Timeline for Some question about cupidal automorphic representation and supercuspidal representation
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| Mar 15, 2018 at 7:52 | vote | accept | Cooler Panda | ||
| Mar 2, 2018 at 14:29 | comment | added | David Loeffler | It is not true that all of the $\pi_v$ are generic. (You claim that "they all generic by Shalike"; presumably you mean the theorem of Piatetski-Shapiro and Shalika which states that cuspidal automorphic representations of $GL(n)$ are generic, but this does not generalise to groups other than $GL(n)$.) E.g. there are well-known examples of cuspidal automorphic reps of $GSp(4)$ coming from Saito--Kurokawa lifts which are not locally generic at any finite place. | |
| Mar 2, 2018 at 9:01 | comment | added | Olivier | I think you'll get a better understanding of Question 2 if you proceed in the reverse direction, that is if you ask yourself what conditions the $\pi_{v}$ must satisfy if they are the local components of an automorphic $\pi$. Then you should realize how rare global automorphic $\pi$ are compared to local ones. Then you'll see that no list of purely local conditions on the $\pi_v$ can ensure that they are the local components of a global $\pi$ (even for $\operatorname{GL}_{2}$, how could such a list guess that some weights are globally forbidden?). | |
| Mar 1, 2018 at 18:24 | answer | added | krl | timeline score: 6 | |
| Mar 1, 2018 at 11:45 | history | edited | Cooler Panda | CC BY-SA 3.0 |
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| Mar 1, 2018 at 11:31 | history | asked | Cooler Panda | CC BY-SA 3.0 |