Julien Chevallier
email: julien.chevallier@dauphine.fr

Price relationships in the EU emissions trading system

Keywords: Kyoto Protocol, Clean Development Mechanism, EU Emissions Trading Scheme, Greenhouse Gases Reductions, Emissions Markets, CDM, EU ETS

The Emissions Trading Scheme (ETS) constrains industrial polluters to buy/sell CO 2 allowances depending on a regional depolluting objective of -8% of CO 2 emissions by 2012 compared to 1990 levels. Companies may also buy carbon offsets from developing countries, funding emissions cuts there instead, under a Kyoto Protocol Clean Development Mechanism (CDM). This article critically analyzes the price relationships in the EU emissions trading system. The United Nations Framework Convention on Climate Change (UNFCCC) delivers credits that may be used by European companies for their compliance needs. Certified Emissions Reductions (CERs) from CDM projects are credits flowing into the global compliance market generated through emission reductions. EUAs (EU Allowances) are the tradable unit under the EU ETS. Besides, the EU Linking Directive allows the import for compliance into the EU ETS up to 13.4% of CERs on average. This article details the idiosyncratic risks affecting each emissions market, be it in terms of regulatory uncertainty, economic activity, industrial structure, or the impact of other energy markets. Besides, based on a careful analysis of the EUA and CER price paths, we assess common risk factors by focusing more particularly on the role played by the CER import limit within the ETS.

I. INTRODUCTION

The Emissions Trading Scheme (ETS) is the EU's flagship climate policy, forcing industrial polluters to buy/sell CO 2 allowances above a pre-specified emissions cap.

Companies may cut the costs imposed on the industry by buying relatively cheaper carbon offsets from developing countries, funding emissions cuts there instead, under the Kyoto Protocol's Clean Development Mechanism (CDM). As the latest Intergovernmental Panel on Climate Change (IPCC, 2007) report pointed out the huge potential for the growth of CO 2 emissions and associated pollutants in non-Annex B countries of the Kyoto Protocol, this article critically analyses to what extent the link between the EU ETS and the CDM will contribute to cut CO 2 emissions by 2020.

According to the article 12 of the Kyoto Protocol, projects under the Clean Development Mechanism consist in achieving greenhouse gases emissions reduction in non-Annex B countries. After validation, the United Nations Framework Convention on Climate Change (UNFCCC) delivers credits that may be used by Annex B countries for use towards their compliance position. Certified Emissions Reductions (CERs) from CDM projects are credits flowing into the global compliance market generated through emission reductions.

EUAs (EU Allowances) are the tradable units under the EU ETS. Albeit being determined on distinct emissions markets, CERs and EUAs may be exchanged based on their representative trading unit. One CER is equal to one ton of CO 2 -equivalent emissions reduction, while one EUA is equal to one ton of CO 2 emitted in the atmosphere.

Besides, the EU Linking Directive 2 allows the import of CERs into the EU ETS up to 13.4% of their compliance needs on average. The import limit is equal to 1.7 billion tonnes of offsets being allowed into the EU ETS from 2008-2020, that is, an absolute maximum of 50% of the depolluting effort fixed by the scheme will be achievable through the CDM.

Overall, our results shed light on the importance of the link between the EU ETS and the CDM to foster investments in infrastructure technology in developing countries, thereby facilitating the transition to a low-carbon future.

The remainder of the article is organized as follows. First, we detail the idiosyncratic risks affecting each emissions market, be it in terms of regulatory uncertainty or economic factors. Second, based on a careful analysis of the EUA and CER price paths, we assess common risk factors by focusing more particularly on the role played by the CER import limit within the EU ETS.

II. IDIOSYNCRATIC RISKS

In this section, we comment first on the price developments of EUAs and CERs.

Second, we examine the idiosyncratic risk factors for each emissions market.

Price development

Primary CERs (pCERs) have a delivery risk, while secondary CERs (sCERs) are already generated and issued by the CDM Executive Board, and are hence risk-free. The risks attached to primary CERs are linked to the United Nations' "International Transaction Log" (ITL) connection, the import limit, and the performance for operating projects, to which we may add a high volume of registered projects, as well as registration and methodological risks for proposed projects. The risks attached to secondary CERs are the ITL connection, the import limit, and eligibility criteria to be met for transfer of a CER from one EU registry to another. In the exchange contract (sCER), the seller agrees to pay EUAs or cash in case of non-delivery. The CER and EUA price series in Figure 1 shows that we are in presence of correlated emissions markets. The CER-EUA spread presented at the bottom of Figure 1 represents arbitrage opportunities for traders who are able to identify pricing anomalies between the quoted spread price and the two emissions markets. 3 The EUA spread over the secondary CER widened to nearly €10 in Figure 1, and was even higher for most primary CER contracts. We may also observe that the CER-EUA spread varies in the range of €-1.5 on 20 November 2008, and has been as high as €-9 during 2008. A broad spread means that there is a high risk premium in the purchase of CERs compared to EUAs. At the same time, it means that there are large arbitrage opportunities to use CERs for compliance within the EU ETS, since they are a lot cheaper than EUAs. A narrow CER-EUA spread could also mean lower demand for CERs, which may discourage project developers from investing in cleaner energy facilities.

This analysis brings us to investigate the respective risk factors of CERs and EUAs.

2. Risk factors specific to CDM credits CDM projects add value to an investment through CER revenues. CDM projects are registered by the CDM Executive Board, and displayed in the CDM pipeline4 along the several steps from registration to validation. One particularly striking source of uncertainty concerning the validation of CDM projects lies in the so-called "additionality" of GHG emissions reductions claimed; that is, project developers need to demonstrate the GHG emissions reductions claimed would not have been achieved in the absence of the project.

According to the World Bank (2008), the CDM projects' distribution is skewed toward a small group of developing host countries: five countries (China, India, Brazil, Mexico, South Korea) concentrate around 80% of the total project pipeline. These countries have been able to streamline their CDM project identification and approval procedures at an early stage.

Moreover, these countries offered several large-scale project opportunities which are generally attractive as they enable investors to spread project transaction costs across a larger number of CERs. Most projects are in the category of renewable energy and biomass energy production: 26% of CDM projects involve hydropower technology, 15% biomass-based energy production, and 14% wind energy. China concentrates 66% of hydropower projects, and India 68% of biomass energy projects. Together, those countries host 90% of wind energy projects. With such a geographic concentration, CER prices are sensitive to a large inflow of validated credits (e.g. from China or India). 5The risk factors specific to primary and secondary CERs are as follows. On the primary market, there needs to be (i) an increased predictability of issuance and frequency of transfer CERs by the CDM EB, (ii) an ITL operational and linked to the EU registry system, and (iii) the development of a robust options market. On the secondary market, we need to accound for (i) the acceptance into EU registries of CDM credits, (ii) the development of new emissions trading schemes which increase sCER demand and thus push CER prices to converge with other carbon prices under compliance schemes, and (iii) limitations on the use of CERs compliance under the EU ETS. Finally, both pCERs and sCERs are affected by the uncertainties concerning the post-2012 climate regime.

Risk factors specific the European carbon market

There are three main sources of risk in the EU ETS. First, the free permits distributed to existing firms on a ''first-come, first-served'' basis represent a market value of €40 billion that was created at the same time as CO 2 emissions were capped. This allocation methodology is also known as grandfathering. Third, installations do not need to physically hold allowances during the year to produce, but only to match the required number of allowances with verified emissions for their yearly compliance report to the European Commission. Consequently, the probability of a potential illiquidity trap exists if market participants face a market squeeze during the compliance event. Another specificity of emission allowances may be highlighted: compared to stocks which are valid during the entire lifetime of the firm, emission allowances are vintaged for a given compliance year and cannot be used for future compliance periods, unless intertemporal flexibility mechanisms are authorized. During Phase I of the EU ETS, the inter-period transfer of allowances has been restricted by all Member States [START_REF] Alberola | European Carbon Prices and Banking Restrictions: Evidence from Phase I (2005-2007)[END_REF].

III. COMMON RISK FACTORS

Similarly, we may identify several common risk factors between EUAs and CERs.

The ITL-CITL connection

On 19 October 2008 the European Commission's "Community Independent Transaction Log" (CITL) connected to the United Nations' "International Transaction Log" (ITL). The ITL-CITL connection involved the EU's 27 Member States disconnecting their national registries, and re-connecting them to the ITL. Simultaneously, the CITL connected directly to the ITL. These operations allowed the delivery of sellers' international credits from the Kyoto Protocol (such as CERs) into buyers' EU national registries. Before that date, issued CERs remained in the UN CDM Registry, waiting for the connection with the EU registry to be completed.

During Phase II of the EU ETS (2008)(2009)(2010)(2011)(2012), the EU Commission has announced that the use of CERs by industrial installations will be capped at 270 million tons per year. For Phase III (2013-2020), the limit for existing installations will be 40 million tons per year.

Altogether, Phase III CER imports will add up to 300 million tons. Thus, the post-2008 EU ETS CER import limit has been fixed at 1.7 billion tons.7 Therefore, the ITL-CITL connection has filled the risk factor common to CDM project developers and investors. First, it has provided stimulus to CER and EUA trading, as the barriers to transfer allowances between UN and EU registries have been removed. Second, it has offered to EU investors in CDM projects the possibility to monetize their assets, by using CERs towards their own compliance or by exchanging them on the European market as secondary CERs. However, it seems worth noting that some types of CERs credits are restricted from use in the EU ETS, such as Land Use, Land Use Change and Forestry (LULUCF) projects.

The import limit of CERs within the EU emissions trading system

Since the ITL-CITL connection, Kyoto credits may be imported into the EU ETS, and are valid for compliance up to 13.4% on average. As detailed in Section 2.1, the EUA-CER price arbitrage becomes possible up to that limit. Investors will maximize their profits by buying the maximum volume of CERs allowed for compliance and selling the same amount EUAs when the EUA-CER spread is at its maximum, i.e. in the range of €6 to €9. [START_REF] Alberola | The EUA-sCER Spread : Price Arbitrage and Compliance Strategies in the European Carbon Market[END_REF] provide a detailed analysis of the determinants of EUA and CER prices. Besides, they study the determinants of the EUA-CER spread with regard to market microstructure mechanisms (such as the volume of allowances exchanged, etc).

Concerning the role played by the CER import limit for compliance within the EU trading system, we may proceed with the following counter-factual exercise. Eighty million tons of CERs have been used for compliance to cover verified emissions in 2008 (according to the EU Commission).8 At the same time, 250 million tons of CERs have been issued (according to the CDM Pipeline). If we assume the CER issuance rate to be constant,9 we obtain the path for the stock of CERs available for compliance within the EU ETS over time as depicted in Figure 2. (2008)(2009)(2010)(2011)(2012). The solid orange lines denote the stock of CERs available corrected for various issuance uncertainties, while the solid green lines indicate the total gross supply of CER issued. We observe on this graph that, due to technical and institutional constraints, firms could not use more than 1.6 Gt of CERs by April 2013 [START_REF] Trotignon | How Many CERs by 2013? Mission Climat Working Paper #[END_REF]). Besides, the yearly CER import limit does not seem to be binding before the 2010 compliance event. 10 The path of CER delivery displayed in Figure 2 critically depends on the criteria for CDM project validation.

In terms of expected CER delivery, HFC, PFC, CH 4 , and N 2 O emission-reduction projects have a relatively large market share, which is mainly due to these gases' high global warming potential. It is also important to look at the distribution of projects and technologies in terms of expected GHG emission reductions. Twenty-two projects involving HFC emissions reductions constitute 17% of expected emissions reductions by 2012.11 Sixty-five projects involving N 2 O emissions reductions constitute 9% of expected CERs12 (World Bank, 2008). By contrast, the 1,098 hydro projects will deliver just 17% of expected CERs. The distribution of issued CERs issued by technology is important to keep in mind, since HFCprojects (and other gases for which CERs are heavily issued) may not allowed for import in the EU ETS towards Phase III.

