Ms. Lagarde and her team have been adamant about providing debt relief to near-bankrupt Greece, putting the fund at odds with Germany. And philosophically, the fund, since its inception after World War II, has ardently advocated the types of free-trade policies Mr. Trump has criticized so aggressively.  The legal issues have dogged Ms. Lagarde’s work at the fund, since she was appointed in 2011. She took over as managing director after Dominique Strauss-Kahn resigned following accusations that he had sexually assaulted a maid in a New York City hotel.  The case against Ms. Lagarde centered on Bernard Tapie, a former entertainer and owner of Adidas who had previously been jailed on corruption charges. Mr. Tapie accused the lender Crédit Lyonnais, in which the French state had a stake at the time, of cheating him in the sale of his share in the sportswear empire in 1993. Years of costly legal battles ensued.  In 2007, Ms. Lagarde sent the dispute to a three-person private arbitration authority. And the arbitration panel, in turn, awarded Mr. Tapie more than 400 million euros, or $420 million at current exchange rates, in damages and interest, to be paid by the state.  The arbitration decision was derided as politically tainted. And Ms. Lagarde was charged with negligence for allowing the arbitration in the first place and for declining to appeal the panel’s verdict.  While the court did not fault Ms. Lagarde for permitting the arbitration, it ruled that she had been negligent for not appealing the decision. Martine Ract-Madoux, the presiding judge in the case against Ms. Lagarde, also rejected the claim that the case was politically motivated.  But the court took a softer stance on the punishment, noting that a judge had previously invalidated the payout to Mr. Tapie and that she had “national and international” stature. The tribunal did not fine or sentence Ms. Lagarde, and it spared her a criminal record.  Speaking to reporters after the hearing, Ms. Lagarde’s lawyer, Patrick Maisonneuve, said he had a “mixed” reaction to the verdict.  “On the one hand, she is found responsible, but given the circumstances, given the responsibilities that Ms. Lagarde had at the time — in 2008, we were in a major economic crisis — the court decided that it would not sentence Ms. Lagarde to anything,” he said.  Ms. Lagarde’s lawyers can appeal the verdict before France’s highest criminal court, the Cour de Cassation, on procedural grounds. But Mr. Maisonneuve suggested she might not, because no punishment was meted out.  The verdict comes as the criticism of the fund increases.  For years, the I.M.F. has automatically awarded the top job to a senior European official. It has continued to do so, even as the influence of emerging economies like China, Brazil and Mexico has grown, economically and otherwise.  Of the last six leaders of the I.M.F., four have come from France. The last two, Ms. Lagarde and Mr. Strauss-Kahn, have encountered serious legal problems.  In a statement issued on Monday after the ruling, the French finance minister, Michel Sapin, said that Ms. Lagarde was heading the I.M.F. “with success” and that the French government had “complete confidence in her ability to carry out her responsibilities.”  The trial also revives concerns in France about high-level corruption, shining a spotlight on intimate ties between politicians and businesspeople, and on the large sums of money that are sometimes used to grease the country’s political wheels. Many people associated with it are under investigation, for accusations as varied as embezzlement and organized fraud.  Witnesses at Ms. Lagarde’s trial described a system in which influential, wealthy members of the French elite, like Mr. Tapie, had easy access to government officials and parlayed those relationships to the advantage of both parties, sometimes at the expense of taxpayers.  Bruno Bézard, a former director of the French Treasury, described “curious relationships” at the Finance Ministry when he worked there with Ms. Lagarde, saying that Mr. Tapie was often seen walking around the hallways, presumably to visit officials — “which was rather unexpected.”  Mr. Bézard, who led a government body overseeing state holdings, including a bank set up to take over Crédit Lyonnais’ bad assets, said Ms. Lagarde had ignored repeated warnings from members of his staff not to proceed with arbitration. He suggested that Ms. Lagarde had further erred by not challenging the amount of the payout.  Ms. Lagarde testified that she had declined to do so because it could have resulted in new lawsuits from Mr. Tapie and additional costs to the state.  Follow Liz Alderman @LizAldermanNYT and Aurelien Breeden @aurelienbrd on Twitter.  Liz Alderman and Aurelien Breeden reported from Paris, and Landon Thomas Jr. from New York.
The judges did not impose a fine or jail time in the case, linked to Ms. Lagarde’s time as France’s finance minister, but the verdict adds to the debate over the International Monetary Fund’s role.