Extension of Schur-Weyl duality for principal series in $SL(2, \mathbb{R})$

Question

In the case of $SU(N)$ all unitary irreps can be obtained from reducing tensor products ($V^{\otimes n}$) of the fundamental representation ($V$). Then given the set of all $SU(N)$ Young diagrams with $n$ boxes $\mathcal Y_n$, $V^{\otimes n}$ is completely reducible to the form $V^{\otimes n} \sim \sum_{Y\in \mathcal Y_n} R(Y)\otimes r(Y)$, where R(Y) is the SU(N) irrep associated to $Y$ and $r(Y)$ is the rep in the symmetric group $S_n$ also associated to $Y$. I am interested if there is a similar relationship between the representations in the principal series of $SL(2, \mathbb R)$ (where of course there is no fundamental rep, no Young diagrams and the irreps are continuous) and those of a different group?