In terms of risk-assessment, it is also interesting to note that the delivery of credits, and thus the performance of projects, greatly varies: only 10 to 15% of methane reduction credits could be verified and therefore issued (World Bank, 2008). This ratio is 75% for HFC 23 emissions reduction projects, while N 2 O projects sometimes deliver more CERs than expected.

Uncertainties concerning Post-Kyoto agreements

Of course, the list of common risk factors between the CDM and EU ETS markets would not be complete without a careful outlook of negotiations for a post-Kyoto treaty.

The European Commission has underlined its will to wait for the outcome of the December 2009 Copenhagen Summit before taking final decisions concerning CERs use in Phase III of the EU ETS (2013-2020). For instance, on technological grounds, CERs generated through HFC destruction or large hydro projects may be banned from import within the EU ETS.

IV. CONCLUSION

This article critically examines the role played by Clean Development Mechanism projects for compliance into the EU Emissions Trading Scheme. We have highlighted risk factors specific to each emissions market -additionality and predictability for CERs, grandfathering, compliance events and banking provisions for EUAs -as well as common risk factors -the ITL-CITL connection, the role played by the 1.7Gton import limit, and uncertainties concerning future international agreements on climate change.

My conclusion cautions towards a prudent approach concerning the use of CERs for compliance within the EU trading system. On the one hand, CERs foster investments in nonpolluting technologies in non-Annex B countries. These project thereby contribute to the "global public good" of fighting climate change by cutting greenhouse gas emissions with a maximum potential (as for HFC destruction or large hydro projects) at lowest cost, and should be encouraged from both economic and environmental viewpoints. On the other hand, the unlimited use of CERs for compliance within the EU ETS would set a price floor for EUAs, and drive down their price. This solution does not appear compatible with the necessary "price signal" of emitting one ton of CO 2 in the atmosphere in Europe, which should be high enough to provide incentives for industries to cut back emissions compared to their business-as-usual scenario.

The bottom line is that the restricted use of CERs for compliance within the EU ETS appears a wise choice from the EU Commission's viewpoint, in order 1) to limit design inefficiencies that have already affected the price path of EUAs during Phase I (2005-2007) and 2) to foster investments at the EU level in less CO 2 -intensive production processes.

Figure 1 :

 1 Figure 1: EUA, CER, and EUA-CER Spread Prices

  Since 1 January 2005, carbon allowances form another asset in commodities against which industrials and brokers need to hedge. As the volume of transaction on the EU ETS has increased steadily from 262 million tons in 2005 to 1,443 million tons in 2007, this trading activity reflects market participants' progressive learning of this new financial market. Second, during Phase I of the EU ETS (2005-2007), EUAs experienced a high level of volatility around each compliance event. Industrial installations have the obligation to surrender to the EC the exact number of allowances that matches their verified emissions each year around end of March. The official report by the European Commission is disclosed by mid-May, but installation operators have already a fair amount of information between the publication of their own report and the compilation of verified emissions by the European Commission to approximate the total level of emissions relative to allowances allocated and to adjust their anticipations 6 .

Figure 2 :

 2 Figure 2: CER Stock available for EU ETS Compliance

See http://ec.europa.eu/environment/climat/emission/linking_en.htm

Note that CERs traded on exchanges are precisely guaranteed secondary CERs (sCERs), but for simplicity we use the common denomination term of CERs throughout the article.

See http:\\www.cdmpipeline.org

To conserve space, we do not comment here on the concept of "programmatic" CDM, which has been developed for small-scale projects.

The data in Figure1reveals a sharp break in EUA price series during the 2005 compliance event. By the end of April 2006, this price correction of 54% within four days followed the announcement by the EC that CO 2 emissions were approximately 3% lower than the allocated allowances during the 2005 compliance period[START_REF] Alberola | Price drivers and structural breaks in European Carbon Prices 2005-07[END_REF].

See http://ec.europa.eu/environment/climat/emission/ets_post2012_en.htm

See http://ec.europa.eu/environment/climat/emission/citl_en.htm

Note this assumption is not overly restrictive. As detailed in Section 2.3, this rate is even likely to be decreasing.

As is characteristic of the functioning of the EU ETS, there is a one-year delay in the accounting of verified emissions. Thus, the 2010 compliance event will occur in May 2011.

Indeed, the HFC global warming potential is 11,700 times higher than CO 2 , and thus each ton of HFC abated delivers 11,700 CERs.

The N 2 O global warming potential is 310 times higher than CO 2 .


Julien Chevallier
email: julien.chevallier@dauphine.fr

Emilie Alberola

Benoit

Simon Buckle

Derek Bunn

Thierry Brechet

Denny Ellerman

Pierre-Andre Jouvet

Valerie Mignon

Christian De

The European carbon market (2005-2007): banking, pricing and risk-hedging strategies

Keywords: Climate Change Policy, Emissions Trading, EU ETS, European carbon market, Banking Borrowing, Carbon Pricing, Spot Prices, Futures Prices, Option Prices, Risk-Hedging Strategies

At the stage of international post-Kyoto negotiations, the adoption of ambitious public policies raises an increasing interest, as society has a whole is more concerned by the scale of damages and the potential irreversibility linked to climate change. The introduction of a tradable permits market in Europe on January 1, 2005, in order to provide incentives to Member-States to take early abatement measures, may be seen as a decisive first step towards that direction.

The creation of the EU ETS has indeed revealed the key role played by the European Union in the preservation of the global public good that constitutes the climate. This article reviews the market rules of the European carbon market during 2005-2007. More particularly, it synthesizes theoretical and empirical analyses of banking and borrowing provisions, price drivers and riskhedging strategies attached to tradable quotas, which were introduced to cover the CO 2 emissions of around 10,600 installations in Europe.

Introduction

The European Union Emissions Trading Scheme (EU ETS) has been created on January 1, 2005 to reduce by 8% CO 2 emissions in the European Union by 2012, relative to 1990 emissions levels. This aggregated emissions reduction target in the EU has been achieved following differentiated agreements, sharing efforts between Member States based on their potential of decarbonization of their economy. The introduction of a tradable permits market has been decided to help Member States in achieving their targets in the Kyoto Protocol, entered into force on February 2005 following the ratification of Iceland, and which aims at reducing the emissions of six greenhouse gases (GHG) considered as the main cause of climate change.

Among the Members of Annex B, these agreements include CO2 emissions reductions for 38 industrialized countries, with a global reduction of CO 2 emissions by 5.2% (see Table 2). These agreements have been fostered by the United Nations Framework Convention on Climate Change (UNFCCC (2000)) which recognizes three principles: the precautionary principle1 , the principle of common but differentiated responsibilities2 , and the principle of the right to development 3 . 174 countries, Australia being the latest on December 3, 2007, have ratified the Protocol, with the notorious exception of the United States. The first commitment period of the Kyoto Protocol goes from January 1, 2008 to December 31, 2012. This political will has been reaffirmed at the international level during the UN Conference that took place in Bali on December 2007, where a roadmap of negotiations that should lead to a post-Kyoto agreement has been adopted. The United States are expected to cooperate, given the initiatives of emissions reduction introduced at the regional level 4 . The next round of negotiations will take place in Copenhagen on December 2009. As the Clean Development Mechanism (CDM) 5 has revealed the strong potential of CO 2 emissions abatement in countries such as Brazil, China or India, the main issue of these negotiations is linked to achieving the largest possible level of cooperation, in order to avoid the well-known free riders behaviours, and to preserve the global public good that constitutes the climate. On this matter, the European Union has clearly adopted a leadership position, which contrasts with its early reluctance during the first steps of the negotiation of the Kyoto Protocol.

On January 2008, the European Commission has extended the scope of its action against global warming by 2020 with the "energy and climate change" package. This package aims at reducing GHG emissions by 20%, at increasing the use of renewable energy in energy consumption to 20%, and at saving 20% of energy by increasing energy efficiency. The European carbon market, which has currently entered its Phase II (2008II ( -2012)), has been confirmed until 2020 also. Its scope has been extended to major sectors in terms of CO 2 emissions growth, such as aviation and petro-chemical industries during 2013-2020. These repeated public policies in favour of climate protection aim at correcting the negative externality attached to the release of uncontrolled GHG emissions in the atmosphere and thus, according to the well-known principle in economics, at internalizing the social cost of carbon. At the same time, these initiatives reveal the difficulty to create a scarcity condition regarding CO 2 emissions. These emissions indeed were not limited in the pre-existing institutional environment, and thus could not be considered as a scarce resource.

The European Union being at the forefront of environmental regulation dedicated to climate policies, this article reviews the market rules of the European carbon market during Phase I. It investigates the role played by the regulator, among the various choices at stake when creating a tradable permits market, on the behaviour of firms. Thus, this article contributes to the literature on the "birth" of the European carbon market [START_REF] Convery | The European Carbon Market in Action: Lessons from the First Trading Period[END_REF][START_REF] Convery | Reflections -The Emerging Literature on Emissions Trading in Europe[END_REF], [START_REF] Ellerman | Over-Allocation or Abatement? A Preliminary Analysis of the EU ETS Based on 2005 Emissions Data[END_REF]), by focusing our attention on the study of several key provisions in a moving institutional context, and by identifying learning effects.

The remainder of the article is organized as follows. Section 2 presents the key design issues of the EU ETS. Section 3 examines the effects of banking restrictions between 2007 and 2008. Section 4 discusses the price fundamentals of CO 2 allowances. Section 5 details market participants' risk behaviour. Section 6 concludes.

Key Design Issues of the EU Emissions Trading Scheme

This section reviews the scope, allocation methodologies, calendar, transactions levels, and penalties associated to non-compliance on the EU ETS.

Scope

The Directive 2003/87/CE defines the scope of the EU ETS 6 . This scheme concerns around 10,600 installations in Europe, mainly in the production sectors of combustion, iron and steel, pulp and paper, refineries, and cement. Installations in these sectors are eligible to emissions trading when their energy consumption is superior to the threshold of 20MWTh. This threshold has been decided by the European Commission so as to target the most energyintensive industries during the first Phases of the program. This choice has been justified initially by the will of the European Commission to minimize political resistance, and to enforce a quick implementation of the scheme in 2005. To increase the environmental performance of the scheme, the debate is now centred on the progressive extension of its scope. The EU ETS Review7 has revealed that other sectors will soon be included, such as aviation as of 2013.

Allocation

The CO 2 emissions reduction target of each Member State has been converted into National Allocation Plans (NAPs). Each government is in charge of deciding the amount of quotas available for trading, after negotiating with industrials, and after the validation by the European Commission. The role of the Environment DG is central in this scheme in order to harmonize NAPs among Member States, and to recommend stricter NAPs validation criteria.

The NAPs submissions may be rejected by the European Commission, and sent back to Member States for revision before the final decision. The sum of NAPs determines the number of quotas distributed to installations in the EU ETS. 2.2 billion of quotas per year have been distributed

Calendar

The EU ETS Phase I may be considered as a warm-up phase, during 2005-2007. Phase II corresponds to the commitment period of the Kyoto Protocol, i.e. 2008Protocol, i.e. -2012. . Phase III is supposed to correspond to a post-Kyoto agreement, i.e. 2013-2020. During each of these Phases, the delivery of allowances is made on a yearly basis, and follows a precise calendar:

• on February 28 of year N, European operators receive their allocation for the commitment year N; • March 31 of year N is the deadline for the submission of the verified emissions report during year N-1, from each installation to the European Commission;

• April 30 of year N is the deadline for the restitution of quotas utilized by operators during year N -1;

• May 15 of year N corresponds to the deadline of the official publication by the European Commission of verified emissions for all installations covered by the EU ETS during year N -1.

The annual frequency of verified emissions, imposed by the European Commission, corresponds thus to a central event, structuring the diffusion of reliable information at the aggregated level on the European carbon market.

