Consider two unitary irreducible representations on vector spaces $V_1$ and $V_2$ of a Lie group $G$. For $G$ is compact and $V_1$ and $V_2$ finite dimensional there is a unique decomposition of $V_1 \otimes V_2$ into a set of irreducible representations $V_k$, $k \in K$. The Clebsch-Gordan coefficients can then be viewed as *intertwiners*, so $G$-equivariant maps from $V_1 \otimes V_2 \to V_k$. Conversely, I think that nontrivial intertwiners $V_1 \otimes V_2 \to V_m$ can only be nonzero if $m \in K$ and then they are essentially the Clebsch-Gordan coefficients.

Now if $G$ is non-compact (but $V_1$ and $V_2$ still unitary) then it seems that I can define intertwiners from $V_1 \otimes V_2$ to another $V_m$ which does not appear in the decomposition of $V_1 \otimes V_2$. For example, if $V_1$ and $V_2$ are in the principal series representation of $SL(2,\mathbb R)$ then I can obtain such intertwiners by analytic continuation of the Clebsch-Gordan coefficients.

I was first of all wondering if the above story is true. If so, is there a way to understand the domain of these intertwiners? Do we understand for which triplet of representations they exist? Is there a general theory of their properties? If not, maybe just for $SL(2,\mathbb R)$?