3 corrected key counterexample; added 127 characters in body; added 36 characters in body; added 65 characters in body

If $G$ is simply connected semisimple and admits holomorphic discrete series, then the tensor product of the holomorphic discrete series of two weights decomposes as a direct sum of other holomorphic discrete series, each appearing with finitely mulitplicity. (In short, holomorphic discrete series comport themselves in general just as in the case of $SL_2(\mathbb R)$.)

I forget the reference now; sorry! [Added: Actually, section 7 of the Kobyashi paper linked to by BR above gives the result, I think.]

I think that for other Harish-Chandra modules, tensor products have infinite multiplicities (or in more analytic terms, involve direct integrals rather than direct sums). Even the generic principal discrete series for $Sp_4(\mathbb R)$ should demonstrate this behaviour, if I remember correctly. (And tensoring a holomorphic discrete series by an anti-holomorphic discrete series would also give infinite multiplicities.)

2 edited body

If $G$ is simply connected semisimple and admits holomorphic discrete series, then the tensor product of the holomorphic discrete series of two weights decomposes as a direct sum of other holomorphic discrete series, each appearing with finitely mulitplicity. (In short, holomorphic discrete series comport themselves in general just as in the case of $SL_2(\mathbb R)$.)

I forget the reference now; sorry!

I think that for other Harish-Chandra modules, tensor products have infinite multiplicities (or in more analytic terms, involve direct integrals rather than direct sums). Even the generic principle principal series for $Sp_4(\mathbb R)$ should demonstrate this behaviour, if I remember correctly.

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If $G$ is simply connected semisimple and admits holomorphic discrete series, then the tensor product of the holomorphic discrete series of two weights decomposes as a direct sum of other holomorphic discrete series, each appearing with finitely mulitplicity. (In short, holomorphic discrete series comport themselves in general just as in the case of $SL_2(\mathbb R)$.)

I forget the reference now; sorry!

I think that for other Harish-Chandra modules, tensor products have infinite multiplicities (or in more analytic terms, involve direct integrals rather than direct sums). Even the generic principle series for $Sp_4(\mathbb R)$ should demonstrate this behaviour, if I remember correctly.