algebra there exists a Lagrangian subspace V of n • p 0 that contains n • p 0 . As we have noted above, the preimage n := Φ -1 (V) is an Abelian subalgebra of maximal dimension of n that contains the totally real subalgebra n .

In the case that n is not Abelian, we choose any maximally totally real subspace V of n • p 0 that contains n • p 0 . Since n • p 0 contains ζ(p 0 ), the same holds for V. Hence, it follows that n := Φ -1 (V) has the required properties.

THE CASE OF THE LIE BALL

In this section we first describe the structure of the normal j-algebra l n of the Lie ball L n in order to establish in particular the existence of a holomorphic submersion from L n onto the unit disk B 1 whose fibers are biholomorphic to B n-1 . It turns out that this submersion is crucial for the proof of Theorem 1.4 since the analogon of Proposition 3.1 does not hold true for l n .

4.1. The normal j-algebra of the Lie ball. The n-dimensional Lie ball is biholomorphically equivalent to the tube domain over the symmetric cone

Ω := y ∈ R n ; y 2 n -(y 2 1 + • • • + y 2 n-1 ) > 0, y n > 0 .
For n = 1 this is just the upper half-plane H + , while for n = 2 we get H + × H + . Therefore, we will concentrate on the case n 3.

In the notation of [16, Table V, Chapter X] the Lie ball L n corresponds to the item BD I with p = 2 and q = n. The connected component of its automorphism group is isomorphic to G = SO 0 (2, n) and the subgroup of affine automorphisms is

G(Ω) R n with G(Ω) = R >0 SO 0 (1, n -1), see [10, Chapter X.5].
Remark. Since the group G(Ω) acts by matrix multiplication on C n , the vector field corresponding to an element x of the Lie algebra of G(Ω) can be obtained by computing

d dt 0 exp(tx)z for z ∈ C n .
In view of [START_REF] Kaneyuki | Homogeneuous bounded domains and Siegel domains[END_REF]Proposition 2.8], in order to describe the normal j-algebra l n = a ⊕ n of the Lie ball, it is sufficient to find a maximal triangular subalgebra of so(1, n -1). By doing so, we can choose the basis of n given by the vector fields

ξ k = ∂ ∂z k (1 k n -2), ξ k = (z n -z n-1 ) ∂ ∂z k + z k ∂ ∂z n-1 + z k ∂ ∂z n (1 k n -2), ζ = ∂ ∂z n-1 + ∂ ∂z n , η = ∂ ∂z n - ∂ ∂z n-1 .
The only non-vanishing Lie brackets are

[ξ k , ξ k ] = ζ and [η, ξ k ] = 2ξ k for all 1 k n -2.
In particular, we see that Rζ is the center of n and that the Abelian Lie algebra n = n-2 k=1 Rξ k ⊕ Rζ is its derived algebra. A basis of a is given by

δ := z 1 ∂ ∂z 1 + • • • + z n ∂ ∂z n
(the Euler field), and

α := z n ∂ ∂z n-1 + z n-1 ∂ ∂z n .
The action of a on n is determined by

[δ, ξ k ] = -ξ k , [δ, ζ] = -ζ, [δ, η] = -η for all 1 k n -2 and by [α, ξ k ] = -ξ k , [α, ζ] = -ζ, [α, η] = η for all 1 k n -2.
To obtain the complex structure j : l n → l n , let us fix the base point p 0 = ie n in L n = R n + iΩ. This yields

j(ξ k ) = ξ k , j(ζ) = α + δ =: α 1 , j(η) = δ -α =: α 2 for all 1 k n -2.
4.2. The equivariant fibration of L n . It follows directly from the bracket relations described above that the subalgebra

b n-1 := Rα 1 ⊕ n-2 k=1 Rξ k ⊕ n-2 k=1
Rξ k ⊕ Rζ is a j-invariant ideal in l n isomorphic to the normal j-algebra of the (n -1)-dimensional unit ball. Moreover, the quotient b 1 ∼ = l n /b n-1 is isomorphic to the j-subalgebra Rα 2 ⊕ Rη of l n . In other words, l n decomposes into the semi-direct product

l n = b n-1 ⊕ b 1 . Let L n ∼ = B n-1 B 1
be the corresponding decomposition at the group level.