Transactions

One allowance exchanged on the EU ETS corresponds to one ton of CO2 released in the atmosphere, and is called a European Union Allowance (EUA). Allowance trading is recorded electronically by national registries. The information contained in these registries is centralized by the European Commission in the European registry, called the Community Independent Transaction Log (CITL) 9 . The CITL contains exhaustive information on CO 2 emissions for all installations covered by the EU ETS, and is used to account the compliance position of each firm. The information contained in the CITL is available at the installation level. As a first step, data compilation appears necessary to reconstruct the ownership structures between subsidiaries and parent companies, which yield to a more precise analysis for the evaluation of the scheme [START_REF] Mcguinness | Technical Memorandum on Analysis of the EU ETS Using the Community Independent Transaction Log[END_REF]).

To comply with their emissions target, installations may exchange quotas either overthe-counter, or through brokers and market places. Bluenext, formerly Powernext Carbon, is the market place dedicated to CO 2 allowance trading based in Paris. The European Climate Exchange is the market place based in London, which is leader for derivatives products.

NordPool represents the market place common to Denmark, Finland, Sweden, Norway, and is based in Oslo. The price of products exchanged on these market places are strongly correlated, which is conform to other market places like stock markets. Moreover, the European carbon market is characterised by an increasing sophistication of financial instruments using a quota of CO 2 as the underlying asset, and the development of option prices or swaps10 .

Penalties

During 2005-2007, if an installation does not meet its emissions target during the compliance year under consideration, the penalty is equal to €40/ton in excess, plus the restitution of one allowance during the next compliance period. During 2008-2012, this amount corresponds to €100/ton, following the same principle.

Following this review of the institutional context on the European carbon market, we detail in the next section the allowance price development and associated banking strategies.

Banking Provisions

This section details the banking borrowing provisions adopted in the EU ETS.

Intertemporal emissions trading allows firms to smooth their emissions overtime, and offers a greater flexibility in order to meet the emissions target. Therefore, banking and borrowing allow firms to achieve their depolluting objectives at least cost, if these provisions are adequately configured by the regulators and their effects has been sufficiently discussed, evaluated and

Price Developments

Banking Restrictions

On the EU ETS, allowances are valid during a specific compliance year. However, an installation may have banked allowances during year N to cover its emissions during year N+1, if years N and N +1 correspond to the same Phase. The same mechanism applies for allowances borrowed from year N + 1 in order to comply with the emissions target of the installation during To further develop our analysis of price developments in the EU ETS, we conduct in the next section a review of the main price fundamentals of EUAs during 2005-2007.

CO 2 Price Fundamentals

This section focuses on the price fundamentals of CO 2 allowances. These fundamentals are mainly linked to regulatory decisions, energy prices and extreme temperatures events [START_REF] Christiansen | Price determinants in the EU emissions trading scheme[END_REF]).

Institutional Decisions

First, it is worth noting that political and institutional decisions on the overall cap stringency have an impact on the carbon price setting through initial allocation. Also, any decision or announcement from regulators may induce changes in market players' behaviour.

From this perspective, official communications by the European Commission are essential to reach a better information flow on installations' net short/long positions11 [START_REF] Ellerman | Over-Allocation or Abatement? A Preliminary Analysis of the EU ETS Based on 2005 Emissions Data[END_REF]).

Whereas on energy markets the question of price formation is closely related to commodity storage, on the EU ETS the essential issue is the expected "emission shortfall" during each compliance year. The emission shortfall, defined as the difference between verified emissions during the compliance year and allocated allowances, depends on the actual amount of emissions abatements required (which are unknown but estimable based on reliable recent data) by the stringency of the cap (which is known). This information is publicly disclosed each year by the European Commission by mid-May as detailed in Section 2.3. It has a strong market effect on allowance price changes of all maturities, as it provides market participants with reliable information to update their expectations about future market developments.

Energy Prices

Second, energy prices are the most important price drivers on the short term of the EUA demand due to the ability of power generators to switch between their fuel inputs. This fuelswitching behaviour at the installations level applies especially in the power sector, which was endowed with more than 50% of EUAs during 2005-2007. The EU ETS price formation is indeed largely influenced by the electricity power market, since its participants are the main traders on the carbon market. 12 CIF ARA denotes the price of coal inclusive of freight and insurance delivered to the large North West European ports, e.g. Amsterdam, Rotterdam or Antwerp. 13 As calculated by the Mission Climat of the Caisse des Depots for Tendances Carbone. The methodology is available on the website:

The Clean Dark Spread (in €/MWh) represents the difference between the price of electricity at peak hours and the price of coal used to generate that electricity, corrected for the energy output of the coal plant and the costs of CO 2 :

) * 1 * ( coal t coal EF p coal elec spread dark clean + - = ρ (1)
with ρ coal the net thermal efficiency of a conventional coal-fired plant 14 , and EF coal the CO 2 emissions factor of a conventional coal-fired power plant 15 .

The Clean Spark Spread (in €/MWh) represents the difference between the price of electricity at peak hours and the price of natural gas used to generate that electricity, corrected for the energy output of the gas-fired plant and the costs of CO 2 :

) * 1 * ( ngas t ngas EF p ngas elec spread spark clean + - = ρ (2)
with ρ ngas the net thermal efficiency of a conventional gas-fired plant 16 , and EF ngas the CO 2 emissions factor of a conventional gas-fired power plant 17 .

During 2005-06, the use of coal appeared more profitable than gas. Since the beginning of 2007, the difference between the clean dark and clean spark spreads has been narrowing. This situation encourages consequently electric companies to decrease the use of coal to the profit of natural gas.

Figure 7 shows the price development of brent prices from July 2005 to November 2007. The oil price (in $/baril) is the daily brent crude futures Month Ahead price negotiated on the Intercontinental Futures Exchange. [START_REF] Alberola | Price Drivers and Structural Breaks in European Carbon Prices 2005-2007[END_REF] show that energy prices forecast errors have basically driven the CO 2 price over 2005-2007, but their influences changed over the period depending on regulatory changes. High levels of natural gas lead power operators to realise a switch in fuel utilization from gas to coal. The natural gas price got higher from October 2005 to April 2006 and thereby positively influenced the EUA price. As the most CO 2 -intensive variable, coal plays a negative role on carbon price changes: when confronted to a rise of the price of coal relative to other energy markets, firms have an incentive to adapt their energy mix towards less CO 2intensive energy sources. Brent prices have a positive effect on EUA price changes, which channels through the natural gas price [START_REF] Kanen | Carbon Trading and Pricing[END_REF]). These energy influences on carbon prices are also in line with [START_REF] Mansanet-Bataller | CO 2 Prices, Energy and Weather[END_REF] 18 .

Extreme Weather Events

Weather conditions have an impact on EUA price changes by influencing energy demand. Previous literature focuses on the most important dimension of weather: extremely hot and cold degree-days [START_REF] Roll | Orange Juice and Weather[END_REF]). Besides, we discuss the non-linearity of the relationship between temperatures and carbon price changes.

Weather influences may be captured by using the daily data of Powernext Weather indices (expressed in °C) for four countries: Spain, France, Germany and the United Kingdom.

These indices are computed as the temperature average at the representative regional weather station weighted by regional population:

∑ ∑ = = Θ = Θ N i i N i i i pop pop 1 1 * (3)
with N the number of regions in the country under consideration, pop i the population of region i, and Θ i the average temperature of region i during the month under consideration in °C.

The European temperature index published by Tendances Carbone 19 may also be used. affected negatively carbon price changes. The economic rationale behind this analysis is that when extremely cold events are colder (hotter) than expected, power generators have to produce more (less) than they forecasted, which conducts to an increase (decrease) of allowances demand and finally to an increase (decrease) of CO 2 price changes.

It is equal to the average of national temperatures indices provided by

In this section, we have identified as CO 2 price drivers energy prices, weather events, and institutional decisions during 2005-2007. Linked to the influence of political, energy, climatic and economic uncertainties on CO 2 price changes, we discuss in the next section adequate risk-hedging strategies on the EU ETS.

Risk-Hedging Strategies

This section deals with the risk-hedging strategies used by firms. Investors naturally attempt at hedging against a variation of the risk attached to allowance trading, especially given the institutional amendments to the functioning of the scheme. We discuss first the introduction of option prices in the EU ETS, and second the consequences on market participants' hedging strategies.

Carbon-based derivatives products

Investors need to manage the risk of holding CO 2 allowances on the European carbon market among a portfolio of diversified investments. As on financial markets, the uses of derivatives products allow to reduce the risk of a position on emissions markets. Indeed, the European Climate Exchange launched on October 2006 derivatives products trading carbon allowances as the underlying asset.

Source: ECX

Since option prices transfer the risk of financial exposure between market agents, we further detail in the next section agents' behaviour with respect to risk on this newly created commodity derivatives market. [START_REF] Chevallier | Risk Aversion and Institutional Information Disclosure on the European Carbon Market: A Case-Study of the 2006 Compliance Event[END_REF] estimate changes in investors' risk aversion on the European carbon market around the 2006 compliance event20 . They recover investors' risk aversion by using the existing relationship with the risk-neutral and historic probabilities. This methodology has proved to be robust for stock markets. First, the risk-neutral distribution is recovered from ECX option prices. Second, the historical distribution is approximated by the historical return distribution of futures allowance prices. Third, the risk aversion is obtained as a by-product [START_REF] Leland | Who Should Buy Portfolio Insurance?[END_REF]). Overall, this study provides an efficient tool to quantify the effects of risk aversion on the European carbon market which, during the period under consideration, has been higher than on the stock market. This situation underlines the necessity for investors to manage adequately the risk attached to holding CO 2 allowances. With the start of Phase II on a sound institutional framework, risk aversion on the European carbon market is likely to tend progressively towards the values found on stock markets.

Investors' risk aversion

Concluding Remarks

This article reviews the market rules of the European carbon market during 2005-2007.

The synthesis of theoretical and empirical approaches developed here has been fruitful for the analysis of banking, pricing and risk-hedging strategies. These results teach us that institutional learning has indeed occurred within Phase I, both from the viewpoint of market agents and the regulator.

The banking restrictions enforced between December 2007 and January 2008 in the EU ETS led to the disconnection between spot prices valid during Phase I, which plummeted to zero, and futures prices valid during Phase II, which remained stable around €20 throughout the period. This particular episode of the EU ETS highlights the necessity to understand the underlying mechanisms of CO 2 price changes.

Like other commodity markets, the amount of allowances available for trading and thus the EUA price are driven by the balance between supply and demand (energy prices, weather variables, etc.), and other factors related to market structure and institutional policies. Decision changes within the regulatory environment have a sharp market effect on allowance prices of all maturities. These structural breaks may be statistically identified within the time series of carbon prices, and are linked to yearly compliance events and to official communications by the European Commission. Besides, it is possible to isolate the influence of carbon price fundamentals linked to energy prices and extreme weather events, which varies before and after institutional events.

Carbon allowances therefore form another asset in commodities against which industrials and brokers need to hedge. Studies based on methods used on stock markets prove to be robust in quantifying changes in investors' anticipations.

