“We respect the integrity of the Czech judicial process,” said a spokesman for the State Department, who spoke on condition of anonymity because he was not authorized to be identified. He declined to comment on how the American government felt about the arrest of someone invited to a conference it was co-hosting.
Ms. Lagronova said she would not speculate on how the events might reflect on the Czech Republic, saying that similar arrests had taken place in other countries.
It was, however, a very unusual occurrence. Mr. Muslim, formerly the first co-president of the Democratic Union political party in the Kurdish areas of northern Syria, was an invited speaker at the conference on regional security in the Middle East. The coalition to which he belongs is an informal ally of the United States-led coalition fighting in Syria. The Czech Republic and Turkey are also NATO allies.
The conference was convened by the Center for Middle East Development of the University of California, Los Angeles, and it was co-hosted by the Czech Republic, the Netherlands, Sweden and the United States. The United States Institute for Peace, a congressionally chartered organization, was also a sponsor, according to fliers issued at the conference.
The gathering was held at the Marriott Hotel in Prague, where Mr. Muslim was arrested a few hours after the last session ended.
Officials at the U.C.L.A. center could not immediately be reached for comment on Mr. Muslim’s arrest.
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The conference was held in secrecy under “Chatham House rules,” by which participants pledge not to publicize the proceedings; journalists who attend do so on an off-the-record basis.
On Friday, the Turkish pro-government newspaper Sabah published a picture that it said showed Mr. Muslim at the Marriott Hotel; the paper did not mention that the photograph was from the conference proceedings there. The Turkish extradition request was issued the following day.
Mr. Muslim’s political organization said in a statement: “What happened in the Czech state is an immoral act and contrary to the values of international norms. It reflects how the Turkish M.I.T. members have the ability to penetrate into the European arena and don’t observe any laws or even respect the sovereignty of states.” M.I.T. refers to the Turkish national intelligence agency.
Shahoz Hasan, the co-president of the Democratic Union party, called Mr. Muslim’s detention “unrightful and immoral.” He was interviewed over WhatsApp from Syria. “Those who invited Mr. Muslim to the conference expressed their sorrow and promised to solve the matter,” he said.
Karel Schwarzenberg, a former foreign minister of the Czech Republic, was scathing about Mr. Muslim’s treatment. “To arrest a politician who arrived in good faith to Prague for a conference is problematic,” he said. “We should also take into account the sad fact that the Turkish government coins every political opponent a terrorist.”
The Turkish ambassador to Prague, Ahmet Necati Bigali, said in a televised interview that, “Unfortunately, the Czech court made a verdict against our dual relationship, our relationship as allies,” adding, “This decision may have negative reflections on our dual relations.”
Prime Minister Binali Yildirim of Turkey said that, regardless of the Czech court’s decision, Mr. Muslim would no longer be able to wander freely in Europe, according to the Daily Sabah. Turkey says that the organizations Mr. Muslim has been affiliated with are linked to the P.K.K., the Kurdistan Worker’s Party, which Europe and the United States consider a terrorist organization. Mr. Yildirim refers to Mr. Muslim as a P.K.K. leader.
The Kurds in Syria, however, say they are not connected to the P.K.K., and the United States considers them allies in the fight against the Islamic State.
Mr. Muslim, who is based in Brussels, has left Prague, according to aides, who declined to say where he was going because of security concerns.