Geometrically, this corresponds to the L n -equivariant holomorphic submersion π : L n → C given by π(z) = z nz n-1 . In fact, an elementary argument shows that for z ∈ L n we have z nz n-1 ∈ H + and z n + z n-1 ∈ H + . Moreover, for a := z nz n-1 ∈ H + and w := z n + z n-1 ∈ H + , we obtain

π -1 (a) ∼ = (z 1 , . . . , z n-2 , w) ∈ C n-1 Im(w) > Im(z 1 ) 2 + • • • + Im(z n-2 ) 2 Im(a) ,
which is a realization of the (n -1)-dimensional unit ball as an unbounded tube domain. Hence, all π-fibers are isomorphic to B n-1 .

Remark. Since the Lie ball L n is irreducible for n 3, the holomorphic submersion π is not holomorphically locally trivial unless n = 2, as follows from [START_REF] Royden | Holomorphic fiber bundles with hyperbolic fiber[END_REF].

Proof of Theorem 1.4 for the Lie ball.

Let Γ be a discrete subgroup of Aut 0 (L n ) such that Γ ⊂ N. Let N Γ be the Zariski closure of Γ in N and suppose that n Γ is totally real, i.e., that n Γ ∩ j(n Γ ) = {0}.

We start by giving an example that shows that Proposition 3.1 does no longer hold true for the Lie ball.

Example. Let us consider the 6-dimensional normal j-algebra l 3 = a ⊕ n of the 3-dimensional Lie ball. We claim that the totally real subalgebra Rξ 1 ⊂ n is not contained in a maximal totally real subalgebra of n.

Since b 2 l 3 is 4-dimensional and not totally real, a maximal totally real subalgebra of n must contain an element x = aξ 1 + bξ 1 + cζ + dη with a, b, c, d ∈ R and d = 0. Thus it also must contain [x, ξ 1 ] = aζ + 2dξ 1 and [x, ξ 1 ], ξ 1 = 2dζ. Therefore it contains ζ and ξ 1 = -j(ξ 1 ) as well, which is impossible.

The following proposition is weaker than Proposition 3.1 but will still allow us to prove Theorem 1.4 for the Lie ball. For its statement and proof we decompose the nilradical n of l n as n = n n-1 ⊕ Rη where the ideal n n-1 := b n-1 ∩ n is isomorphic to the nilradical of b n-1 . Proposition 4.1. Let n be a totally real subalgebra of the nilradical n = n n-1 ⊕ Rη. If n is not contained in n n-1 , then there exists a maximal totally real subalgebra n of n which contains n .

Proof. Since by assumption n is not contained in n n-1 , there is an element of the form

x 0 = ∑ a k ξ k + ∑ b l ξ l + cζ + η which belongs to n . Moreover, we have dim(n ∩ n n-1 ) = dim n -1.
As a first step, we claim that n is totally real if and only if n ∩ n n-1 is so. In order to see this, suppose that n ∩ n n-1 is totally real, the other implication being trivial. Choose an element x ∈ n ∩ j(n ) and decompose it as x = x 1 + tx 0 where x 1 ∈ n ∩ n n-1 and t ∈ R.

By assumption, we have j(x) = j(x 1 ) + tj(x 0 ) ∈ n . Then, from j(η) = α 2 we deduce t = 0, hence x = x 1 ∈ n ∩ n n-1 . Since the latter algebra is assumed to be totally real we obtain x 0, as was to be shown.

Next, observe that if ζ / ∈ n , then n ⊕ Rζ is again a totally real subalgebra, for the following reason. If ζ / ∈ n , then n ∩ n n-1 is Abelian. Therefore, the Lie algebra (n ∩ n n-1 ) ⊕ Rζ is also Abelian, hence totally real, which due to the previous step implies that n ⊕ Rζ is totally real.

Consequently, we may assume that ζ ∈ n . Let us consider the projection π :

n n-1 → n n-1 onto k Rξ k ⊕ Rζ with kernel l Rξ l . If the restriction π| n ∩n n-1 : n ∩ n n-1 → k Rξ k ⊕ Rζ is surjective, then dim(n ∩ n n-1 ) n -1, hence dim n = n,
and we are done.

