Timeline for On the notion of cuspidality
Current License: CC BY-SA 4.0
11 events
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| Jun 5, 2023 at 17:52 | comment | added | GH from MO | @MatyMangoo You are right, and I have now restricted the multiplicity one property to cuspidal automorphic representations. I think this property extends to unitary isobaric automorphic representations, but I don't know a reference off-hand. | |
| Jun 5, 2023 at 17:46 | history | edited | GH from MO | CC BY-SA 4.0 |
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| Jun 5, 2023 at 14:27 | comment | added | Maty Mangoo | What do you exactly mean by "if you change finitely many local factors $\pi_{\nu}$ in a given automorphic representation $\pi=\bigotimes'_{\nu} \pi_{\nu}$ , the resulting global admissible representation will no longer be automorphic"? Due to strong multiplicity I lose 'cuspidality', but not necessarily automorphy (if that word exists) | |
| Jun 5, 2023 at 13:22 | comment | added | GH from MO | @MatyMangoo Also, from a representation theoretic point of view, there is not much difference between $\mathbb{Q}$ and a number field. | |
| Jun 5, 2023 at 13:22 | comment | added | GH from MO | @MatyMangoo The $L$-function of a (local or global) admissible representation is defined even when the representation is not generic. In the generic case, this $L$-function can also be obtained from the Whittaker function of a (local or global) newform. At any rate, the $L$-function of an automorphic representation of $\mathrm{GL}_n$ is a product of (global) cuspidal representations of $\mathrm{GL}_{n_j}$ with $n=\sum_j n_j$. It is conjectured that this decomposition is unique. See arxiv.org/abs/math/0609460 more detail, and how this fits into the Langlands program. | |
| Jun 5, 2023 at 12:59 | comment | added | Maty Mangoo | @GH I honestly don't agree on that; an admissible representation is generic (embeddable into some induced representation of an additive char $\psi = \otimes \psi_{\nu} \colon k \backslash \mathbb{A} \to \mathbb{C}^{\times}$, if and only if the local components are for $\psi_{\nu}$. There are plenty of non-generic representations, but maybe you can show me a reference for that? | |
| Jun 5, 2023 at 12:38 | comment | added | Maty Mangoo | Oh yes, those two books are quite nice, although they only treat the case $k = \mathbb{Q}$. | |
| Jun 5, 2023 at 12:34 | history | edited | GH from MO | CC BY-SA 4.0 |
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| Jun 5, 2023 at 12:33 | vote | accept | Maty Mangoo | ||
| Jun 5, 2023 at 12:33 | comment | added | Maty Mangoo | Thanks @GH. I see. And what is then the difference between cuspidal automorphic and just automorphic in relation with their $L$-functions? I thought one could attach $L$-fcts to cuspidal reps (as to cuspidal modular forms), as they are generic (i.e. have Whittaker models). A general automorphic form does not neet to be generic. | |
| Jun 5, 2023 at 12:27 | history | answered | GH from MO | CC BY-SA 4.0 |