  The gamma kernels are a family of projection kernels
$K^{(z,z')}=K^{(z,z')}(x,y)$ on a doubly infinite $1$-dimensional lattice. They
are expressed through Euler's gamma function and depend on two continuous
parameters $z,z'$. The gamma kernels initially arose from a model of random
partitions via a limit transition. On the other hand, these kernels are closely
related to unitarizable representations of the Lie algebra
$\mathfrak{su}(1,1)$. Every gamma kernel $K^{(z,z')}$ serves as a correlation
kernel for a determinantal measure $M^{(z,z')}$, which lives on the space of
infinite point configurations on the lattice.
  We examine chains of kernels of the form $$ \ldots, K^{(z-1,z'-1)}, \;
K^{(z,z')},\; K^{(z+1,z'+1)}, \ldots, $$ and establish the following
hierarchical relations inside any such chain:
  Given $(z,z')$, the kernel $K^{(z,z')}$ is a one-dimensional perturbation of
(a twisting of) the kernel $K^{(z+1,z'+1)}$, and the one-point Palm
distributions for the measure $M^{(z,z')}$ are absolutely continuous with
respect to $M^{(z+1,z'+1)}$.
  We also explicitly compute the corresponding Radon-Nikod\'ym derivatives and
show that they are given by certain normalized multiplicative functionals.
