Intertwinings for general $\beta$-Laguerre and $\beta$-Jacobi processes

We show that for $\beta \ge 1$ the semigroups of $\beta$-Laguerre and $\beta$-Jacobi processes of different dimensions are intertwined in analogy to a similar result for $\beta$-Dyson Brownian motion recently obtained by Ramanan and Shkolnikov. These intertwining relations generalize to arbitrary $\beta \ge 1$ the ones obtained for $\beta=2$ by the author, O'Connell and Warren between $h$-transformed Karlin-McGregor semigroups. Moreover they form the key step towards constructing a multilevel process in a Gelfand-Tsetlin pattern leaving certain Gibbs measures invariant. Finally as a by product we obtain a relation between general $\beta$-Jacobi ensembles of different dimensions.