For a root system $\Phi$ of rank $n$ with Weyl group $W$ and a dominant integral weight $\lambda$ consider the Hall-Littlewood polynomial $$P_\lambda(x_1,\ldots,x_n;t)=\frac1{W_\lambda(t)}\sum_{w\in W}w\left(e^\lambda\prod_{\alpha\in\Phi^-}\frac{1-te^\alpha}{1-e^\alpha}\right).$$ (The normalizing factor $W_\lambda(t)$ is the Poincaré series of the stabilizer $W_\lambda$.)
The branching rule for classical, i.e. $\mathfrak{gl}_n$, Hall-Littlewood polynomials is essentially contained in Macdonald's "Symmetric Functions and Hall Polynomials", it appears, for instance, here: https://arxiv.org/pdf/1405.7035.pdf as (4). My question is whether similar branching rules are known in the symplectic and orthogonal cases.
(As pointed out in the above paper, when iterated such a branching rule is equivalent to a formula for $P_\lambda$ in the form of a sum over the Gelfand-Tsetlin basis.)
Update. It's worth mentioning that the branching rule for $\mathfrak{gl}_n$ can be generalized to Macdonald polynomials (see, for instance, Theorem 1.1 in https://arxiv.org/pdf/1412.0714.pdf). Of course, an analog of this result for other classical types would more than answer my question.