Overall, this article highlights the inefficiencies following the creation of the European carbon market that prevented the emergence of a price signal leading to effective emissions reductions by industrials. The early design inefficiencies of the European carbon market, linked to initial allocation or the inter-period transfer of allowances, seem to have been corrected for the period 2008-2012, thereby limiting the transfer of inefficiencies towards Phase II.

  during 2005-2007. 2.08 billion of quotas per year will be distributed during 2008-2012, which corresponds to a more restrictive allocation, given some changes in the scope of the market with the inclusion of new Member States. Figures 1 and 2 represent, respectively, the repartition of quotas (in million tons of CO 2 ) between Member States during the commitment periods 2005-2007 and 2008-2012 8 . Germany, Poland, Italy, the UK and Spain total around two thirds of allowances distributed. The allocation methodology consists in a free distribution of quotas in proportion of recent emissions, also known as grandfathering. With a value of around €20 per quota, the launch of the EU ETS corresponds to a net creation of wealth of around €40 billion. The environmental constraint during 2005-2007 has not been considered as sufficiently binding for most market observers, and the allocation methodology has been criticised for distributing rents to pre-existing market players, as some of them may make a net profit simply by selling their unused allowances. During 2005-2007, allowances distributed have more than covered verified emissions, with a net cumulated surplus of 156 million tons. This surplus has however decreased, going from 83 million tons in 2005 to 37 million tons in 2006, and finally 36 million tons in 2007. Emissions have increased by 0.4% in 2007 compared to 2006, and reached 2,043 million tons with respect to 2,080 million allowances distributed.

Figure 1 :Figure 2 :

 12 Figure 1: EU ETS National Allocation Plans -Phase 1 (2005-2007) Source: CITL (2007) and CDC (2006)

Figure 3 :

 3 Figure 3: Volume exchanged for the spot price valid during 2005-2007 from June 24, 2005 to April 25, 2008 in tons of CO 2 Source: BlueNext

  understood. Let us first examine the allowance price development in the EU ETS during 2005-2007.

Figure 4 :

 4 Figure 4: EUA Spot and Futures Prices from July 2005 to May 2007 Source: BlueNext and ECX

  year N. Thus, allowances banked or borrowed are fungible within the same Phase. However, allowances distributed during Phase I are not valid during Phase II. Allowances distributed during Phases II and III are fungible between the different Phases. Phase I is characterized by a full intertemporal flexibility, like Phases II and III. Yet, given the simultaneity of the commitment periods between the Kyoto Protocol and Phase II, the intertemporal transfer of allowances has been strictly limited between Phases I and II, in effect banning the transfer of allowances between December 31, 2007 and January, 1, 2008. Alberola and Chevallier (2009) develop a statistical analysis showing that the disconnection between Phase I prices, decreasing towards zero, and Phase II prices, stabilized around 20€/ton, may be explained by the restriction on the inter-period transfer of allowances enforced during Phase I. Indeed, the cost-of-carry relationship between EUA spot and futures prices for delivery during Phase II does not hold after the enforcement of the inter-period banking restrictions around October 2006.The inefficiency of the EUA price signal to reflect correctly the social value of carbon until the end of Phase I may be explained by the restrictions enforced by Member-States concerning the transfer of quotas, banked or borrowed, from Phase I to Phase II. This sacrifice of the intertemporal flexibility mechanism may be interpreted by the will of the European Commission to limit the transfer of inefficiencies from the creation of the allowance market to Phase II, which simultaneously corresponds the Kyoto Protocol commitment period. Between Phases II and III of the EU ETS, the transfer of allowances has been authorized. Therefore, it appears possible to identify institutional learning effects between Phases I and II, as the early inefficiencies due to the youth of the European carbon market during 2005-2007 do not seem to have been transferred to the subsequent periods.Moreover, preliminary analyses of the 2005-2007 data concerning the extent of the use of banking in the EU ETS may be found in[START_REF] Ellerman | Compliance Behavior in the EU ETS: Cross Border Trading, Banking and Borrowing[END_REF] and[START_REF] Chevallier | Bankable Pollution Permits under Uncertainty and Optimal Risk-Management Rules: Theory and Empirical Evidence[END_REF].

  [START_REF] Alberola | Price Drivers and Structural Breaks in European Carbon Prices 2005-2007[END_REF] develop an original method to identify structural breaks in the CO 2 price series. They provide statistical evidence that two institutional events on April 2006, following the disclosure of 2005 verified emissions, and on October 2006, following the European Commission announcement of stricter Phase II, occurred during 2005-2007. Those events have a sharp effect on market participants' expectations changes, and further allow isolating distinct energy and weather influences on carbon prices as discussed below.

Figure 5 :

 5 Figure 5: Gas Zeebrugge and Coal CIF ARA Prices from July 2006 to June 2007 Source: Thomson Financial Datastream

Figure 6 :

 6 Figure 6: Clean Spark Spread, Clean Dark Spread and Electricity Powernext Prices from July 2006 to June 2007 Source: Thomson Financial Datastream

Figure 7 :

 7 Figure 7: Brent ICE Prices from July 2005 to November 2007 Source: Thomson Financial Datastream

  Powernext weighted by the share of each NAP in the previous four countries: the number of allowances allocated by the NAP in country j, and Θ j the national temperature index of country j. The national share of allocation during Phase I in total allocation of EUAs are equal to 14.55% for France, 46.40% for Germany, 22.82% for the UK, and 16.23% for Spain, according to the European Commission.

Figure 8 :

 8 Figure 8: European Temperatures Index from July 2005 to May 2007 Source: Mission Climat -Caisse des Dépôts

Figure 9

 9 Figure 9 displays option prices available along with several strikes from October 2006 to October 2007 on ECX. As detailed in Section 2.4, ECX is the most liquid trading platform with approximately 86.5% of the total exchange-based trades of allowances. The underlying assets of the contracts are first and second period spot prices. The maturity of the contracts typically range from December 2008 to 2013, Phase II contracts (2008-2012) being more actively traded than post-Kyoto contracts. Option prices on the carbon market lead to pricing errors that are usual for commodity or equity markets. On such a commodity market, the easiest way to hedge against the risk of allowance price changes is by selling calls: call prices are more actively traded than puts.

Figure 9 :

 9 Figure 9: Option prices available along with several strikes from October 2006 to October 2007.

Figure 10 :Figure 11 :Figure 12 :

 101112 Figure 10: Changes in the Risk Neutral Distribution for the December 2008 Futures Contract.Source:[START_REF] Chevallier | Risk Aversion and Institutional Information Disclosure on the European Carbon Market: A Case-Study of the 2006 Compliance Event[END_REF]

Figure 13 :

 13 Figure 13: Changes in the Implied Volatility for the December 2009 Futures Contract. Source: Chevallier et al. (2009) Note: the blue line denotes the risk neutral density before the 2006 compliance event, while the red line denotes the risk neutral density after this institutional event.

Figures

  Figures 10 and 11 represent changes in the risk neutral distribution for the futures contracts of maturity, respectively, December 2008 and December 2009. For both figures, the blue line denotes the risk neutral density before the 2006 compliance event, while the red line denotes the risk neutral density after this institutional event. The blue line has a steeper slope than the red line, which induces more volatility. These results are consistent with the role of information in lowering volatility on financial markets. Figures 12 and 13 represent changes in the implied volatility for the futures contracts of maturity, respectively, December 2008 and December 2009. These two figures illustrate the dramatic changes in investor's risk aversion around the 2006 compliance event, as the implied volatilities exhibit dramatically different slopes depending on the sample considered. By extracting the information contained in option and futures prices, these results uncover a dramatic shift in investors' anticipation around the 2006 compliance event.

Scientific uncertainty concerning the precise impacts of climate change does not justify delaying immediate action.

Each signatory country recognizes the impact of its GHG emissions on climate change. The most industrialized countries carry a heavier historical responsibility, given their prior GHG intensive development, which translates into tighter targets.

Action will be taken in accordance with the economic development of each country.

We may cite the Regional Greenhouse Gas Initiative (RGGI), which contains several GHG reduction objectives in nine North-Eastern States, and the Assembly Bill 32 in California which aims at reducing CO 2 emissions by 25% by 2020 relative to 1990 emissions levels, and by 80% by 2050. At the federal level, the Climate Stewardship Act introduced by the Senator Lieberman-McCain did not find sufficient political support to become legally binding.

According to the article 12 of the Kyoto Protocol, CDM projects consist in achieving GHG emissions reduction in non-Annex B countries. After validation, the UNFCCC delivers credits that may be used by Annex B countries for use towards their compliance position.

See the European Commission Environment DG website at http://ec.europa.eu/environment/ets/

Available at http://ec.europa.eu/environment/climat/emission/reviewen.htm

The data comes from the Mission Climat Caisse des Depots, available at: http://www.caissedesdepots.fr

Available at: http://ec.europa.eu/environment/ets

Note there exists also financial instrument with a CDM credit on the secondary market as the underlying asset, stemming from the Kyoto Protocol and fungible with quotas traded in the EU ETS with a maximum limit of around 13.4%.

Note an installation is defined as short (long) when it records a deficit (surplus) of allowances allocated with respect to actual emissions.

Given the central role played by the 2005 compliance event highlighted in Section 3.1, we focus on the 2006 compliance event, which is the only event empirically observable following the introduction of options trading on ECX.

to acknowledge financial support from the French Ministry of Research and Superior Education. The usual disclaimer applies.


Christian Défarge

Pascale Gautret

Joachim Reitner

Jean Trichet

Keywords: Organominerals, Organomineralization, Terminology

The best reply to [START_REF] Perry | Reply to comments on defining biominerals and organominerals: direct and indirect indicators of life[END_REF] reply to comments by [START_REF] Altermann | Defining biominerals and organominerals: direct and indirect indicators of life[END_REF] and [START_REF] Défarge | Defining organominerals: comments on 'Defining biominerals and organominerals: direct and indirect indicators of life[END_REF] on the paper by Perry et al. (2007, Defining biominerals and organominerals: direct and indirect indicators of life, Sedimentary Geology 201, 157-179) still remains these very comments, which Perry and Sephton do not seem to have carefully read. However the authors' reply is so remote from the customary practice of scientific discussion that a further answer is called for.

First of all, contrary to all academic custom, Perry and Sephton's reply does not include the least excuse for not having quoted, in [START_REF] Perry | Defining biominerals and organominerals: direct and indirect indicators of life[END_REF], former definitions of organominerals, in particular those given by (Défarge & Trichet, 1995) and (Trichet & Défarge, 1995), which they were aware of ( [START_REF] Altermann | Defining biominerals and organominerals: direct and indirect indicators of life[END_REF] and [START_REF] Défarge | Defining organominerals: comments on 'Defining biominerals and organominerals: direct and indirect indicators of life[END_REF]). Moreover they repeat this cavalier attitude in their reply, in which the title of Défarge et al.'s (2009) comment is cited without its incipit "Defining organominerals: [Comments on 'Defining biominerals and organominerals: direct and indirect indicators of life']". Apparently Perry and Sephton find it hard to accept that organominerals have been highlighted, named and defined before them.

As in their original paper [START_REF] Perry | Defining biominerals and organominerals: direct and indirect indicators of life[END_REF], the examples chosen by [START_REF] Perry | Reply to comments on defining biominerals and organominerals: direct and indirect indicators of life[END_REF] to illustrate their 'new' definition of organominerals are unconvincing: glauconite mineral, which is cited as an example of organomineral, can be considered so only when the existence of an organic control on mineral formation has been proved. Glauconite in general is no more an organomineral than is, e. g., calcite, which is represented in nature both by organominerals and biominerals, sometimes within the same sediment ( [Défarge & Trichet, 1995] and [START_REF] Défarge | Texture of microbial sediments revealed by cryo-scanning electron microscopy[END_REF]), and by minerals formed through inorganic processes.

Finally, [START_REF] Perry | Reply to comments on defining biominerals and organominerals: direct and indirect indicators of life[END_REF] abruptly claim, without any discussion of former definitions of organominerals or of critical remarks addressed to [START_REF] Perry | Defining biominerals and organominerals: direct and indirect indicators of life[END_REF], that the term 'organomineral' is "now newly defined" (by [START_REF] Perry | Defining biominerals and organominerals: direct and indirect indicators of life[END_REF], although both [START_REF] Altermann | Defining biominerals and organominerals: direct and indirect indicators of life[END_REF] and [START_REF] Défarge | Defining organominerals: comments on 'Defining biominerals and organominerals: direct and indirect indicators of life[END_REF] have shown that Perry et al.'s paper does not contain a single clear definition of these objects. In accord with [START_REF] Altermann | Defining biominerals and organominerals: direct and indirect indicators of life[END_REF], we persist in considering that Perry et al.'s (2007) 'new' definition of organominerals is unnecessary, unfounded, and introduces confusion that is detrimental to a fine understanding of geobiological and astrobiological processes and products. In these scientific domains, the only valid definition of organominerals that has proved useful until now remains that of (Défarge & Trichet, 1995) and (Trichet & Défarge, 1995), as recapped and specified by [START_REF] Défarge | Defining organominerals: comments on 'Defining biominerals and organominerals: direct and indirect indicators of life[END_REF] and (Défarge, in press).


