If $\lambda,\mu$ are two dominant weights for a Lie group $G$, then
there is a canonical surjection $V_\lambda \otimes V_\mu \to V_{\lambda+\mu}$
(up to scale, perhaps) 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.)