Is there a "Cartan product" of Harish-Chandra modules? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T23:08:44Z http://mathoverflow.net/feeds/question/78979 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/78979/is-there-a-cartan-product-of-harish-chandra-modules Is there a "Cartan product" of Harish-Chandra modules? Allen Knutson 2011-10-24T14:08:08Z 2011-10-25T06:12:59Z <p>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".</p> <p>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.</p> <blockquote> <p>I want to know what analogues exist of the Cartan projection out of $V\otimes W$ if $V,W$ are two Harish-Chandra modules <em>with the same associated $K$-orbit</em> (other than the open orbit case above). </p> </blockquote> <p>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.) </p> <p>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 $G_0$.</p> http://mathoverflow.net/questions/78979/is-there-a-cartan-product-of-harish-chandra-modules/79029#79029 Answer by Emerton for Is there a "Cartan product" of Harish-Chandra modules? Emerton 2011-10-24T21:24:23Z 2011-10-25T06:12:59Z <p>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 $SL_2(\mathbb R)$.) </p> <p>I forget the reference now; sorry! [Added: Actually, section 7 of the Kobyashi paper linked to by BR above gives the result, I think.]</p> <p>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.)</p>