The general reader might kiss off jazz in the Soviet Union as a subject for that mythical old series of Thin Books including ''Snowstorms on the Equator'' and ''The Wit and Humor of Adolf Hitler.'' But last year an American jazzman back from Moscow reported that the best Soviet jazz players were the equal of those anywhere, though their gigs were assigned with a heavy bureaucratic hand. Then along swung the new Soviet leader, former KGB chief Yuri Andropov, to a public relations tune of fondness for trumpeter Miles Davis. Now ''Red and Hot'' makes clear that Mr. Andropov is by no means the first spy or leader to go for America's native art form in a communist country where jazz stars have been rewarded beyond the dreams of capitalism.
Take Eddie Rosner, a cornet virtuoso remembered from World War II days as if he were Louis Armstrong and Benny Goodman rolled into one. He was the leader of the State Jazz Orchestra of the Byelorussian Republic. He is officially a nonperson these days. But: ''It is doubtful that any jazz musician on earth has ever been recompensed more generously within his society than Eddie Rosner in the Soviet Union in wartime.''
Who says so? It is author S. Frederick Starr, the president of Oberlin College who is both a jazz musician and a scholar in Russian studies. Not to mention a skilled writer. His documented volume deals with jazz not only for its own sake but as a prism held up to Soviet history. While jazz slips in and out of Kremlin favor, Red Army officers and others who can afford it keep on building up their record collections. It all adds up to a deft illumination of the paradoxes in a totalitarian government's efforts to impose or deny ''from above'' the kind of popular culture that by definition depends on the taste of the consumers.
The late George Balanchine, before he became a famed American choreographer, appeared in a 1920s Leningrad revue with Leonid Utesov, who was to be called Russia's richest man and king of jazz. When the wind turned against foreign influences, Utesov thoughtfully informed his audiences that jazz had not been invented in New Orleans but in Odessa, by street musicians improvising at Jewish weddings. Everyone knew he was joking, except for foreign newspapers that imagined Moscow was rewriting history again.
Jazz musicians have not had it easy in a land that once banned the saxophone. But they have nurtured domestic musical resources as well as drawn appreciatively on jazz's homeland across the Atlantic. (When bandleader Bob Tsfasman was compelled to conduct political education courses for his musicians, he would begin his lectures with ''America is not something the cat dragged in!'')
And the flow of jazz has not been all one way. During the 1970s, according to Mr. Starr, more than 60 well-established jazz musicians emigrated from the Soviet Union. A half-century earlier, Joseph Schillinger left Leningrad for New York, and his Schillinger system of composition influenced a host of American jazz musicians.
In short, jazz in the Soviet Union is not the subject for a Thin Book, after all.