Luzin was accused of plagiarism and other misconduct by Kolmogorov and other students during the Luzin Affair of 1936. Were these allegations actually true?
It was part of Stalin's political witch hunt, the Great Purge. Luzin was denounced in Pravda as the author of "would-be scientific papers" who "felt no shame in declaring the discoveries of his students to be his own achievements", anonymously. Then the corroborating "testimony", from Aleksandrov and Kolmogorov among others, was obtained by the usual methods of NKVD (later renamed into KGB): the "witnesses" were threatened and blackmailed, if not "persuaded" physically. Then Luzin was declared "enemy under the mask of a Soviet citizen", and he was lucky not to get imprisoned or executed.
"The methods of political insinuations and slander had been used against the old Muscovite professorship already several years before the article in Pravda". Luzin's advisor, Egorov, was named among "active counter-revolutionaries among mathematicians" back in 1930, arrested, and died in prison after a hunger strike in 1931. He had wrong religious beliefs. Luzin's fault, aside from the usual guilt by association, was likely that he published major results in foreign journals.
Wikipedia has a detailed account of the Luzin 1936 affair, another source is Kutateladze's The Tragedy of Mathematics in Russia, which quotes many contemporary documents:
"It is customary to emphasize that Luzin was not so great a mathematician as his students that had persecuted him. Some moral fault is persistently incriminated to Luzin in the untimely death of M. Ya. Suslin (1894–1919) from typhus fever. Luzin is often blamed for all his disasters at least partly. He is ascribed such traits of character as theatricality, envy of others’ success, hypocrisy, plagiarism, and inclination to intriguing. He is said to deserve all punishments and if not all then it is not his student s’ fault but stalinism and the curse of the epoch.
No person with these defects of personality could ever become the founder of “Lusitania” — the most successful mathematical school in the history of world science. Therefore, there exists as rather likely theory of “two Luzins”: one of the epoch of Lusitania and the other of the times of the Luzin case... If Luzin were guilty then his fault would belong to the sphere of the personal mathematical relations between a teacher and a student. No convincing evidence of Luzin’s plagiarism was ever submitted."
In particular it quotes Lebesgue's preface to Luzin's book on analytic sets:
"Anyone will be astonished to find out from Luzin’s book that I had incidentally invented the sieve method and was the first to construct an analytic set. But nobody could be more amazed than me. Mr. Luzin feels himself happy only when he has managed to ascribe his own discoveries to someone else."
Curiously enough, even this is sometimes turned against Luzin, he was supposedly trying to "flatter" Lebesgue.