TITLE: Intertwiners and Clebsch-Gordan coefficients
QUESTION [8 upvotes]: 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 $SL(2,\mathbb R)$?

REPLY [2 votes]: I have some trouble understanding notation (what is $K$?), so here is a very general answer. 
The case of ${\rm SL}(2,\Bbb{R})$ was studied in the old work of Repka and, if I am not mistaken, described in Lang's eponymous book and Lyon-Vergne's book. Much more is known, in particular, through the theory of theta correspondence in the case of classical groups. You may want to focus your question more.