Is there a "Cartan product" of Harish-Chandra modules?

2011-10-24T14:08:08Z

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 $G_0$.

Answer by Emerton for Is there a "Cartan product" of Harish-Chandra modules?

2011-10-24T21:24:23Z

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.)

Is there a "Cartan product" of Harish-Chandra modules? - MathOverflow

Is there a "Cartan product" of Harish-Chandra modules?

Answer by Emerton for Is there a "Cartan product" of Harish-Chandra modules?