Therefore, suppose that ξ k 0 / ∈ π(n ∩ n n-1 ) for some k 0 . Note that in particular

ξ k 0 / ∈ n . From ζ ∈ n it follows that (n ∩ n n-1 ) ⊕ Rξ k 0 is a subalgebra of n ⊕ n n-1 .
Since moreover, η and ξ k 0 commute, we see that n ⊕ Rξ k 0 is a subalgebra of n. We claim that n ⊕ Rξ k 0 is totally real. In order to see this it is sufficient to show that (n ∩ n n-1 ) ⊕ Rξ k 0 is totally real. This, however, follows from the fact that x 0 , j(ξ k 0 ) has a non-zero contribution from ξ k 0 .

Iterating this procedure we eventually construct a maximal totally real subalgebra of n that contains n , as was to be shown.

Since N C

Γ /Γ is Stein due to [23, Théorème 1], the same is true for N C Γ • L n /Γ. Hence, the theorem of Docquier-Grauert [START_REF] Docquier | Levisches Problem und Rungescher Satz f ür Teilgebiete Steinscher Mannigfaltigkeiten[END_REF] implies that L n /Γ is Stein as well.

THE CASE OF THE SIEGEL DISK

In this section we present an example that shows that the analoga of Theorem 1.4 and Proposition 4.1 do not hold true for arbitrary bounded homogeneous domains. This example will be constructed on a 5-dimensional bounded homogeneous domain holomorphically embedded in the 6-dimensional Siegel disk S 3 . Therefore we describe as a first step the normal j-algebra s n of S n .

5.1. The normal j-algebra of Siegel's upper half-plane. Recall that the Siegel disk S n can be realized as Siegel's upper half plane which is the symmetric tube domain associated with the cone of positive definite real symmetric matrices. The automorphism group of S n is isomorphic to the symplectic group Sp(n, R) and S n corresponds to the hermitian symmetric space of type C I in [START_REF] Helgason | Differential geometry, Lie groups, and symmetric spaces[END_REF]. For n = 1, Siegel's upper half-plane is the usual upper half-plane, while, for n = 2, it is isomorphic to the 3-dimensional Lie ball.

The linear automorphism group of the symmetric cone of positive definite real symmetric matrices is GL(n, R) acting by g • A := gAg t , see [10, p. 213]. Hence, the normal j-algebra s n of S n can be determined as before using [10, Proposition X.5.4] and [START_REF] Kaneyuki | Homogeneuous bounded domains and Siegel domains[END_REF]Proposition 2.8]. Our description of s n follows [START_REF] Ishi | On symplectic representations of normal j-algebras and their application to Xu's realizations of Siegel domains[END_REF].

Let us denote by u n the solvable Lie algebra of lower triangular real n × n matrices. We have the decomposition u n = a n ⊕ u n where a n is the Abelian Lie algebra of diagonal matri- ces in R n×n . The normal j-algebra of Siegel's upper half plane can be realized as

s n := A B 0 -A t A ∈ u n , B ∈ Sym(n, R) . Note that ϕ : u n → Sym(n, R), ϕ(A) = A + A t
, is a linear isomorphism. The complex structure j : s n → s n is given by

j A B 0 -A t = ϕ -1 (B) -ϕ(A) 0 -ϕ -1 (B) t .
The linear form λ ∈ s * n corresponding to the Bergman metric is given by

λ A B 0 -A t = Tr(B).
The elements of the solvable Lie group S n are of the form A B

0 (A t ) -1
where A is lower triangular and B is symmetric. An element of the form A 0 0 (A t ) -1 acts on S n by Z → AZA t while the elements of the form I n B 0 I n act by translation. The vector field induced by x ∈ s n is given by

d dt 0 exp(tx) • Z for Z ∈ S n .
One verifies directly that the subspace of s n consisting of all matrices A = (a kl ) and B = (b kl ) such that a kl = b kl = 0 for all 1 k, l n -1 is a j-invariant ideal isomorphic to b n , while the subspace consisting of all matrices where a nl = b nl = 0 for all l is a jinvariant complementary subalgebra isomorphic to s n-1 . Consequently, s n decomposes as a semi-direct sum s n-1 ⊕ b n . Geometrically, this decomposition corresponds to the equivariant fibration π n : S n → S n-1 where π n (Z) is the submatrix of Z consisting of the first n -1 lines and columns.

