The Complex Man Behind the Duke : BIOGRAPHY : LUSH LIFE: A Biography of Billy Strayhorn.<i> By David Hajdu (Farrar, Straus and Giroux: $27.50, 306 pp.)</i>

To jazz fans, Billy Strayhorn is a much-admired but enigmatic figure whose name is inevitably linked with that of Duke Ellington. To the wider musical public, he is a virtual unknown.

Yet it was Strayhorn, not Ellington, as is generally believed, who wrote the Ellington Orchestra’s classic theme song, “Take the ‘A’ Train.” And it was Strayhorn who composed such other definitive items as “Passion Flower,” “Chelsea Bridge,” “Raincheck” and “Johnny Come Lately” and co-composed “Satin Doll” as well as many of the Ellington Suites (including “A Drum Is a Woman” and “Such Sweet Thunder”).

Why, then, the unfamiliarity of Strayhorn’s name? The pat answer is that Strayhorn, who died of cancer in 1967 at the age of 51, was eclipsed by the bright glare of Ellington’s public presence. And the fact that Ellington’s name appears on Strayhorn tunes such as “Day Dream” and “Something to Live For” hasn’t exactly helped facilitate the emergence of Strayhorn as an independent entity.

The real answer, however, lies somewhere within the complicated life described in David Hajdu’s compelling and extraordinarily detailed biography.

The most revealing segment of the book encompasses Strayhorn’s pre-Ellington years as a young man in Pittsburgh in the early ‘30s. Hajdu’s extensive research--primarily via interviews with friends and musical associates from the period--makes it clear that Strayhorn was a prodigy from the very beginning. Before his 20th birthday, he had written words, music and orchestrations for a full-scale musical, “Fantastic Rhythm” (the title an obvious tribute to Strayhorn’s affection for George Gershwin). And “Lush Life,” the great anthem to the angst of sophisticated urban romance that gives Hajdu’s book its title, was started, incredibly, when Strayhorn was 17 and finished before he turned 21.

Hajdu generally avoids the tricky labyrinth of psycho-biography, but his depiction of the close relationship between Strayhorn and his culturally oriented mother provides, at the very least, some insight into Strayhorn’s fascination with the external accouterments of a life of material refinement. And the choice of an openly gay lifestyle--remarkable for anyone in the ‘40s and ‘50s, doubly so for a black man in the world of jazz--reveals another, courageous aspect of Strayhorn’s complex personality.

His multilayered relationship with Ellington is a different matter, one that manages to escape Hajdu’s most meticulous research--and with good cause. It is almost as hard to imagine Strayhorn’s and Ellington’s puzzling temperaments coming together in such a strikingly intimate connection as it is to explain it. It may be, paradoxically, that the emotional elusiveness, the internal privacy, that each brought to their long alliance was the factor that allowed their association to realize its fullest potential for personal and creative longevity. Hajdu, preferring to stick with his data, makes no assumptions on that count.

But he does make a seemingly ironclad case, with precise chapter and verse, to clarify one critical aspect of the Ellington-Strayhorn creative bond: the fundamental significance of Strayhorn’s contributions to Ellington’s music, often in works attributed to Ellington, over the course of their nearly three decades of collaboration. And that fact alone should justify comprehensive reevaluation of Strayhorn’s importance, in his own right, as a 20th century composer.

If, despite all of Hajdu’s research and detail, the forces that drove Strayhorn still remain somewhat mysterious, “Lush Life” has illuminated many of the shadows, bringing light and perspective to one of the sadly underrated artists in the history of jazz.

(Hajdu has supervised a superb, representative collection of Strayhorn’s music, also titled “Lush Life,” and released by Verve Records in association with the publication of the book. It includes interpretations of Strayhorn songs by Sarah Vaughan, Oscar Peterson, Ella Fitzgerald and Stan Getz, among others. Equally fascinating is “Portrait of A Silk Thread,” on Kokopelli Records, a collection of newly discovered Strayhorn works performed by the Dutch Jazz Orchestra.)