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 $SL(2,\mathbb R)$?


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.

  • $\begingroup$ With $K$ I meant the set of irreducible representations that appear in the decomposition of the tensor product, and $V_k$ the corresponding vector space. I do not seem to find a discussion of tensor products in the works by Lion-Vergne or Lang. Repka (and this paper by Martin which he refers to) does discuss tensor products. However I am asking about the existence of intertwiners from $V_1 \otimes V_2 \to V_m$ where irrep $m$ does not appear in the decomposition of the tensor product. $\endgroup$ – fanfare Sep 26 '15 at 10:36
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    $\begingroup$ You have to keep in mind that for noncompact Lie groups, the tensor product of irreducible representations rarely decomposes discretely. In general (for type I groups, including semisimple Lie groups), one gets a direct integral decomposition over a non-atomic measure. Thus, it is common for $V\otimes W$ not to contain any irreducible subrepresentations; at the same time, there are many irreducible quotients that, as you point out, can be described by means of appropriate intertwining operators. $\endgroup$ – Victor Protsak Sep 27 '15 at 3:01
  • $\begingroup$ I recommend papers of Kobayashi for the state of the art on the branching problem for unitary/admissible representations of semisimple Lie groups (this problem includes tensor product problem as a special case corresponding to the diagonal embedding of $G$ into $G\times G$). $\endgroup$ – Victor Protsak Sep 27 '15 at 3:08

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