Suppose that we have a probability distribution $\rho(\mathbf x)$ on a manifold $\mathcal M$, which is invariant under the action of a Lie group $G$, $\rho(g\mathbf x)=\rho(\mathbf x)$ for all $\mathbf x\in\mathcal M$ and $g\in G$. I am interested in the properties of the function $\mathcal Z(\mathbf z)$, defined by $$ \mathcal Z(\mathbf z)=\int_{\mathbf x\in\mathcal M}\rho(\mathbf x)e^{\mathbf z\cdot\mathbf x} $$ with a $G$-invariant integration measure on $\mathcal M$, for generally complex values of the argument $\mathbf z$. (For real $\mathbf z$, this is the moment generating function of the probability distribution, and for imaginary $\mathbf z$ its characteristic function.) Thanks to $G$-invariance, we can decompose the integral into one over orbits of $G$ and a sum over the orbits, $$ \mathcal Z(\mathbf z)=\sum_{\mathrm{orbits}\,\Phi}\rho(\Phi)\int_{\mathbf x\in\Phi}e^{\mathbf z\cdot\mathbf x}\equiv\sum_{\mathrm{orbits}\,\Phi}\rho(\Phi)\mathcal Z_\Phi(\mathbf z). $$ The latter integral depends just on the geometry of the group $G$, since the orbit is homeomorphic to the coset space $G/H$, where $H$ is the isotropy group of some $\mathbf x_0\in G$.
What is known about the functions $\mathcal Z(\mathbf z)$ or $\mathcal Z_\Phi(\mathbf z)$? I can calculate $\mathcal Z_\Phi(\mathbf z)$ in some special cases, but I am interested in the general properties. What literature or keywords should I look for? I would be particularly interested in the zeros of $Z(\mathbf z)$, which comes from the physical interpretation of the problem: $\rho(\mathbf x)$ is a probability distribution for an order parameter of a system, and $\mathcal Z(\mathbf z)$ its partition function.