In the following we will concentrate on the case n = 3. Here we have

s 3 = b 3 ⊕ b 2 ⊕ b 1 where b 3 is a j-ideal in s 3 and b 2 is a j-ideal in s 2 = b 2 ⊕ b 1 .
The composition of the two equivariant fibrations is the map π := π 2 • π 3 : S 3 → H + given by π(Z) = z 11 . For the rest of this section let D := π -1 (i) be the 5-dimensional bounded homogeneous (nonsymmetric) domain corresponding to the normal j-algebra b 3 ⊕ b 2 . Let us consider the bases (α 3 , ξ 31 , ξ 32 , ξ 31 , ξ 32 , ζ 3 ) of b 3 and (α 2 , ξ 21 , ξ 21 , ζ 2 ) of b 2 . Let us realize these elements explicitly as matrices as well as vector fields in the coordinates of the entries of Z = (z kl ) ∈ Sym(3, C): [START_REF]Lie groups and Lie algebras[END_REF] , [START_REF]Lie groups and Lie algebras[END_REF] , [START_REF]Lie groups and Lie algebras[END_REF] , [START_REF] Loeb | Action d'une forme réelle d'un groupe de Lie complexe sur les fonctions plurisousharmoniques[END_REF] , [START_REF] Loeb | Action d'une forme réelle d'un groupe de Lie complexe sur les fonctions plurisousharmoniques[END_REF] .

ζ 3 = -2 0 E 33 0 0 → -2 ∂ ∂z 33 , α 3 = E 33 0 0 -E 33 → z 13 ∂ ∂z 13 + z 23 ∂ ∂z 23 + 2z 33 ∂ ∂z
ξ 31 = 0 E 13 + E 31 0 0 → ∂ ∂z 13 , ξ 31 = E 31 0 0 -E 13 → z 11 ∂ ∂z 13 + z 12 ∂ ∂z 23 + 2z 13 ∂ ∂z
ξ 32 = 0 E 23 + E 32 0 0 → ∂ ∂z 23 , ξ 32 = E 32 0 0 -E 23 → z 12 ∂ ∂z 13 + z 22 ∂ ∂z 23 + 2z 23 ∂ ∂z
ζ 2 = -2 0 E 22 0 0 → -2 ∂ ∂z 22 , α 2 = E 22 0 0 -E 22 → z 12 ∂ ∂z 12 + 2z 22 ∂ ∂z 22 + z 23 ∂ ∂z
ξ 21 = 0 E 12 + E 21 0 0 → ∂ ∂z 12 , ξ 21 = E 21 0 0 -E 12 → z 11 ∂ ∂z 12 + 2z 12 ∂ ∂z 22 + z 13 ∂ ∂z
For instance, in order to find the vector field belonging to ξ 31 we calculate exp(tξ 31 ) all other brackets being zero.

• Z =   1 0 0 0 1 0 t 0 1   Z   1 0 t 0 1 0 0 0 1   =  

A counterexample.

We present an example that shows that the analoga of Proposition 4.1 and Theorem 1.4 do not hold true for D.

Let n denote the nilradical of b 3 ⊕ b 2 and consider the elements x 1 , x 2 , x 3 ∈ n given by

x 1 := ξ 31 + ζ 3 + ξ 21 , x 2 := -ξ 31 + ξ 21 , x 3 := ζ 3 + ζ 2 .
Since [x 1 , x 2 ] = x 3 , they generate a Lie subalgebra n Γ := Rx 1 ⊕ Rx 2 ⊕ Rx 3 that is isomorphic to the 3-dimensional Heisenberg algebra and projects surjectively onto the nilradical of b 2 . Moreover, the corresponding group N Γ of automorphisms of D admits a cocompact discrete subgroup, which justifies the notation n Γ . We shall see that all N Γ -orbits in D are totally real, while n Γ is not contained in any maximal totally real subalgebra of n. This shows that the analogon of Proposition 4.1 does not hold true for the domain D, even under the stronger assumption that all N Γ -orbits are totally real in D.

Remark. A subalgebra of codimension 1 in a nilpotent Lie algebra is automatically an ideal. We apply this result in the following way: If y ∈ n is an element such that n Γ ⊕ Ry is a subalgebra, then y normalizes n Γ .

Lemma 5.1. The Lie algebra n Γ is not contained in a maximal totally real subalgebra of n. Hence, the analogon of Proposition 4.1 does not hold true for the domain D.

