Timeline for Characters on $PGL(2)$
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 28, 2018 at 21:06 | comment | added | LSpice | In general one replaces the notion of conductor by the notion of depth (for which see the papers of Moy and Prasad), and the depth of $\chi \circ \det$ is $r - 1$ in the situation you describe. (I am assuming that $p \ne 2$.) | |
| Feb 28, 2018 at 21:05 | comment | added | LSpice | It is not quite clear what "the conductor of $\chi \circ \det$" should mean a priori (hence how to compare it to the conductor of $\chi$), since $\mathrm{PGL}(2, F)$ carries multiple natural filtrations, unlike $F^\times = \mathrm{GL}(1, F)$ which carries only a single natural filtration. One possible definition would be just to define the conductor of $\chi \circ \det$ to be that of $\chi$. | |
| Feb 28, 2018 at 21:02 | comment | added | LSpice | Since $\mathrm{SL}(2, F)$ is generated by its unipotents, any character of $\mathrm{PGL}(2, F)$ is trivial on the image of $\mathrm{SL}(2, F)$. Since $\mathrm{PGL}(2, F)/\mathrm{SL}(2, F)$ is isomorphic to $F^\times/(F^\times)^2$ via the determinant, indeed the characters of $\mathrm{PGL}(2, F)$ are as you describe. | |
| Feb 28, 2018 at 21:01 | comment | added | LSpice | I think also "a group such that $F^2(1 + p^r\mathscr O_p)$" is meant to be "a group such as $(F^\times)^2(1 + p^r\mathscr O_p)$". | |
| Feb 27, 2018 at 21:30 | comment | added | D_S | Somewhat related: mathoverflow.net/questions/292424/… | |
| Feb 27, 2018 at 18:39 | comment | added | Qfwfq | Ah okay, I suspected that, but.. who knows! :) | |
| Feb 27, 2018 at 16:47 | comment | added | Desiderius Severus | @Qfwfq That I made a typo, now corrected :) | |
| Feb 27, 2018 at 16:47 | history | edited | Desiderius Severus | CC BY-SA 3.0 |
Details and typos
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| Feb 27, 2018 at 16:47 | comment | added | Qfwfq | I'm curious about the notation $PGL(é)$: what does it mean? | |
| Feb 27, 2018 at 16:31 | history | asked | Desiderius Severus | CC BY-SA 3.0 |