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

We prove a limit theorem concerning the sequence of star-generators of [Formula: see text], where the expectation functional is provided by a character of [Formula: see text] with weights [Formula: see text] 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 [Formula: see text] satisfy the commutation relation [Formula: see text]. The special case [Formula: see text] yields the law of a bona fide traceless GUE matrix, and we retrieve a result of Köstler and Nica from 2021, which in turn extended a result of Biane from 1995.