Proof. Suppose for a moment that n Γ is a maximal totally real subalgebra of n with n Γ ⊂ n Γ and let y ∈ n Γ \ n Γ . Writing

y = a 1 ξ 31 + a 2 ξ 32 + a 1 ξ 31 + a 2 ξ 32 + c 3 ζ 3 + b 1 ξ 21 + b 1 ξ 21 + c 2 ζ 2 we find [y, x 1 ] = a 2 ξ 31 + (a 1 -b 1 )ξ 32 + a 1 ζ 3 -b 1 ζ 2 and [y, x 1 ], x 1 = a 2 ζ 3 .
If a 2 = 0, then n Γ would contain ζ 3 and ζ 2 and consequently ξ 31 + ξ 21 = -j(x 2 ) as well, thus contradicting the fact that n Γ is totally real. Hence, we must have a 2 = 0.

It follows that the elements

[y, x 1 ] = (a 1 -b 1 )ξ 32 + a 1 ζ 3 -b 1 ζ 2 and
[y,

x 2 ] = -(a 1 + b 1 )ξ 32 + a 1 ζ 3 + b 1 ζ 2
belong to n Γ . Consequently, n Γ contains also the element

(a 1 + b 1 )[y, x 1 ] + (a 1 -b 1 )[y, x 2 ] = (a 2 1 -a 1 b 1 + a 1 b 1 + a 2 1 )ζ 3 -(b 2 1 -a 1 b 1 + a 1 b 1 + b 2 1 )ζ 2 .
As above, since n Γ is totally real, neither ζ 3 nor ζ 2 belong to n Γ . This gives

a 2 1 -a 1 b 1 + a 1 b 1 + a 2 1 = -b 2 1 + a 1 b 1 -a 1 b 1 -b 2 1 , which is equivalent to (a 1 + b 1 ) 2 + (a 1 -b 1 ) 2 = 0.
Hence, we obtain a 1 = -b 1 and a 1 = b 1 . In summary, this yields

y = a 1 (ξ 31 -ξ 21 ) + a 2 ξ 32 + a 1 (ξ 31 + ξ 21 ) + c 3 ζ 3 + c 2 ζ 2 .
In particular y normalizes n Γ . Adding a 1 x 2a 1 x 1c 2 x 3 to y, we conclude that n Γ \ n Γ contains an element of the form (5.1) y τ := ξ 32 + τζ 3 for some τ ∈ R. Observe that y τ centralizes n Γ and that n Γ ⊕ Ry τ is a totally real subalgebra of n. According to the above remark, n Γ ⊕ Ry τ must be a normal subalgebra of n Γ . Therefore, we calculate its normalizer and find

N n (n Γ ⊕ Ry τ ) = n Γ ⊕ Rξ 32 ⊕ Rζ 3 = R(ξ 31 + ξ 21 ) ⊕ R(-ξ 31 + ξ 21 ) ⊕ Rξ 32 ⊕ Rζ 3 ⊕ Rζ 2 .
Since the normalizer is 5-dimensional, it must coincide with n Γ . This however contradicts our assumption because the normalizer is not totally real.

Let N Γ be the connected Lie group having Lie algebra n Γ ⊕ Ry τ where y τ is defined in Equation (5.1). Since y τ centralizes n Γ , the group N Γ admits cocompact discrete subgroups. We will show that the quotient of D with respect to any cocompact discrete subgroup of N Γ is not Stein, hence that the analogon of Theorem 1.4 does not hold for D. For this it is enough to find an N Γ -orbit in D which is not totally real. Note that this implies again that n Γ ⊕ Ry τ cannot be contained in a maximal totally real subalgebra. Indeed, for every α > (1 + τ 2 ) -2 the matrix

Z 0 =   i τ+i 1+τ 2 0 τ+i 1+τ 2 iα 0 0 0 i  
belongs to D, and one sees without difficulty that N Γ • Z 0 is not totally real.

Nevertheless, we shall prove in the rest of this subsection that D/Γ is a Stein manifold where Γ is any cocompact discrete subgroup of N Γ . Using the description of the elements of s n as matrices, we can realize N C Γ as the matrix group (5.2) Conversely, suppose that the actions of H on M and of G/H on M/H are proper. Let (g n ) and (p n ) be sequences in G and M, respectively, such that (g n • p n , p n ) converges to (q 0 , p 0 ). We must show that (g n ) has a convergent subsequence.

