TITLE: Is there a "Cartan product" of Harish-Chandra modules?
QUESTION [5 upvotes]: If $\lambda,\mu$ are two dominant weights for a Lie group $G$, then
there is a canonical (up to scale, perhaps) 
surjection $V_\lambda \otimes V_\mu \to V_{\lambda+\mu}$
of finite-dimensional representations, 
which I have occasionally heard called the "Cartan projection".
To each irreducible (or standard) Harish-Chandra module for
$({\mathfrak g},K)$, one can associate a $K$-orbit on $G/B$. 
For finite-dimensional representations, this orbit is the open $K$-orbit.

I want to know what analogues exist of the Cartan projection out of $V\otimes W$ if
  $V,W$ are two Harish-Chandra modules with the same associated $K$-orbit (other than 
  the open orbit case above). 

The answer may be something like "every H-C quotient of $V\otimes W$ has
the wrong Gel$'$fand-Kirillov dimension for it to again
have that associated $K$-orbit," in which case I'd appreciate references
that make that most clear. (I will be sad, but not overly surprised,
if that is the case.) 
EDIT: For consistency of notation, let's take $G,K$ to be the complexifications of $G_0,K_0$, where $K_0$ is a maximal compact in $G_0$.

REPLY [4 votes]: 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 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.)