Is the real Jacquet module of a Harish-Chandra module still a Harish-Chandra module? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T19:29:38Z http://mathoverflow.net/feeds/question/14921 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/14921/is-the-real-jacquet-module-of-a-harish-chandra-module-still-a-harish-chandra-modu Is the real Jacquet module of a Harish-Chandra module still a Harish-Chandra module? unknown (google) 2010-02-10T17:23:49Z 2010-02-10T20:11:04Z <p>Casselman defined the real Jacquet module for a Harish-Chandra module, if we view the Jacquet module as a module corresponding to the Levi subgroup, the question is is it still a Harish-Chandra module? In particular is it still admissible?</p> http://mathoverflow.net/questions/14921/is-the-real-jacquet-module-of-a-harish-chandra-module-still-a-harish-chandra-modu/14926#14926 Answer by Emerton for Is the real Jacquet module of a Harish-Chandra module still a Harish-Chandra module? Emerton 2010-02-10T17:44:05Z 2010-02-10T20:11:04Z <p>As I understand it, the Jacquet module for $(\mathfrak g, K)$-modules is defined so as to again be a $\mathfrak g$-module, and in fact it is a Harish-Chandra module, not for $({\mathfrak g},K)$, but rather for $(\mathfrak g,N)$ (where $N$ is the unipotent radical of the parabolic with respect to which we compute the Jacquet module). (I am probably assuming that the original $(\mathfrak g, K)$-module has an infinitesimal character here.)</p> <p>I am using the definitions of <a href="http://www.math.northwestern.edu/~emerton/pdffiles/geom%5Fjacquet.pdf" rel="nofollow">this paper</a>, in particular the discussion of section 2. This in turn refers to Ch. 4 of Wallach's book. So probably this latter reference will cover things in detail.</p> <p>Added: I may have misunderstood the question (due in part to a confusion on my part about definitions; see the comments below), but perhaps the following remark is helpful:</p> <p>If one takes the Jacquet module (say in the sense of the above referenced paper, which is also the sense of Wallach), say for a Borel, then it is a category {\mathcal O}-like object: it is a direct sum of weight spaces for a maximal Cartan in ${\mathfrak g},$ and any given weight appears only finitely many times. (See e.g. Lemma 2.3 and Prop. 2.4 in the above referenced paper; no doubt this is also in Wallach in some form; actually these results are for the geometric Jacquet functor of that paper rather than for Wallach's Jacquet module, but I think they should apply just as well to Wallach's. </p> <p>Maybe they also apply with Casselman's definition; if so, doesn't this give the desired admissibility? </p>