Guillaume Tomasini
email: tomasini@math.u-strasbg.fr

In this article, we study the decomposition of weight-sl 2n -modules of degree 1 to a dual pair (sl 2 , sl n ). We show that in some generic cases we have an explicit branching rule leading to a Howe-type correspondence between simple highest weight modules. We also give a Howe-type correspondence in the non-generic case. This latter involves some (non simple) Verma modules.

. All these articles deal with the minimal representation of some Lie algebra or Lie group.

. The vector space of the representation is some kind of polynomial algebra and the action is via differential operators.

The correspondence is completely explicit (see theorems 3.3, 3.7 and 3.8).

In the first part of this article we give the construction of the representation and some of its properties. The second part is devoted to the description of the dual pair (sl 2 , sl n ) of sl 2n and its action on the representation of the first part. In the last part we prove the Howe-type correspondence for this module with respect to our dual pair.

Acknowledgements.-I thank gratefully Professor H. Rubenthaler for many helpful conversations and valuable comments concerning the writing of this article.

Simple weight-modules of degree 1

Let m be a positive integer greater than 1. Let g denote the complex Lie algebra sl m of traceless m × m matrices. Let h denote its standard Cartan subalgebra, consisting of traceless diagonal matrices. In [START_REF] Benkart | Modules with bounded weight multiplicities for simple Lie algebras[END_REF], Benkart, Britten and Lemire described all the simple infinite dimensional weight g-modules of degree 1. Recall that a weight module is a module for which the action of h is semisimple with finite multiplicities. A weight module is of degree 1 if all its non-trivial weight spaces are 1-dimensional.

The definition of the representations which we are interested in uses the Weyl algebra W m which is the associative algebra with generators q i and p i for i ∈ {1, . . . , m} subject to the relations [q i , q j ] = 0 = [p i , p j ] and [p j , q i ] = δ i,j .

Let a ∈ C m . Set P a = {b ∈ C m such that b i -a i ∈ Z for all i ∈ {1, . . . , m} and b i < 0 ⇐⇒ a i < 0}.

We consider the following vector space W (a) := b∈Pa Cx(b) whose basis elements x(b) are parametrized by the set P a . Now we give a structure of W m -module to W (a). To do so, we should think of the element x(b) as a monomial x b1 1 • • • x bm m and of the q i and p i as operators of multiplication and derivation. Let ǫ i denote the m-tuple whose all entries are 0 except i-th entry which is 1. We then define the (almost) natural representation of W m on W (a) by

q i • x(b) = (b i + 1)x(b + ǫ i ) if b i ∈ Z - x(b + ǫ i ) otherwise ( 1a
)
p j • x(b) = x(b -ǫ j ) if b j ∈ Z - b j x(b -ǫ j ) otherwise (1b)
From theorem 2.9 in [START_REF] Benkart | Modules with bounded weight multiplicities for simple Lie algebras[END_REF], we know that the W m -module W (a) is simple.

We can embed g (and more generally gl m ) into W m by sending the elementary matrix E i,j to q i p j . We now restrict our set of parameters. Set P a := {b ∈ P a : A m-uple b ∈ P a is called admissible and a vector x(b) associated to b ∈ P a is called an admissible vector. Now we have the following Theorem 1.1 (Benkart, Britten, Lemire). [1, proposition 2.12, theorem 5.8]

i b i = i a i }
The vector subspace N (a) of W (a) is a simple weight g-module of degree 1.

Conversely if M is an infinite dimensional simple weight g-module of degree 1, then there exists a = (a 1 , . . . , a m ) ∈ C m , there exist two integers k and l with -

a i = -1 for i = 1, . . . , k -1, -a i ∈ C \ Z for i = k, . . . , l, -and a i = 0 for i = l + 1, . . . , m, such that M ∼ = N (a).
Recall that a g-module M is cuspidal if the action on M of E i,j is injective for all (i, j) with i = j. Using the theorem 1.1 and equations [START_REF] Benkart | Modules with bounded weight multiplicities for simple Lie algebras[END_REF], one shows that the cuspidal simple weight g-modules of degree 1 are those N (a) for which a ∈ (C -Z) m .

From now on, we suppose m = 2n with n > 1. In what follows we consider only the modules of the form

N (-1, . . . , -1 n-1 , a 1 , a 2 , 0, . . . , 0 n-1 ) with a 1 , a 2 ∈ C -Z.
We denote N a1,a2 this module for short. We set a = (-1, . . . , -1, a 1 , a 2 , 0, . . . , 0). Remark that if b is admissible then b i < 0 if i < n and b j ≥ 0 if j > n + 1. We let α 1 , . . . , α 2n-1 denote the standard set of simple roots for the root system R of (g, h). Then the conditions a 1 and a 2 ∈ C -Z ensure that the root vectors X ±αn act injectively on N a1,a2 . From the action of q i and p j on W (a) we derive the action of g on N a1,a2 . For example we have:

X -αi • x(b) =    (b i+1 + 1)x(b -ǫ i + ǫ i+1 ) if i < n -1 x(b -ǫ n-1 + ǫ n ) if i = n -1 b i x(b -ǫ i + ǫ i+1 ) if i ≥ n (2a) X αi • x(b) =    (b i + 1)x(b + ǫ i -ǫ i+1 ) if i < n -1 b n (b n-1 + 1)x(b + ǫ n-1 -ǫ n ) if i = n -1 b i+1 x(b + ǫ i -ǫ i+1 ) if i ≥ n (2b) H αi • x(b) = (b i -b i+1 )x(b) (2c)
Remark from this action that the root vectors X αi with i = n act trivially on elements of the form x(-1, . . . , -1, a 1 + b, a 2 -b, 0, . . . 0), with b ∈ Z.

Highest weight vectors for the action of b on N a1,a2

Set θ := {α 1 , . . . , α 2n-1 } \ {α n } and let θ denote the roots in R which are linear combinations of the simple roots in θ. To each root α ∈ R, we denote by g α the associated root space. Then we associate the following Levi subalgebra l θ := h ⊕ ⊕ α∈ θ g α . This Levi subalgebra is the following set of matrices:

l θ = A 0 0 B , A, B ∈ M n (C), tr(A + B) = 0 .
Remark that the semisimple part of l θ is the sum of two copies of sl n . We denote by l + θ the set of all the matrices A 0 0 B in l θ with A and B upper triangular with zeros on the diagonal. We denote by h(θ) the Cartan subalgebra of this semisimple part consisting of diagonal matrices. From this decomposition we can construct a dual pair (a, b) (which is C-admissible in the terminology of [START_REF] Rubenthaler | Les paires duales dans les algèbres de Lie réductives[END_REF]). Define

b := A 0 0 A , with A ∈ sl n .
Let h n denote the following Cartan subalgebra of b:

D 0 0 D , with D diagonal and traceless.
The commutant of b is easily seen to be a := X, H, Y , where

X = n i=1 X -(αi+•••+αn+i-1) , Y = n i=1 X αi+•••+αn+i-1 ,
and

H = H α1 + 2H α2 + • • • + nH αn + (n -1)H αn+1 + • • • + 2H α2n-2 + H α2n-1 .
The Lie algebra a is isomorphic to sl 2 while b is isomorphic to sl n . It is easy to see that (a, b) is a dual pair. Note that b ⊂ l θ and that the center of l θ is the Cartan subalgebra h θ := CH of a.

Now we describe the action of l θ on the module N a1,a2 .

Proposition 2.1. As a l θ -module, the module N a1,a2 decomposes into a direct sum of infinite dimensional simple highest weight modules. Moreover the decomposition is multiplicity free and the highest weight vectors are the x(-1, . . . , -1, 1. The action of E n,n+1 and E n+1,n on M is injective.

a 1 + b, a 2 -b, 0, . . . , 0) for b ∈ Z, whose highest weight under h(θ)×h θ is (0, . . . , 0 n-2 , -1- a 1 -b, a 2 -b, 0, . . . , 0 n-2 ) ⊗ (-(n -1) + a 1 -a 2 + 2b).
2. As a l θ -module, M is a direct sum of simple highest weight modules.

See [START_REF] Tomasini | On a generalisation of Bernstein-Gelfand-Gelfand category O[END_REF] for a general definition of the categories O ∆,θ , and their properties.

For later use, let us compute the action of the root vectors occuring in X and Y ∈ a: Lemma 2.3. For i ∈ {1, . . . , n}, we have

X αi+...+αn+i-1 x(b) = (b i + 1)b n+i x(b -ǫ n+i + ǫ i ) if i < n, b 2n x(b -ǫ 2n + ǫ n ) if i = n, (3a) X -(αi+...+αn+i-1) x(b) = x(b + ǫ n+i -ǫ i ) if i < n, b n x(b + ǫ 2n -ǫ n ) if i = n, ( 3b
)
Proof. The root vector X αi+...+αn+i-1 correspond to the elementary matrix E i,n+i . The root vector X -(αi+...+αn+i-1) correspond to the elementary matrix E n+i,i . Now the lemma follows from equations [START_REF] Benkart | Modules with bounded weight multiplicities for simple Lie algebras[END_REF].

⊓ ⊔

The first step toward a correspondence is to understand the action of b on N a1,a2 . We set b + the subalgebra of b consisting of the matrices A 0 0 A in b such that A is upper triangular with zero diagonal. From proposition 2.1, we obtain that the action of b + is locally finite. We will now investigate the subspace

M 0 := N b + a1,a2 = {x ∈ N a1,a2 : X • x = 0, ∀ X ∈ b + }. Define X i := X αi + X αn+i , X -i := X -αi + X -αn+i , and
H i = [X i , X -i ].
From equation (2b) we get

X i •x(b) = (b i + 1)x(b + ǫ i -ǫ i+1 ) + b n+i+1 x(b + ǫ n+i -ǫ n+i+1 ) if i < n -1 b n (b n-1 + 1)x(b -ǫ n + ǫ n-1 ) + b 2n x(b + ǫ 2n-1 -ǫ 2n ) if i = n -1 (4) Lemma 2.4. Let x = k λ k x(b k ) ∈ M 0 be a weight vector for h θ ⊕ h n . Then there exist indices k 0 and k 1 such that x(b k0 ) = x(-1, . . . , -1, b k0 n , . . . , b k0 2n ) and x(b k1 ) = x(b k1 1 , . . . , b k1 n+1 , 0, . . . , 0). Proof. If b is admissible, then b 1 < 0. Let i 1 be an index such that λ i1 = 0 and b i1 1 is maximal among the possible values of the different b k 1 occuring in x. Suppose b i1 1 = -1. Then applying X 1 to x(b i1
) gives according to equation ( 4) the following sum of two vectors:

(b i1 1 + 1)x(b i1 + ǫ 1 -ǫ 2 ) + b i1 n+2 x(b i1 + ǫ n+1 -ǫ n+2 ).
The first summand is a vector

x(b ′ ) such that b ′ 1 = b i1 1 + 1 > b i1 1 . The second summand is of the form x(b ′′ ) with b ′′ 1 = b i1 1 .
But by our hypothesis on x, we have X 1 •x = 0. We look at the other occurences of the vector x(b ′ ) in X 1 •x. By the maximality of b i1

