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 $G_0$.

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    $\begingroup$ I'm not sure I understand, so can we go through an example? What should be the answer for $G=SL_2(\mathbb R)$? Here, the tensor product of two holomorphic discrete series, $D_k$ and $D_l$, is the direct sum of all holomorphic discrete series with weight at least $k+l$, that is: $D_k\otimes D_l\simeq \bigoplus_{i=0}^\infty D_{k+l+2i}$. (On the other hand, the tensor product of unitary principal series is a direct integral of principal series plus holomorphic discrete series.) $\endgroup$ – B R Oct 24 '11 at 16:13
  • $\begingroup$ (Note added, to make $SL_2({\mathbb R})$ to be $G_0$ not $G$.) Here $K=SO(2)$, and the $K$-orbit is the north pole. Then I would want the map $D_k \otimes D_l \to D_{k+l}$. $\endgroup$ – Allen Knutson Oct 24 '11 at 16:29
  • $\begingroup$ ...which is to say, yes! this is exactly the sort of thing I want. $\endgroup$ – Allen Knutson Oct 24 '11 at 16:30
  • $\begingroup$ Does your "direct integral" comment say that I can't get principal $\otimes$ principal onto principal? $\endgroup$ – Allen Knutson Oct 24 '11 at 16:47
  • $\begingroup$ Well, I wish I knew more, then :) The $SL_2(\mathbb R)$ case is in Repka's early work (the papers with "Tensor products" in the title). Section 7 of Kobayashi's paper (ms.u-tokyo.ac.jp/~toshi/texpdf/jfa98-full.pdf) may point you to some references (and Theorem 7.4 is useful). And Section 4 of Weissman's paper (people.ucsc.edu/~weissman/MultMod.pdf) may do the same (in fact, Prop 4.2 may be exactly what you want). $\endgroup$ – B R Oct 24 '11 at 17:02

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.)

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