N C Γ =                        1 0 0 0 a -b b 1 0 a -2c -a 2 +b 2 2 a 0 1 -b a 2 +b 2 2 -2(a + c) 0 0 0 1 -b -a 0 0 0 0 1 0 0 0 0 0 0 1         a, b, c ∈ C                . The action of N C Γ on Sym(
For this, note that g n H

(h n ) in H such that h -1 n g n → g 0 in G, from which we conclude that h n • (h -1 n g n • p n ), h -1 n g n • p n → (q 0 , g 0 • p 0 )
. Since H acts properly on M, it follows that (h n ) and hence (g n ) have convergent subsequences.

We first apply Lemma 5. Note that the projection onto the first three coordinates is equivariant; therefore we obtain a free algebraic C-action on C 3 of the form t • w = w 1 , w 2 + t f (w 1 ), w 3 + tg(w 1 ) where f (w 1 ) = -(w 2 1 + 1) and g(w 1 ) = -2w 1 . Due to a result of Kaliman, see [START_REF] Kaliman | Free C + -actions on C 3 are translations[END_REF], every free algebraic C-action on C 3 is conjugate under a biregular map to a translation. Hence, it follows that C b acts properly on C 3 Remark. In our case it is possible to give an elementary proof of Kaliman's result which applies in a slightly more general setup. For this, let f , g ∈ O(C) be two entire functions and consider the holomorphic C-action on C 3 given by t • z = z 1 , z 2 + t f (z 1 ), z 3 + tg(z 1 ) . This action is free if and only if f and g have no common zeros. In this case, there exist entire functions ϕ, ψ ∈ O(C) such that ϕ f + ψg = 1, see [START_REF] Helmer | Divisibility properties of integral functions[END_REF]. Then the biholomorphic map Φ : C 3

Open problems and concluding remarks.

Let Γ ⊂ N be a discrete subgroup with Zariski closure N Γ . As we have seen, in general the condition that only one N Γ -orbit is totally real in D does not imply that D/Γ is Stein. It is natural to ask, however, whether D/Γ is Stein if all N Γ -orbits in D are totally real. In this case, we obtain a free algebraic action of N C Γ on the domain N C Γ • D ⊂ C n . As we have observed in the above examples, even if all N Γ -orbits are totally real in D, the hypothesis of Proposition 1.3 does not need to hold true. Then, in order to answer the above question, one could try a direct approach similar to the one carried out in the previous subsection. This, however, poses two major problems. First of all, one has to show that N C Γ acts properly on N C Γ • D ⊂ C n . Secondly, if N C Γ does indeed act properly, one must prove that N C Γ • D /N C Γ is a Stein manifold. These questions are far from being trivial, bearing in mind that there is an example of a proper algebraic C 2 -action on C 6 by affine-linear transformations such that C 6 /C 2 is quasi-affine but not Stein, see [START_REF] Winkelmann | On free holomorphic C-actions on C n and homogeneous Stein manifolds[END_REF].

3 . 3 .

 33 and [α, ζ] = -2ζ for all 1 k n -1, and where the complex structure j : b n → b n is given by j(ζ) = α and j(ξ k ) = ξ k for all 1 k n -1. Proof of Theorem 1.4 for the unit ball. Let b n = a ⊕ n be the normal j-algebra of B n .In this subsection we are going to show that every totally real subalgebra of n is contained in a maximal totally real subalgebra, which generalizes[START_REF] Miebach | Quotients of bounded homogeneous domains by cyclic groups[END_REF] Lemma 4.1]. Consequently, we can apply Proposition 1.3 in order to prove Theorem 1.4 for B n .

 z 11 z 12 z 13 z 12 z 22 z 23 z 13 z 23 z 33 2 +

 332 2 to the action of the center of N C Γ on Sym(3, C). The center of N C Γ is given by all matrices of the form (5.2) having a = b = 0 and acts on Sym(3, C) by c •  corresponding C-principal bundle is holomorphically trivial and given byπ : Sym(3, C) → C 5 , π(z kl ) = (z 11 , z 12 , z 13 , z 22z 33 , z 23 ). The induced action of C 2 ∼ = N C Γ /Z(N C Γ ) on C 5 is given by (a, b) • z = bz 1 + a z 3 + az 1b z 4 + 2bz 2 -2az 3 + (b 2a 2 )z 1 + 2ab + 2a z 5 + az 2 + bz 3 + abz 1 + (a 2b 2)/2 step we consider only the action of the one parameter group C a of elements (a, 0). A direct calculation shows that the C a -action on C 5 becomes a translation in the first coordinate after conjugation by the biregular map Φ :C 5 → C × C 4 , z → (z 2 , z 1 , z 3z 1 z 2 , z 4 + 2z 2 z 3 -2z 1 z 5 -2z 2 , z 2 2 -2z 5). It follows that C a acts properly on C5 and that the corresponding C a -principal bundleC 5 → C 5 /C a ∼ = C 4 is holomorphically trivial. Moreover, the induced C b -action on C 4 is of the form b • w =     w 1 w 2b(w 2 1 + 1) w 3 -2bw 1 w 4 -2bw 2 + b 2 (w 2 1 + 1)

