  We study zonal characters which are defined as suitably normalized
coefficients in the expansion of zonal polynomials in terms of power-sum
symmetric functions. We show that the zonal characters, just like the
characters of the symmetric groups, admit a nice combinatorial description in
terms of Stanley's multirectangular coordinates of Young diagrams. We also
study the analogue of Kerov polynomials, namely we express the zonal characters
as polynomials in free cumulants and we give an explicit combinatorial
interpretation of their coefficients. In this way, we prove two recent
conjectures of Lassalle for Jack polynomials in the special case of zonal
polynomials.
