  In a series of papers [22-24] by Bufetov and Gorin, Schur generating
functions as the Fourier transforms on the unitary group $U(N)$, are introduced
to study the asymptotic behaviors of random $N$-particle systems. We introduce
and study the Jack generating functions of random $N$-particle systems. In
special cases, this can be viewed as the Fourier transforms on the Gelfand
pairs $(GL_N(\mathbb R), O(N))$, $(GL_N(\mathbb C), U(N))$ and $(GL_N(\mathbb
H), Sp(N))$. Our main results state that the law of large numbers and the
central limit theorems for such particle systems, is equivalent to certain
conditions on the germ at unity of their Jack generating functions. Our main
tool is the Nazarov-Sklyanin operators [50], which have Jack symmetric
functions as their eigenfunctions. As applications, we derive asymptotics of
Jack characters, prove law of large numbers and central limit theorems for the
Littlewood-Richardson coefficients of zonal polynomials, and show that the
fluctuations of the height functions of a general family of nonintersecting
random walks are asymptotically equal to those of the pullback of the Gaussian
free field on the upper half plane.
