Timeline for On the notion of cuspidality
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
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| Jun 6, 2023 at 19:55 | comment | added | Maty Mangoo | I think I the statemente @paulgarrett meant is Theorem 5.6 and Cor.5.7 in Corvallis 2.3 (Borel-Jacquet). It appears that one does not need cuspidality for it. | |
| Jun 6, 2023 at 18:51 | comment | added | paul garrett | Also, by "the theory of the constant term", a not-necessarily-$L^2$, but moderate growth (and sufficient finiteness under center of enveloping algebra, and Hecke algebras at finite places) is in fact of rapid decay, so is in $L^2$. In particular, the repns generated by such cfms are unitary. | |
| Jun 6, 2023 at 17:00 | comment | added | paul garrett | Two threads: unitary repns of reductive groups over local fields, where Harish-Chandra proved admissibility for real, Bernstein for p-adic. These results were anticipated (at least by optimists) because of the theorem (sketched in the Gelfand-PiatetskiShapiro book, treated more carefully by Godement in the Boulder conference) that the induced action of test functions on $L^2$ cuspforms is by compact operators, and that there's an orthonormal basis $\{f_i\}$ such that there are test functions $\varphi_i$ such that $\varphi_i\cdot f_i=f_i$. This gives the admissibility. | |
| Jun 6, 2023 at 9:14 | comment | added | Maty Mangoo | @GHfromMO thanks a lot! | |
| Jun 6, 2023 at 8:38 | comment | added | GH from MO | @MatyMangoo In general, an irreducible unitary representation of a linear connected reductive Lie group is admissible. See Theorem 8.1 in Knapp: Representation theory in semisimple Lie groups. This result is due to Harish-Chandra (1951) who even treated Banach space representations. | |
| Jun 6, 2023 at 8:23 | comment | added | Maty Mangoo | @paulgarrett do you have a reference for that 'non-trivial' theorem? | |
| Jun 5, 2023 at 18:55 | comment | added | Maty Mangoo | Ok, so it also does not make sense to talk about 'cuspidality' in general admissible representations in the sense of Flath, right? If one says a cuspidal representation, one necessarily means an admissible automorphic representation (realized in the space of cusp forms) | |
| Jun 5, 2023 at 18:17 | comment | added | paul garrett | Just to be clear on one point: it is a non-trivial theorem (perhaps declared a definition in the mid-to-late 1970's) that the (irreducible) adele-group repns generated by (suitably-strong-sense) cuspforms are admissible. So proving theorems about factoring, etc., of irreducible admissible repns of adele groups is relevant to the (irreducible) repns generated by (suitable) cuspforms. But, for general and uninteresting reasons, most admissible adele-group repns are not automorphic. Unsurprising, much like saying that not every real number is algebraic... | |
| Jun 5, 2023 at 17:08 | history | became hot network question | |||
| Jun 5, 2023 at 12:33 | vote | accept | Maty Mangoo | ||
| Jun 5, 2023 at 12:27 | answer | added | GH from MO | timeline score: 8 | |
| Jun 5, 2023 at 11:23 | comment | added | Maty Mangoo | *Let us just stick to the $\mathrm{GL}_n$-case. | |
| Jun 5, 2023 at 11:21 | comment | added | Maty Mangoo | @Echo thank you for your answer! Well, yes, he abstracts the notion of an admissible automorphic representation (i.e. he leaves out the part of being realizable as space of automorphic forms) and calls such representations just admissible (not automorphic), that's true. Hence my question; if the definition of Flat is more general, or if any admissible representation in the sense of Flath can be realized as automorphic admissible representation as introduced e.g. in Bump's book. | |
| Jun 5, 2023 at 11:06 | history | edited | YCor | CC BY-SA 4.0 |
added top-level tag
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| Jun 5, 2023 at 9:35 | comment | added | user473423 | I think Flath speaks of general representations, not automorphic representations. | |
| Jun 5, 2023 at 9:05 | history | asked | Maty Mangoo | CC BY-SA 4.0 |