1 , this vector only occur as the first summand of

X 1 • x(b i1 ). Thus its coefficient in X 1 • x is b i1 1 + 1 which is non zero by our hypothesis on b i1 1 , contradicting the fact that x ∈ M 0 . Hence b i1 1 = -1. If b is admissible, then we also have b 2 < 0. So let now i 2 denote an index such that λ i2 = 0, b i2 1 = -1 and b i2 2 is maximal among the possible values of the different b k
2 occuring in x and subject to the condition that b k 1 = -1. We apply the same reasonning using equation ( 4) for the action of X 2 to prove that b i2 2 = -1. Applying then X 3 , . . . X n-1 we get an index k 0 satisfying the condition of the lemma, i.e. b k0

1 = • • • = b k0 n-1 = -1. If b is admissible, then b 2n ≥ 0.
Therefore to find k 1 we do the same thing starting from the action of X n-1 to a vector x(b j1 ) such that λ j1 = 0 and b j1 2n is minimal among the possible b k 2n . We prove that necessarily b j1 2n = 0. Applying successively X n-2 , . . . , X 1 we obtain an index k 1 satisfying the condition of the lemma, i.e. b k1

2n = • • • = b k1 n+1 = 0. ⊓ ⊔
Corollary 2.5. Let the notations be as in lemma 2.4. Then there are integers c n , c n+1 , c 2n with c 2n ≥ 0 such that

x(b k0 ) = x(-1, . . . , -1, a 1 + c n , a 2 + c n+1 , 0, . . . , 0, c 2n ) and x(b k1 ) = x(-1 -c ′ 1 , -1, . . . , -1, a 1 + c ′ n , a 2 + c ′ n+1 , 0, . . . , 0), with c n + c n+1 + c 2n = 0, c ′ 1 = c 2n , c ′ n = c n + c 2n and c ′ n+1 = -c n . Proof. Set b k0 n = a 1 + c n , b k0 n+1 = a 2 + c n+1 , b k0 n+i = c n+i for 2 ≤ i ≤ n. Set also b k1 n = a 1 + c ′ n , b k1 n+1 = a 2 + c ′ n+1 , b k1 i = -1 -c ′ i for 1 ≤ i ≤ n -1. Therefore, b k0 = (-1, . . . , -1, a 1 + c n , a 2 + c n+1 , c n+2 , . . . , c 2n ), b k1 = (-1 -c ′ 1 , . . . , -1 -c ′ n-1 , a 1 + c ′ n , a 2 + c ′ n+1 , 0, . . . , 0
). The two vectors x(b k0 ) and x(b k1 ) should be admissible vectors and should have the same weight with respect to the Cartan subalgebras of a (since it commutes with b) and to the Cartan subalgebra of b (generated by H 1 , . . . , H n-1 ). This gives rise to the following equations admissibility of x(b k0 ) :

c n + c n+1 + . . . + c 2n = 0 admissibility of x(b k1 ) : -(c ′ 1 + . . . + c ′ n-1 ) + c ′ n + c ′ n+1 = 0 h θ -weight of x(b k0 ) and x(b k1 ) : c n -(c n+1 + . . . + c 2n ) = -(c ′ 1 + . . . + c ′ n-1 ) + c ′ n -c ′ n+1 weight of x(b k0 ) and x(b k1 ) under H 1 , . . . , H n-1 :            c n+1 -c n+2 c n+2 -c n+3 c 2n-2 -c 2n-1 -c n + c 2n-1 -c 2n = = . . . = = c ′ 2 -c ′ 1 + c ′ n+1 c ′ 3 -c ′ 2 c ′ n-1 -c ′ n-2 -c ′ n-1 -c ′ n Some calculations show that the unique solution in the c ′ i 's variables of this system is        c ′ 1 = -c n -c n+1 c ′ i = -c n+i , 1 < i < n c ′ n = c 2n + c n c ′ n+1 = -c n
But now as the vectors x(b k0 ) and x(b k1 ) must be admissible, we should have

c n+i ≥ 0 for 2 ≤ i ≤ n and c ′ j ≥ 0 for 1 ≤ j ≤ n -1. This imposes that c n+i = c ′ i = 0 for 2 ≤ i ≤ n -1. Then taking into account that c n + c n+1 + • • • + c 2n = 0
again by admissibity of b k0 , we finally obtain

c n+1 = -c n -c 2n , c 2n ≥ 0, c ′ 1 = c 2n , c ′ n = c n + c 2n , c ′ n+1 = -c n (5)

⊓ ⊔

From now on, we shall write x(b k0 ) = x(-1, . . . , -1, a 1 +b, a 2 -b-c, 0, . . . , 0, c) where b and c are integers and c ≥ 0. Let us investigate which admissible vectors x(b) have the same weight with respect to h θ ⊕ h n than x(b k0 ). This is the following lemma:

Lemma 2.6. Set x := x(-1-b 1 , . . . , -1-b n-1 , a 1 +b n , a 2 +b n+1 , b n+2 , . .

. , b 2n ) an admissible vector. Then x has the same weight than x(b k0 ) under the action of h θ ⊕ h n if and only if

       b n+i = b i , 2 ≤ i ≤ n -1 b n = b + b 1 + • • • + b n-1 b n+1 = b 1 -(b + c) b 2n = c -(b 1 + • • • + b n-1 )
Proof. As in the proof of corollary 2.5, we write down the equations obtained by expressing the admissibility of x and the fact that x and x(b k0 ) have the same weight under h θ and h n :

-(b 1 + • • • + b n-1 ) + b n + • • • + b 2n = 0 (6) -(b 1 + . . . + b n-1 ) + b n -(b n-1 + • • • + b 2n ) = 2b (7)            b 2 -b 1 + b n+1 -b n+2 b 3 -b 2 + b n+2 -b n+3 b n-1 -b n-2 + b 2n-2 -b 2n-1 -b n-1 -b n + b 2n-1 -b 2n = = . . . = = -b -c 0 0 -b -c (8)
Then we set bi = b i -b n+i for 1 ≤ i ≤ n -1 and bn = b n + b 2n . We rewrite equations ( 6) and ( 8) in the new variables bi :

-( b1 + • • • + bn-1 ) + bn = 0            b2 -b1 b3 -b2 bn-1 -bn-2 -bn-1 -bn = = . . . = = -b -c 0 0 -b -c
The unique solution of this system in the bi 's variables is b2 =

• • • = bn-1 = 0, b1 = b + c, bn = b + c. Therefore, we have b n + b 2n = c + b, b n+1 = b 1 -c -b, and b n+i = b i for 2 ≤ i ≤ n -1.
Then using equation [START_REF] Rubenthaler | Les paires duales dans les algèbres de Lie réductives[END_REF], we express b n+i for i ≥ 0 in the b j 's variables for 1 ≤ j ≤ n -1, which gives the lemma.

⊓ ⊔ Corollary 2.7. Let x ∈ M 0 be a weight vector with respect to h θ ⊕ h n . Then there are two integer b and c such that c ≥ 0 and

x = ki≥0, |k|≤c λ k x(-1-k 1 , . . . , -1-k n-1 , a 1 +b+|k|, a 2 -b-c+k 1 , k 2 , . . . , k n-1 , c-|k|),
where

k = (k 1 , . . . , k n-1 ) ∈ N n-1 , |k| = i k i and λ k ∈ C. If n > 2, its h n -weight is (a 2 -b-c, 0, . . . , 0, -1-a 1 -b-c) and its h θ -weight is a 1 -a 2 + 2b -(n -1). If n = 2 then its h n -weight is (-1 -a 1 + a 2 -2(b + c)) and its h θ -weight is a 1 -a 2 + 2b -1.
Proof. From lemma 2.4 and corollary 2.5 we know that x = i λ i x(b i ) and that there is an index i 0 such that b i0 = (-1, . . . , -1, a 1 + b, a 2 -b -c, 0, . . . , 0, c) for some integers b and c with c ≥ 0. Then the lemma 2.6 asserts that the others

x(b i ) occuring in x are of the form x(b i ) = x(-1 -k 1 , . . . , -1 -k n-1 , a 1 + b + (k 1 + • • • + k n-1 ), a 2 + k 1 -b -c, k 2 , . . . , k n-1 , c -(k 1 + • • • + k n-1 )
). These vectors should also be admissible. Therefore we must have

k i ∈ N, and c -(k 1 + • • • + k n-1 ) ≥ 0.
This is the corollary.

⊓ ⊔ Proposition 2.8. Let x be as in the corollary 2.7. Write

x = ki≥0, |k|≤c λ k x(-1-k 1 , . . . , -1-k n-1 , a 1 +b+|k|, a 2 -b-c+k 1 , k 2 , . . . , k n-1 , c-|k|),
for some integers b and c with c ≥ 0. Then λ k = κ(k)λ 0 , where λ 0 ∈ C and

κ(k) = k 1 + k 2 k 1 • • • k 1 + • • • + k n-1 k 1 + • • • + k n-2 k1+•••+kn-1 j=1 (c + 1 -j) (k 1 + • • • + k n-1 )! k1+•••+kn-1 j=1 (a 1 + b + j) . Conversely, if x = ki≥0, |k|≤c λ k x(-1-k 1 , . . . , -1-k n-1 , a 1 +b+|k|, a 2 -b-c+k 1 , k 2 , . . . , k n-1 , c-|k|) with λ k = κ(k)λ 0 , then x ∈ M 0 .
Proof. From equation ( 4) we have

X 1 • x = k λ k -k 1 x(-k 1 , -2-k 2 , -1-k 3 , . . . , -1-k n-1 , a 1 +b+|k|, a 2 -b-c+k 1 , k 2 , . . . , k n-1 , c-|k|) +k 2 x(-1-k 1 , . . . , -1-k n-1 , a 1 +b+|k|, a 2 -b-c+k 1 +1, k 2 -1, k 3 , . . . , k n-1 , c-|k|) . Let k ∈ {(k i ) 1≤i≤n-1 : k i ≥ 0 and i k i ≤ c}. Suppose k 1 > 0. Let k ′ = (k ′ i ) 1≤i≤n-1 be such that k ′ 1 = k 1 -1, k ′ 2 = k 2 + 1, and k ′ i = k i otherwise. We look at the coefficient of x(-k 1 , -2-k 2 , -1-k 3 , . . . , -1-k n-1 , a 1 +b+|k|, a 2 -b-c+k 1 , k 2 , . . . , k n-1 , c-|k|) in the expression of X 1 • x. We find -k 1 λ k + (k 2 + 1)λ k ′ . As x ∈ M 0 , we have X 1 •x = 0. Therefore we should have -k 1 λ k +(k 2 +1)λ k ′ = 0, i.e. λ k = k2+1 k1 λ k ′ . By induction we find that λ k = k1+k2 k1 λ k 1 where k 1 = (0, k 1 + k 2 , k 3 , . . . , k n-1
). We then look at the coefficient of

x(-1, -k 2 -k 1 , -2-k 3 , -1-k 4 , . . . , -1-k n-1 , a 1 +b+|k|, a 2 -b-c+k 1 , k 2 , . . . , k n-1 , c-|k|) in X 2 • x(k 1
). This allows us to express λ k 1 from λ k 2 where k 2 = (0, 0, k 1 + k 2 + k 3 , k 4 . . . , k n-1 ). More precisely, we get λ k 1 = k1+k2+k3 k1+k2 λ k 2 . Then using successively the action of X 3 , . . . , X n-1 on x, we express λ k from λ 0 .

