Timeline for what information of a representation was killed by Jacquet functor?
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 27, 2018 at 7:19 | comment | added | Victor Protsak | However, an analogous computation with $f$ shows that the Jacquet module with respect to the nilradical of the lower Borel is $0.$ The subtlety here is that, unlike in the $p$-adic case, the Jacquet functor depends on the choice of the parabolic: only the action of $\frak{k}$ exponentiates to the group, and so the isomorphism class of the Jacquet module is determined by the $K$-orbit of $\frak{n}$. The Beilinson-Bernstein result mentioned in my answer asserts that for an open dense orbit of $K$ on the flag variety the corresponding $\frak{n}$-homology and Jacquet module are non-zero. | |
| Nov 27, 2018 at 7:12 | comment | added | Victor Protsak | No, at least not generically. For example, consider a holomorphic discrete series representation $V=V_k, k\geq 2$ for $G={\rm SL}(2,\Bbb {R})$. Choose the standard basis $\{e,h,f\}$ for $\frak {g}=\frak {sl}_2$. Then $V$ is a lowest weight module with lowest weight $k$, the subspace $e^{i}V$ is spanned by the vectors of weight at least $k+2i$, and so the Jacquet module with respect to the nilradical $\frak{n}$ of the standard (upper) Borel, as defined above, is the contragredient $V^*$. | |
| Nov 26, 2018 at 16:07 | comment | added | LSpice | Do Jacquet modules in the real case also kill off the discrete series (as they do cuspidals in the non-Archimedean case)? If so, then connecting irreducibility of a representation and its Jacquet module can fail "the other way": the Jacquet module of a reducible module may be irreducible. | |
| Oct 28, 2012 at 7:55 | vote | accept | user1832 | ||
| May 25, 2010 at 4:50 | comment | added | Victor Protsak | I don't think so: Jacquet module of a simple module may have multiplicities. | |
| May 24, 2010 at 13:54 | comment | added | user1832 | By the way, can we test irreducibility from J(V)? | |
| May 1, 2010 at 4:15 | vote | accept | user1832 | ||
| May 1, 2010 at 4:15 | |||||
| May 1, 2010 at 2:55 | history | answered | Victor Protsak | CC BY-SA 2.5 |