A central limit theorem for star-generators of S-infinity, which relates to traceless CCR-GUE matrices

We prove a limit theorem concerning the sequence of star-generators of S-infinity, where the expectation functional is provided by a character of S-infinity with weights (w(1), ..., w(d), 0, 0, ...) in the Thoma classification. The limit law turns out to be the law of a "traceless CCR-GUE" matrix, an analogue of the traceless GUE where the off-diagonal entries g(i,j) satisfy the commutation relation g(i,j)g(j,i) = g(j,i)g(i,j) + (w(j) - w(i)). The special case w(1) = ... = w(d) = 1/d yields the law of a bona fide traceless GUE matrix, and we retrieve a result of of Kostler and Nica from 2021, which in turn extended a result of Biane from 1995.