  We study asymptotics of traces of (noncommutative) monomials formed by images
of certain elements of the universal enveloping algebra of the
infinite-dimensional unitary group in its Plancherel representations. We prove
that they converge to (commutative) moments of a Gaussian process that can be
viewed as a collection of simply yet nontrivially correlated two-dimensional
Gaussian Free Fields. The limiting process has previously arisen via the global
scaling limit of spectra for submatrices of Wigner Hermitian random matrices.
  This note is an announcement, proofs will appear elsewhere.
