The intertwining property for Laguerre processes with a fixed parameter

We investigate the intertwining of Laguerre processes of parameter $\alpha$ in different dimensions. We introduce a Feller kernel that depends on $\alpha$ and intertwines the $\alpha$-Laguerre process in $N+1$ dimensions and that in $N$ dimensions. When $\alpha$ is a non-negative integer, the new kernel is interpreted in terms of the conditional distribution of the squared singular values: if the singular values of a unitarily invariant random matrix of order $(N+\alpha +1) \times (N+1)$ are fixed, then the those of its $(N+\alpha) \times N$ truncation matrix are given by the new kernel.