# Tensor products of unitary irreducible representations of $SU(2,2)$

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.

• For admissible irreducible representations only the groups $SO(n,1)$ have finite tensor product multiplicities. If you are interested I could try to find references. Oct 14 '17 at 15:04
• @FriedrichKnop, are you saying that any pair of admissible representations tensors with infinite multiplicities? I thought that $SO(n,2)$ admit unitary highest weight representations which definitely tensor with finite multiplicities (at least that's my physicists intuition). I guess I'm interested in some examples where the tensor products are worked out, with any kind of multiplicities. Oct 14 '17 at 15:50
• no, not every pair but some pair. Oct 14 '17 at 19:33

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