The converse is easy. ⊓ ⊔ From now on, we denote by x(b, c) the vector in M 0 obtained in proposition 2.8 such that λ 0 = 1. We also denote by

x k (b, c) = x(-1-k 1 , . . . , -1-k n-1 , a 1 +b+|k|, a 2 -b-c+k 1 , k 2 , . . . , k n-1 , c-|k|) for k = (k 1 , . . . , k n-1 ) such that k i ∈ N and |k| = k 1 + • • • + k n-1 ≤ c. Corollary 2.9. For 2 ≤ i ≤ n -2 we have X -i • x(b, c) = 0.
Proof. From equation (2b), we see that X -i acts trivially on x k (b, c) if and only if k i = k i+1 = 0. If the action is non trivial, we have

X -i • x k (b, c) = -(k i+1 )x k ′ (b, c) + k i x k ′′ (b, c) where k ′ j =    k j if j = i or i + 1 k i -1 if j = i k i+1 + 1 if j = i + 1
, and

k ′′ j =    k j if j = n + i or n + i + 1 k n+i -1 if j = n + i k n+i+1 + 1 if j = n + i + 1
.

We now look at the occurences of x k ′ (b, c) in X -i • x(b, c). It appears in the expression of X -i •x k (b, c) as we already mentionned and in the second summand of the expression of X -i • x l (b, c) where

l j =    k j if j = n + i or n + i + 1 k n+i + 1 if j = n + i k n+i+1 -1 if j = n + i + 1 . Therefore the coefficient of x k ′ (b, c) in X -i • x(b, c) is (-k i+1 )λ k + l i λ l .
Using the expression of λ k given by proposition 2.8, we see that this coefficient is 0. ⊓ ⊔

A Howe type correspondence for N a1,a2

Generic case

Now to state and prove a Howe-type correspondence for the module N := N a1,a2 we need to compute the action of a on M 0 . Lemma 3.1. Let b and c be two integers with c ≥ 0. Then 1. X • x(b, c) is a non zero element of M 0 which is equal to a multiple of x(b -1, c + 1).

The vector Y •x(b, c) is non zero if and only if c(a

1 -a 2 +2b+c-(n-2)) = 0.
In this case, it is equal to a multiple of x(b + 1, c -1).

Proof. As the action of a commutes with the action of b, we get that Y • x(b, c) and X • x(b, c) both are in M 0 . Let us set b ′ = b + 1 and c ′ = c -1. Then using lemma 2.3, we compute Y • x(b, c). We obtain

Y • x(b, c) = λ k (a 2 -b ′ -c ′ + k 1 )(-k 1 )x(-k 1 , -1 -k 2 , . . . , -1 -k n-1 , a 1 + b ′ + |k| -1, a 2 -b ′ -c ′ + k 1 -1, k 2 , . . . , k n-1 , c ′ -|k| + 1) + n-2 i=2 λ k (-k 2 i )x(-1-k 1 , -1-k 2 , . . . , -1-k i-1 , -k i , -1-k i+1 , . . . , -1-k n-1 , a 1 +b ′ +|k|-1, a 2 -b ′ -c ′ + k 1 , k 2 , . . . , k i-1 , k i -1, k i+1 , . . . , k n-1 , c ′ -|k| + 1) + λ k (c ′ +1-|k|)x(-1-k 1 , . . . , -1-k n-1 , a 1 +b ′ +|k|, a 2 -b ′ -c ′ +k 1 , k 2 , . . . , k n-1 , c ′ -|k|) As Y •x(b, c) ∈ M 0 , it should be a linear combination of some x(b ′′ , c ′′ ). But each x(b ′′ , c ′′ ) contains a vector of the form x 0 (b ′′ , c ′′ ) = x(-1, . . . , -1, a 1 + b ′′ , a 2 - b ′′ -c ′′ , 0, . . . , 0, c ′′ ). The only such vector in the expression of Y • x(b, c) is x 0 (b ′ , c ′ ). Therefore if Y • x(b, c) is non zero then it is a multiple of x(b ′ , c ′ ).
To see when it is zero, it is enough to compute the coefficient of x 0 (b ′ , c ′ ) in the above equation. The first sum gives a contribution equal to -(a 2 -b -c + 1)λ ǫ1 , the second sum gives n-2 i=2 -λ ǫi and the last sum gives c (recall that the coefficient λ 0 of x(b, c) was set equal to 1). Using proposition 2.8 which allows us to express the λ ǫi 's, we find the global contribution:

c a 1 + b + 1 (a 1 -a 2 + 2b + c -(n -2)). Thus Y • x(b, c) = 0 if and only if c(a 1 -a 2 + 2b + c -(n -2)) = 0.
We apply the same method for X • x(b, c).

The coefficient of x 0 (b -1, c + 1) in the expression of X • x(b, c) is a 1 + b, which is non zero since a 1 ∈ C \ Z.
1. If n = 2, N a1,a2 = b∈Z L(-1 -a 1 + a 2 -2b) ⊗ L(a 1 -a 2 + 2b -1). 2. If n > 2, N a1,a2 = b∈Z L(a 2 -b, 0, . . . , 0, -1 -a 1 -b) ⊗ L(a 1 -a 2 + 2b -(n -1)).
Proof. We know from proposition 2.1 that the module N a1,a2 is l + θ -locally finite. Thus it is also b + -locally finite. Therefore for every vector v in N a1,a2 there is an element u of U(b + ) such that u • v is in M 0 . From corollary 3.2, we know that each weight vector in M 0 spans a simple highest weight b-module. Thus N a1,a2 = ⊕ b∈Z,c∈N U(b)x(b, c) is the decomposition of N a1,a2 into simple b-modules. The hypothesis a 1 -a 2 ∈ Z ensures that Y • x(b, c) = 0 if and only if c = 0. Then we get the following chain of b-modules :

0 ⇀ U(b)x(b, 0) ⇋ U (b)x(b -1, 1) ⇋ • • • ⇋ U (b)x(b -k, k) ⇋ • • • ,
where ⇀ stands for the action of X and ↽ for the action of Y . Thanks to lemma 3.1, we conclude that this chain is a simple a ⊕ b-module which is then by corollary 2.7 isomorphic to L(-1 -

a 1 + a 2 -2b) ⊗ L(a 1 -a 2 + 2b -1) if n = 2 and to L(a 2 -b, 0, . . . , 0, -1 -a 1 -b) ⊗ L(a 1 -a 2 + 2b -(n -1)) if n > 2.
⊓ ⊔

As a consequence of this theorem, we find the following Howe-type correspondence in the ′′ generic ′′ case a 1 -a 2 ∈ Z, namely:

L(-1 -a 1 + a 2 -2b) ↔ L(a 1 -a 2 + 2b -1), if n = 2, L(a 2 -b, 0, . . . , 0, -1 -a 1 -b) ↔ L(a 1 -a 2 + 2b -(n -1)), if n > 2.

Non-generic case

Let us now consider the non-generic case a 1 -a 2 ∈ Z. From lemma 3.1, we find that Y • x(b, c) = 0 if and only if c = 0 or a 1 -a 2 + 2b + c = n -2 and that X • x(b, c) is always non zero. We also get that the a-module generated by x(b, 0) is a highest weight module of highest weight a 1 -a 2 + 2b -(n -1). As a vector space, this module is k∈N Cx(b -k, k). The vector x(b -k, k) for k > 0 is annihilated by Y ∈ a if and only if a 1 -a 2 + 2b -k = n -2. Therefore there is at most one k for which x(b -k, k) is a highest weight vector for a. Thus we have shown the following:

Corollary 3.4. Assume a 1 -a 2 ∈ Z. Let b ∈ Z. 1. If a 1 -a 2 + 2b -(n -2) ≤ 0 then the a-module generated by x(b, 0) is irreducible. 2. If a 1 -a 2 + 2b -(n -2) > 0, then the a-module generated by x(b, 0) has length 2: U(a)x(b, 0) ⊃ U(a)x(b-(a 1 -a 2 +2b-(n-2)), a 1 -a 2 +2b-(n-2)) ⊃ {0},
where

U(a)x(b -(a 1 -a 2 + 2b -(n -2)), a 1 -a 2 + 2b -(n -2)) is a simple highest weight a-module (of weight a 1 -a 2 + 2b -(n -1)
) and the quotient

U(a)x(b, 0)/U(a)x(b -(a 1 -a 2 + 2b -(n -2)), a 1 -a 2 + 2b -(n -2))
is a simple highest weight a-module of weight a 1 -a 2 + 2b -(n -1).

In this case the a-module U(a)x(b, 0) is isomorphic to the Verma module

V (a 1 -a 2 + 2b -(n -1)) of highest weight a 1 -a 2 + 2b -(n -1).
We can now use the same method as in corollary 3.2 and prove the following result:

Corollary 3.5. Assume a 1 -a 2 ∈ Z. Let b ∈ Z such that a 1 -a 2 +2b-(n-2) ≥ 0. Then for all c ∈ N, the b-module generated by x(b, c) is a simple highest weight b-module of weight (a 2 -b -c, 0, • • • , 0, -1 -a 1 -b -c)
Proof. The proof of corollary 3.2 can also be applied in this case because the hypothesis on b together with corollary 3.4 ensures that Y • x(b, c) is non zero as soon as c > 0.

⊓ ⊔

In general, the same argument shows that the only vector x(b, c ′ ) that can belong to the b-module generated by x(b, c) satisfies c ′ > c and a 1 -a 2 + 2b + c + c ′ = n -2 (see the proof of corollary 3.2). Now we prove the following

Proposition 3.6. Let b ∈ Z such that a 1 -a 2 + 2b -(n -2) < 0. Let c ∈ N be such that a 1 -a 2 + 2b + c -(n -2) = 0. Then the b-module U(b)x(b, c) is contained in the b-module U(b)
x(b, 0). Moreover, the latter has length 2, with the following composition serie:

U(b)x(b, 0) ⊃ U(b)x(b, c) ⊃ {0}.
Proof. To prove the first assertion, it suffices to find u ∈ U(b) such that u • x(b, 0) = αx(b, c), with α ∈ C * . We define the following element. Let Z ′ ∈ b be the element corresponding to the matrix E n,1 + E 2n,n+1 and let

Z ′′ = n-2 i=1 (E i+1,1 + E n+i+1,n+1 ) (E n,i+1 + E 2n,n+i+1 ) ∈ U(b).
Remark that [Z ′ , Z ′′ ] = 0. For λ ∈ C, we denote by Z λ the vector Z ′ + λZ ′′ . Then [Z λ , Z λ ′ ] = 0 for all λ, λ ′ ∈ C. We list now some computations of brackets :

[X 1 , Z λ ] = (E n,2 + E 2n,n+2 ) (λ(n -2) -1 + λH 1 ) -λ n-2 i=2 (E n,i+1 + E 2n,n+i+1 ) (E i+1,2 + E n+i+1,n+2 ) , [X i , Z λ ] =0, for 2 ≤ i ≤ n -2, [X n-1 , Z λ ] = (E n-1,1 + E 2n-1,n+1 ) (1 + λH n-1 ) + λ n-3 i=1 (E i+1,1 + E n+i+1,n+1 ) (E n-1,i+1 + E 2n-1,n+i+1 ) , [E n,2 + E 2n,n+2 , Z λ ] =λZ ′ (E n,2 + E 2n,n+2 ) , [H 1 , Z λ ] = -Z λ , [E i+1,2 + E n+i+1,n+2 , Z λ ] =0, [E n-1,1 + E 2n-1,n+1 , Z λ ] = -λZ ′ (E n-1,1 + E 2n-1,n+1 ) , [H n-1 , Z λ ] = -Z λ , [E n-1,i+1 + E 2n-1,n+i+1 , Z λ ] =0. For 1 ≤ i ≤ c, set λ i = 1 a1+b+i (note that a 1 + b + i is non zero since a 1 ∈ C \ Z). Set also Z i = Z λi and Z = Z 1 • • • Z c . Now set x = Z • x(
b, 0). We show that x is a highest weight vector for b. We already know that x(b, 0) is a highest weight vector for b. Thus we have to show that ad(X i )(Z) • x(b, 0) = 0 for 1 ≤ i ≤ n -1. From the relations above, we already find that ad(X i )(Z) = 0 for 2 ≤ i ≤ n -2. Let us compute ad(X 1 )(Z) • x(b, 0). We obtain

ad(X 1 )(Z) • x(b, 0) = [X 1 , Z 1 ]Z 2 • • • Z c • x(b, 0) + • • •+ Z 1 • • • Z c-1 [X 1 , Z c ] • x(b, 0).
In the expression of [X 1 , Z i ], appear the vectors (E k+1,2 + E n+k+1,n+2 ). But we have seen that these vectors commute with all the Z j 's. Moreover from corollary 2.9, we get that (E k+1,2 + E n+k+1,n+2 ) acts trivially on x(b, 0). Therefore the only part in the expression of [X 1 , Z i ] that can give a non trivial contribution in the expression of ad(X 1 )(Z)