  and thus on C 4 . Moreover, we have C 4 /C b ∼ = C 3 . Consequently, Lemma 5.2 applies to show that N C Γ acts properly on Sym(3, C) with quotient C 3 .

  1 ) g(z 1 ) 0 -ψ(z 1 ) ϕ(z 1 ) C-action into a translation.

Table 1

 1 , p. 341].

  tz 12 z 13 + tz 11 z 23 + tz 12 z 33 + 2tz 13 + t 2 z 11   and then derive with respect to t. The representation of b 2 on b 3 is defined by [α 2 , ξ 32 ] = ξ 32 [α 2 , ξ 32 ] = -ξ 32 [ξ 21 , ξ 31 ] = -ξ 32 [ξ 21 , ξ 32 ] = -ξ 31 [ξ 21 , ξ 31 ] = ξ 32 [ξ 21 , ξ 32 ] = -ξ 31 [ζ 2 , ξ 32 ] = 2ξ 32 ,

	z 11	z 12	z 13 + tz 11
	z 12	z 22	z 23 +

  12 + bz 11 + a z 13 + az 11b z 12 + bz 11 + a z 22 + 2bz 12 + b 2 z 11 + ab -2c z 23 + az 12 + bz 13 + abz 11 + a 2 -b 2 2 z 13 + az 11b z 23 + az 12 + bz 13 + abz 11 + a 2 -b 2 2 z 33 + 2az 13 + a 2 z 11ab -2(a + c) C 3 . It then follows that the principal bundle Sym(3, C) → C 3 is holomorphically trivial. Consequently, Sym(3, C)/Γ and D/Γ are Stein. In order to prove that the N C Γ -action is proper we shall use the following lemma. Lemma 5.2. Let G be a Lie group acting smoothly and freely on a manifold M and let H be a closed normal subgroup of G. Then G acts properly on M if and only if H acts properly on M and G/H acts properly on M/H. Proof. Firstly, suppose that G acts properly on M. Since H is closed in G, the H-action on M is proper and we get a smooth action of G/H on the quotient manifold M/H. Properness of this latter action was shown in [29, Proposition 1.3.2].

			3, C) is given by
	Z = (z kl ) →
		z 11	z 
			 .
	It is not difficult to verify that N C Γ acts freely on Sym(3, C), which in particular implies that N Γ has only totally real orbits in D, see Lemma 2.1.
	In the following we will show that N C Γ acts properly on Sym(3, C) with quotient man-ifold Sym(3, C)/N C

Γ ∼ =

  • [p n ], [p n ] converges to [q 0 ], [p 0 ] in M/H × M/H where [p] ∈ M/Hdenotes the class of p modulo H. Thus (g n H) has a convergent subsequence. Without loss of generality we assume that g n H → g 0 H. Hence, there is a sequence

