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.

  • $\begingroup$ 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. $\endgroup$ Oct 14 '17 at 15:04
  • $\begingroup$ @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. $\endgroup$ Oct 14 '17 at 15:50
  • $\begingroup$ no, not every pair but some pair. $\endgroup$ Oct 14 '17 at 19:33

People have mostly discussed tensoring of (continued) holomorphic discrete series and “ladder” representations — see all the references in Dvorsky (2007), maybe also Libine (2017).

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 $\mathrm{SL}(n,\mathbf R)$).


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