Timeline for Functional equation and contragredient
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 2, 2017 at 13:59 | answer | added | David Loeffler | timeline score: 8 | |
| Dec 2, 2017 at 9:37 | comment | added | GH from MO | @DavidLoeffler: You comment is truly valuable and highly appreciated! | |
| Dec 2, 2017 at 9:35 | comment | added | GH from MO | Another comment: The functional equation is for the completed version (with the gamma factors inclued). The complete $L$-function should be denoted by $\Lambda(s,\pi)$, while the notation $L(s,\pi)$ should be reserved for the product of local factors over the finite (non-archimedean) places. Just to pay respect to our ancestors and not to confuse analytic number theorists (who discovered these objects). | |
| Dec 2, 2017 at 9:32 | comment | added | GH from MO | Having $\tilde\pi$ on the RHS of the functional equations is just the analogue of having $1-s$ there. So, please, don't be annoyed by it. It serves harmony! In fact twisting $\pi$ by $|\det|^{it}$ has the same effect as shifting $s$ by $it$. On the right hand side this corresponds to twisting $\tilde\pi$ by $|\det|^{-it}$, or shifting $1-s$ by $-it$. It is as nice as it can get. | |
| Nov 30, 2017 at 22:17 | comment | added | David Loeffler | For general $n$, self-dual automorphic representations of $GL(n)$ have been classified by Arthur. They are the ones arising as functorial transfers from orthogonal or symplectic groups (according to whether it is $L(Sym^2 \pi)$ or $L(\bigwedge^2 \pi)$ that has a pole). This is one of the major results of Arthur's book on automorphic representations of the classical groups. | |
| Nov 30, 2017 at 14:00 | comment | added | Peter Humphries | For $n \geq 3$, it is no longer necessarily the case that $\widetilde{\pi} \cong \pi \otimes \omega_{\pi}^{-1}$. For $n = 3$, self-contragredient representations $\Pi$ are nonetheless "easy" to classify: they are, up to a quadratic twist, of the form $\operatorname{ad} \pi$ for some automorphic representation $\pi$ of $\mathrm{GL}_2$. This is a result of Ramakrishnan. For $n \geq 4$, however, there's no nice classification of self-contragredient representations that I know of. | |
| Nov 30, 2017 at 13:57 | comment | added | Peter Humphries | If $\pi \cong \widetilde{\pi}$, then $\pi$ is said to be self-dual or self-contragredient. For $\mathrm{GL}_2$, this is particularly easy to classify: we have that $\widetilde{\pi} \cong \pi \otimes \omega_{\pi}^{-1}$, where $\omega_{\pi}$ is the central character of $\pi$, so self-contragredient automorphic representations are precisely those with trivial central character. | |
| Nov 30, 2017 at 13:15 | history | asked | Wolker | CC BY-SA 3.0 |