Proof of Proposition 1.2. Suppose that D/Γ is a Stein manifold. As a first step we are going to show that N Γ has at least one totally real orbit in D.Since D/Γ is Stein, the domain D admits a Γ-invariant smooth strictly plurisubharmonic function ρ that is exhaustive modulo Γ. We may assume that 0 is a global minimum of ρ. Since N Γ /Γ is compact, we can suppose without loss of generality that the function ρ is invariant under N Γ and is an exhaustion modulo N Γ , see[START_REF] Loeb | Action d'une forme réelle d'un groupe de Lie complexe sur les fonctions plurisousharmoniques[END_REF] Lemme 2.1]. Let N Γ • z be an orbit lying in the minimal set of ρ. Then N Γ • z is totally real due to[START_REF] Harvey | Zero sets of non-negative strictly plurisubharmonic functions[END_REF].Finally, due to Lemma 2.1, it is enough to show that N C Γ acts freely onN C Γ • D ⊂ C n .In order to do this, let z ∈ D and consider the algebraic subgroupH := (N C Γ ) z of N C Γ . Since N C Γ • z is closed in C n , the intersection Ω := D ∩ (N C Γ • z) is Stein and N Γ -equivariantly biholomorphic to an N Γ -invariant Stein open neighborhood Ω of N Γ • eH ∼ = N Γ in N C Γ /H ∼ = C k .As D/Γ is Stein by assumption, Ω/Γ is also Stein. By applying[START_REF] Harvey | Zero sets of non-negative strictly plurisubharmonic functions[END_REF] and[START_REF] Loeb | Action d'une forme réelle d'un groupe de Lie complexe sur les fonctions plurisousharmoniques[END_REF] as above, one sees that N Γ • eH is totally real in N C Γ /H. Then, the analog argument as in the proof of Lemma 2.1 shows that H is trivial, as wished.Remark. Every orbit lying in the minimal set of ρ is isotropic with respect to the Kähler form i∂∂ρ. In general, it is not isotropic with respect to the Bergman metric of D.2.4.A sufficient condition for D/Γ to be Stein. In this subsection we prove Proposition 1.3.As above, let Γ ⊂ N be a discrete group of unipotent automorphisms of a bounded homogeneous domain D and let N Γ be its Zariski closure in N. Realize D as a Siegel domain of the second kind D ⊂ C n such that N acts by affine transformations on D.Proof of Proposition 1.3. By assumption, there exists a maximal totally real subalgebra n Γ ⊂ n which contains n Γ . At the group level we thus find a connected subgroupN Γ ⊂ N containing N Γ such that N Γ • p 0 is a maximal totally real submanifold of N • p 0 . Consider the complexifications N C Γ ⊂ N C Γ ⊂ N Cas well as their algebraic actions by affine transformations on C n . Since N Γ • p 0 is maximally totally real, the orbitN C Γ • p 0 is open in C n and the N C Γ -istropy at p 0 is trivial, see Lemma 2.1. Since N C Γ • p 0 is also closed in C n , we obtain N C Γ • p 0 = C n .In other words, N C Γ acts freely and transitively on C n . It follows that N C Γ acts freely and properly onC n ∼ = N C Γ and that C n /N C Γ ∼ = C m .There- fore the corresponding holomorphic principal bundle is holomorphically trivial. Together with[START_REF] Loeb | Action d'une forme réelle d'un groupe de Lie complexe sur les fonctions plurisousharmoniques[END_REF] Théorème 1] this implies that C n /Γ ∼ = (N C Γ /Γ) × C m is a Stein manifold. Since D/Γ is a locally Stein domain in C n /Γ, the quotient D/Γ is likewise Stein by[START_REF] Docquier | Levisches Problem und Rungescher Satz f ür Teilgebiete Steinscher Mannigfaltigkeiten[END_REF].Remark. The proof of Proposition 1.3 shows that, if n Γ is contained in a maximal totally real subalgebra of n, then N C Γ acts properly and freely on C n . Hence, all N Γ -orbits are totally real in D. Note that this follows also from Proposition 1.2.

Proof of Theorem 1.4 for the Lie ball. We shall prove that L n /Γ is Stein where Γ satisfies the conditions stated at the beginning of this subsection.Let us first assume that n Γ is not contained in the j-ideal b n-1 . Due to Proposition 4.1 there exists a maximal totally real subalgebra n Γ ⊂ n that contains n Γ . Then the result follows from Proposition 1.3.If n Γ is contained in b n-1 , we find a maximal totally real subalgebra n Γ of b n-1 that contains n Γ , see Proposition 3.1. Let N Γ be the corresponding closed subgroup of Aut 0 (L n ). The holomorphic submersion π :L n → H + is N Γ -invariant and extends to a N C Γ -invariant holomorphic submersion π : N C Γ • L n → H + . Then, N CΓ acts freely and properly on N C Γ • L n , and the fibers of π are precisely the N C Γ -orbits, isomorphic to C n-1 . In other words, π defines an N C Γ -principal bundle. Since H + is contractible, this N C Γ -principal bundle is holomorphically trivial, i.e., we haveN C Γ • L n /Γ ∼ = ( N C Γ /Γ) × H + .