What is known about irreducible decomposition of tensor products of (infinite-dimensional) unitary irreducible representations of $SU(2,2)$ (or, more generally, simple groups of split rank greater than $1$)?


I am aware of some general and special results for split rank 1 groups, but I feel that the case of split rank 2 should be considerably more complicated. For example, I believe I can construct infinitely many (more precisely, continuous families of) kernels which at least naively appear to be singlets in a tensor product of any three principal series representations for $SU(2,2)$. This seems to suggest that some multiplicities in the decompositions are going infinite.



Tensor products of unitary irreducible representations of $SU(2,2)$
People have mostly discussed tensoring of (continued) holomorphic discrete series and “ladder” representations — see all the references in Dvorsky ([2007](http://www.ams.org/mathscinet-getitem?mr=2296624)), maybe also Libine ([2017](http://www.ams.org/mathscinet-getitem?mr=3611096)).


For the principal series you ask about, one should probably try the Mackey methods well described in Lipsman ([1974](http://www.ams.org/mathscinet-getitem?mr=372116), Chap. II B); but other than a “very brief consideration” in Neretin ([1986](http://www.ams.org/mathscinet-getitem?mr=831058)), I’m not aware of published details beyond Lipsman’s references: Williams ([1973](http://www.ams.org/mathscinet-getitem?mr=349903)) (last page has infinite multiplicities in complex semisimple groups) and Martin ([1975](http://www.ams.org/mathscinet-getitem?mr=374341)) (last page has them in $\mathrm{SL}(n,\mathbf R)$).


