Timeline for Restriction of cuspidal representations of GL(2) to the Borel subgroup
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 2, 2023 at 0:16 | comment | added | Linda | @PaulBroussous: Thanks! | |
| Dec 1, 2023 at 19:09 | comment | added | Paul Broussous | @Linda. Using the Bruhat decomposition $G=B\cup BwB$, you see that to specify a cuspidal, you have to give the action of $w=\left(\begin{array}{cc} 0 & 1\\ 1 & 0\end{array}\right)$ on the Kirillov space $\mu$. This is the difficult part! | |
| Dec 1, 2023 at 19:06 | comment | added | Paul Broussous | @Linda. There is no simple construction of cuspidal representations. You have Deligne-Lusztig theory which involves very sophisticated tools from algebraic geometry. You also have the construction via the Weil representation (in characteristic not $2$). This latter is simpler. | |
| Dec 1, 2023 at 18:36 | vote | accept | Linda | ||
| Dec 1, 2023 at 18:35 | comment | added | Linda | @SimonWadsley: Ah, I understand what you were getting at now! I am working on my phone, so I didn’t see the edit you made to your comment giving a more precise reference until after I replied. | |
| Dec 1, 2023 at 18:31 | comment | added | Simon Wadsley | If you look closely you'll see that I do construct the cuspidal representations after restriction to $B$ though not on the whole of $GL_2(\mathbb{F}_q)$. They are constructed exactly as Paul Broussous indicates in his answer but more details are provided. You can see from the characters that the $\mu_\theta$ which are constructed as representations coincide with the restrictions of the cuspidal representations. | |
| Dec 1, 2023 at 18:29 | comment | added | Linda | @SimonWadsley: You calculate the character table (which can also be found in many other places), but on page 67 you say re the cuspidal representations “we won’t be able to explicitly construct the representations”. | |
| Dec 1, 2023 at 18:22 | comment | added | Simon Wadsley | You can find a construction in chapter 9 of my notes dpmms.cam.ac.uk/~sjw47/2023Lectures.pdf. More precisely you want the things I call $\mu_\theta$ at the end of section 9.3. | |
| Dec 1, 2023 at 17:19 | answer | added | Paul Broussous | timeline score: 3 | |
| S Dec 1, 2023 at 3:57 | review | First questions | |||
| Dec 1, 2023 at 8:45 | |||||
| S Dec 1, 2023 at 3:57 | history | asked | Linda | CC BY-SA 4.0 |