Schéma : Poor graft function (PGF) after allogeneic hematopoietic cell transplantation (allo-HCT) is a misunderstood complication associated with poor outcome and limited therapeutic options. Despite the lack of standardized diagnostic criteria, PGF is commonly defined as follows: one or several significant cytopenias after allo-HCT persisting or developing after allo-HCT despite full donor chimerism and in the absence of relapse or other causes. Not only PGF can alter patients' quality of life by leading to recurrent transfusions, bleeding events and infections, but it is also associated with poor survival after allo-HCT.

Although PGF is relatively frequent, there is no well-codified behavior in the literature or in the recommendations issued by the various learned societies of transplantation.

The aim objective of the investigator's study is to demonstrate that eltrombopag improve PGF after allo-HCT

Study arm:
Experimental: eltrombopag
Eligible patients will receive the investigational drug eltrombopag
Eltrombopag at the starting dose of 50mg/day. After 2 weeks of eltrombopag initiation and in the absence of platelet response, eltrombopag will be increased every two weeks (50mg increase) up to a maximum dose of 150mg/day (2 maximum escalation from D1, with maximum dose escalation phase of 4 weeks).

Current primary outcome:
Platelet response [ Time Frame: at 12 weeks ]
Platelet response defined as a platelet count ≥ 20G/L at 12 weeks measured on at least two serial measurements performed 1 week apart and sustained for 1 month or more without support of platelet transfusions.

Current secondary outcomes:
- Time to erythroid response [ Time Frame: at 12 and 24 weeks ]
Time to erythroid response defined as an increase of at least 1.5g/dL without transfusion, that is sustained for at least 2 weeks and transfusion requirements at 12 and 24 weeks for BRC as compared with transfusion requirements during the eight weeks preceding study entry
- Time to neutrophil response [ Time Frame: at least 7 days ]
Time to neutrophil response defined as an increase of ANC above 1G/L, which is sustained for at least 7 days,
- Percentage of patients presenting best bone marrow response at 12 and 24 weeks of treatment assessed by bone marrow aspirate and bone marrow biopsy with fibrosis staining. [ Time Frame: at 12 and 24 weeks ]
- Transfusion requirements [ Time Frame: at 12 and 24 weeks ]
Transfusion requirements for BRC and platelets as compared with transfusions requirements during the eight weeks preceding study entry
- Proportion of patients presenting grade 3 or 4 adverse events from the first to the last administration of eltombopag. [ Time Frame: from 1st administration of eltrombopag to 1 month after the last administration of eltrombopag, ]
All adverse events will be reported on the adverse events reporting form of the case report file. Each adverse event will be recorded individually. The severity of the adverse event will be determined as follows :
* Severe (grade 3): significant interference with the patient's daily activity and unacceptable,
* Life-threatening (grade 4).
- Quality of life evaluation using the European Organisation for Research Treatment of Cancer Quality of Life Questionnaire (EORTC QLQ-C30 questionnaire). [ Time Frame: at 12 and 24 weeks ]
The QLQ-C30 consists of thirty items:
* 05 functional scales to explore the physical (1 to 5), executive (6 to 7), social (26 and 27), cognitive (20 and 25), and emotional (21 to 24) dimensions of the quality of life.
* 09 symptomatic scales to explore fatigue (10, 12 and 18), nausea symptoms (14 and 15), pain (9 and 19), dyspnea (8), insomnia (11), anorexia (13), diarrhea (17), constipation (16) and financial difficulties (28).
* 01 scale measuring the overall quality of life of each patient (29 and 30). The results of these different scales allow the calculation of a score that varies from 0 (worst) to 100 (better). A high overall health score reflects good health and a good quality of life. A high score for a symptom scale reflects a high level of symptoms. An average difference of 5 to 10 scores between two visits indicates a minor change, from 10 to 20 a moderate change and a difference of more than 20 points a significant change.
- Immune function (T/B/NK cells counts) [ Time Frame: at 12 and 24 weeks ]
- Overall survival, relapse-free survival and non-relapse mortality [ Time Frame: at 24 weeks of treatment ]
No relapse disease will be investigated using the Fine-Gray Test. No relapse mortality represent all death without relapse of the underlind disease.