• x(b, 0) is (E n,2 + E 2n,n+2 ) (λ i (n -2) -1 + λ i H 1 ). Thus, ad(X 1 )(Z) • x(b, 0) = (E n,2 + E 2n,n+2 ) (λ 1 (n -2) -1 + λ 1 H 1 ) Z 2 • • • Z c • x(b, 0) + • • • + Z 1 • • • Z c-1 (E n,2 + E 2n,n+2 ) (λ c (n -2) -1 + λ c H 1 ) • x(b, 0).
From our previous computations we also get that

H 1 Z k • • • Z c = -(c + 1 -k)Z k • • • Z c + Z k • • • Z c H 1 .
Thus we have

ad(X 1 )(Z) • x(b, 0) =Z 2 • • • Z c (λ 1 (n -2) -1 + λ 1 H 1 -(c -1)λ 1 ) • x(b, 0) + • • • + Z 1 • • • Z c-1 (λ c (n -2) -1 + λ c H 1 ) • x(b, 0). Now from corollary 2.7, we get that H 1 • x(b, 0) = (a 2 -b)x(b, 0). Then using the definition of c, we conclude that (λ k (n -2) -1 + λ k H 1 -(c -k)λ k )•x(b, 0) = 0, which in turn expresses that ad(X 1 )(Z) • x(b, 0) = 0. Now since [Z λ , Z λ ′ ] = 0, we have also that Z = Z c • • • Z 1 .
Then we compute ad(X n-1 )(Z) • x(b, 0) using the same method as above and prove that ad(X n-1 )(Z) • x(b, 0) = 0. Therefore we have proved that Z • x(b, 0) is a highest weight vector for b (note that Z • x(b, 0) is a weight vector because Z is a weight vector in U(b)).

It only remains to show that Z • x(b, 0) = 0. To do so, we compute the coefficient of x 0 (b, c) in the expression of Z • x(b, 0). We have seen that Z ′ commute with Z ′′ . So Z is a homogeneous polynomial of degree c in the two variables Z ′ and Z ′′ . Remark that Z ′′ cannot increase the 2n-th component of the admissible vectors. Therefore the only monomial in the expression of Z that can give the admissible vector x 0 (b, c) when acting on x(b, 0) is Z ′c , whose coefficient in the polynomial Z is 1. After some computations we find that the coefficient of

x 0 (b, c) in Z • x(b, 0) is (a 2 -b)(a 2 -b -1) • • • (a 2 -b -(c -1)
). This is non zero since a 2 ∈ C \ Z. Therefore from proposition 2.8, we conclude that Z • x(b, 0) is a non zero multiple of x(b, c).

From our choice of c and lemma 3. ,c)), which has therefore a composition serie of finite length consisting of highest weight modules. But we have remarked that U(b)x(b, 0) cannot contain any other highest weight vector than the linear combinations of x(b, 0) and x(b, c) (see above the statement of this proposition). From this we conclude that

U(b)x(b, 0) ⊃ U(b)x(b, c) ⊃ {0}
is the composition serie of U(b)x(b, 0).

⊓ ⊔

We can now state and prove the following Howe-type correspondence in the non-generic case:

Theorem 3.7. Assume a 1 -a 2 ∈ Z and n > 2. Let b ∈ Z. Then we have the following correspondence: -If a 1 -a 2 + 2b -(n -2) = 0, then U(b)x(b, 0) is a simple b-module isomor- phic to L(a 2 -b, 0, . . . , 0, -1 -a 1 -b) and we have L(a 2 -b, 0, . . . , 0, -1 - a 1 -b) ↔ L(-1). -If a 1 -a 2 + 2b -(n -2) > 0, then U(b)x(b, 0) is a simple b-module isomor- phic to L(a 2 -b, 0, . . . , 0, -1 -a 1 -b) and we have L(a 2 -b, 0, . . . , 0, -1 - a 1 -b) ↔ V (a 1 -a 2 + 2b -(n -1)). -If a 1 -a 2 +2b-(n-2) < 0, then U(b)x(b, 0) is an indecomposable b-module of length 2 and we have U(b)x(b, 0) ↔ L(a 1 -a 2 + 2b -(n -1)).
Proof. This is a consequence of corollaries 3.4 and 3.5 and of proposition 3. Proof. This is a consequence of corollaries 3.4 and 3.5 and of proposition 3.6. ⊓ ⊔ Remark that in both case we have a correspondence of the infinitesimal caracteres. Let us give an interpretation of the theorem 3. Note that in this correspondence some a-modules are not simple modules. So we can consider the corresponding semisimplification N ss a1,a2 of N s a1,a2 (which we call the bi-semisimplification of N a1,a2 ). In this module, we get the following correspondence:

- L(a 1 -a 2 + 2b -(n -1)). Note that this is no more a one to one correspondence. We can also give an interpretation of theorem 3.8 in the same spirit. The final correspondence in the ′′ bi-semisimplification ′′ of N a1,a2 is in this case the following:

-If a 1 -a 2 + 2b = 0, then we have L(-1) ↔ L(-1).

-If a 1 -a 2 + 2b > 0, then we have L(a 2 -a 1 -2b -1) ↔ L(a 1 -a 2 + 2b -1) ⊕ L(-(a 1 -a 2 + 2b + 1)). -If a 1 -a 2 + 2b < 0, then we have L(a 2 -a 1 -2b -1) ↔ L(a 1 -a 2 + 2b -1).

Proof.Remark 2 . 2 .

 22 From equation (2b), we conclude that the only vectors annihilated by l + θ are exactly the admissible vectors x(b) with b = (-1, . . . , -1, a 1 + b n , a 2 + b n+1 , 0, . . . , 0) such that b n ∈ Z, b n+1 ∈ Z, and b n +b n+1 = 0. By using equation (2a) we check that for every x(b ′ ) there is an element u ∈ U(l + θ ) such that u•x(b ′ ) is a non zero multiple of some x(b) with b as above. Therefore the highest weight module U(l θ )x(b) corresponding to x(b) is in fact simple and N a1,a2 is the direct sum of the U(l θ )x(b) where b is of the form (-1, . . . , -1, a 1 + b n , a 2 -b n , 0, . . . , 0) for some integer b n . The weight of this x(b) is easily computed using equation (2c).⊓ ⊔ The modules N a1,a2 give the exact list of the simple modules in the category O ∆,θ (sl N ) of weight-sl 2n -modules M which satisfy the following:

Corollary 3 . 2 .Theorem 3 . 3 .

 3233 This gives the lemma. ⊓ ⊔ Assume a 1 -a 2 ∈ Z. Then for all integers b and c with c ≥ 0 the b-module generated by x(b, c) is a simple highest weight module. Proof. The b-module U(b)x(b, c) generated by x(b, c) is a highest weight module and is therefore indecomposable. Thus it is simple if and only if x(b, c) is the only highest weight vector in U(b)x(b, c), up to a scalar multiple. Another highest weight vector in U(b)x(b, c) would be of the form x(b ′ , c ′ ). But then the vectors x(b, c) and x(b ′ , c ′ ) would have the same h θ -weight. Using the h θ -weight given in corollary 2.7, we see that necessarily b ′ = b. Thus if x(b, c ′ ) ∈ U(b)x(b, c), then there is an element u ∈ U(b -) such that u • x(b, c) = x(b, c ′ ). This implies first that c ≤ c ′ since the vectors X -i can only increase the 2n-th component of every admissible vector. Hence if c = c ′ , we have c < c ′ . Then from the hypothesis a 1 -a 2 ∈ Z and from lemma 3.1, we get Y c+1 • x(b, c) = 0 and Y c+1 • x(b, c ′ ) = 0. As Y ∈ a commutes with u ∈ U(b), we should have uY c+1 x(b, c) = Y c+1 x(b, c ′ ). This is a contradiction. ⊓ ⊔ Let L(λ) denote the simple highest weight sl 2 -module with highest weight λ and L(a 2 -b, 0, . . . , 0, -1 -a 1 -b) denote the simple highest weight sl n -module with highest weight (a 2 -b, 0, . . . , 0, -1 -a 1 -b). Assume a 1 -a 2 ∈ Z. Then we have the following decomposition of N a1,a2 as a b ⊕ a-module:

1 ,

 1 we show that Y • x(b, c + k) = 0 for k > 0. Then we can apply the same proof as in 3.2 to show that U(b)x(b, c) is a simple b-module. Thus the b-module U(b)x(b, 0) is a highest weight module (containing the simple b-module U(b)x(b

Theorem 3 . 8 .

 38 Assume a 1 -a 2 ∈ Z and n = 2. Let b ∈ Z. Then we have the following correspondence:-Ifa 1 -a 2 + 2b = 0, then U(b)x(b, 0) is a simple b-module isomorphic to L(-1) and we have L(-1) ↔ L(-1). -If a 1 -a 2 + 2b > 0, then U(b)x(b, 0) is a simple b-module isomorphic to L(a 2 -a 1 -2b -1) and we have L(a 2 -a 1 -2b -1) ↔ V (a 1 -a 2 + 2b -1). -If a 1 -a 2 + 2b < 0, then U(b)x(b, 0) is an indecomposable b-module of length 2 isomorphic to V (a 2 -a 1 -2b-1) and we have V (a 2 -a 1 -2b-1) ↔ L(a 1 -a 2 + 2b -1).

7 .

 7 First we have the following decomposition of N a1,a2 as a b-module:N a1,a2 = b∈Z,c∈N U(b)x(b, c).But we have seen that each U(b)x(b, c) is either simple or has length 2. Let us then consider the semisimplification N s a1,a2 of N a1,a2 obtained by changing those U(b)x(b, c) which are indecomposable by their composition factors. The space N s a1,a2 is still a b-module (but not a g-module anymore) and we have the following branching rules:N s a1,a2 = b∈Z,c∈N L(a 2 -b -c, 0, . . . , 0, -1 -a 1 -b -c).But it is clear that N s a1,a2 still caries an action of a induced by the action of a on N a1,a2 . From theorem 3.7, we find the following Howe-type correspondence for this b ⊕ a-module N s a1,a2 :-If a 1 -a 2 + 2b -(n -2) = 0, then we have L(a 2 -b, 0, . . . , 0, -1 -a 1 -b) ↔ L(-1). -If a 1 -a 2 + 2b -(n -2) > 0, then we have L(a 2 -b, 0, . . . , 0, -1 -a 1 -b) ↔ V (a 1 -a 2 + 2b -(n -1)). -If a 1 -a 2 + 2b -(n-2) < 0, then we have L(a 2 -b, 0, . . . , 0, -1 -a 1 -b) ↔ L(a 1 -a 2 + 2b -(n -1)).

If a 1 -

 1 a 2 + 2b -(n -2) = 0, then we have L(a 2 -b, 0, . . . , 0, -1 -a 1 -b) ↔ L(-1). -If a 1 -a 2 + 2b -(n -2) > 0, then we have L(a 2 -b, 0, . . . , 0, -1 -a 1 -b) ↔ L(a 1 -a 2 + 2b -(n -1)) ⊕ L(-(a 1 -a 2 + 2b -(n -3)). -If a 1 -a 2 + 2b -(n -2) < 0, then we have L(a 2 -b, 0, . . . , 0, -1 -a 1 -b) ↔