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Hecke algebras for p-adic groups, the explicit Local Langlands Correspondence and stability
Multi angle
Authors : Xu, Yujie (Author of the conference)
CIRM (Publisher )
11F70 - Representation-theoretic methods; automorphic representations over local and global fields
11S37 - Langlands-Weil conjectures, nonabelian class field theory
20C08 - Hecke algebras and their representations
20G05 - Representation theory of linear algebraic groups
22E50 - Representations of Lie and linear algebraic groups over local fields
CIRM (Publisher )
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Abstract :
I will talk about my joint work with Aubert where we prove the Local Langlands Conjecture for $G_2$ (explicitly). This uses our earlier results on Hecke algebras attached to Bernstein components of (arbitrary) reductive $p$-adic groups, as well as an expected property on cuspidal support, along with a list of characterizing properties (including stability). In particular, we obtain 'mixed' L-packets containing F-singular supercuspidals and nonsupercuspidals. Our methods are inspired by the Langlands-Shahidi method, Deligne-Lusztig and Lusztig theories etc. If time permits, I will explain how to characterize our correspondence using stability of L-packets, by computing character formulae in terms of (generalized) Green functions ; one key input is a homogeneity result due to Waldspurger and DeBacker. Moreover, I will mention how to adapt our general strategy to construct LLC for other reductive groups, such as $G S p(4), S p(4)$, etc. The latter parts are based on recent joint work with Suzuki.
Keywords : Local Langlands Correspondence; Hecke algebras; Representation theory of p-adic groups; Bernstein blocks; L-parameters; character formulas
MSC Codes : Keywords : Local Langlands Correspondence; Hecke algebras; Representation theory of p-adic groups; Bernstein blocks; L-parameters; character formulas
11F70 - Representation-theoretic methods; automorphic representations over local and global fields
11S37 - Langlands-Weil conjectures, nonabelian class field theory
20C08 - Hecke algebras and their representations
20G05 - Representation theory of linear algebraic groups
22E50 - Representations of Lie and linear algebraic groups over local fields
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Information on the Video
Film maker : Hennenfent, GuillaumeLanguage : English Available date : 12/10/2023 Conference Date : 18/09/2023 Subseries : Research talks arXiv category : Number Theory ; Representation Theory Mathematical Area(s) : Algebra ; Lie Theory and Generalizations ; Number Theory Format : MP4 (.mp4) - HD Video Time : 01:04:11 Targeted Audience : Researchers ; Graduate Students ; Doctoral Students, Post-Doctoral Students Download : https://videos.cirm-math.fr/2023-09-18_Xu.mp4 
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Information on the Event
Event Title : Automorphic forms, endoscopy and trace formulas / Formes automorphes, endoscopie et formule des tracesEvent Organizers : Beuzart-Plessis, Raphaël ; Lemaire, Bertrand ; Nicole, Marc-Hubert ; Nyssen, Louise Dates : 18/09/2023 - 22/09/2023 Event Year : 2023 Event URL : https://conferences.cirm-math.fr/2903.html
Citation Data
DOI : 10.24350/CIRM.V.20094203Cite this video as: Xu, Yujie (2023). Hecke algebras for p-adic groups, the explicit Local Langlands Correspondence and stability. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.20094203 URI : http://dx.doi.org/10.24350/CIRM.V.20094203 |
See Also
- [Multi angle] Zeta functions of Shimura varieties: past, present, and the near future / Author of the conference Zhu, Yihang.
- [Multi angle] Counting $l$-adic local systems over a curve over a finite field / Author of the conference Yu, Hongjie.
- [Multi angle] Counting non-tempered automorphic forms using endoscopy / Author of the conference Gerbelli-Gauthier, Mathilde.
- [Multi angle] Motives and automorphic representations / Author of the conference Arthur, James.
Bibliography
- AUBERT, Anne-Marie et XU, Yujie. The Explicit Local Langlands Correspondence for $ G_2$. arXiv preprint arXiv:2208.12391, 2022. - https://doi.org/10.48550/arXiv.2208.12391
- AUBERT, Anne-Marie et XU, Yujie. Hecke algebras for $ p $-adic reductive groups and Local Langlands Correspondence for Bernstein blocks. arXiv preprint arXiv:2202.01305, 2022. - https://doi.org/10.48550/arXiv.2202.01305
- SUZUKI, Kenta et XU, Yujie. The explicit Local Langlands Correspondence for $ G_2 $ II: character formulas and stability. arXiv preprint arXiv:2304.02630, 2023. - https://doi.org/10.48550/arXiv.2304.02630
- SUZUKI, Kenta et XU, Yujie. The explicit Local Langlands Correspondence for $\mathrm {GSp} _4 $, $\mathrm {Sp} _4 $ and stability (with an application to Modularity Lifting). arXiv preprint arXiv:2304.02622, 2023. - https://doi.org/10.48550/arXiv.2304.02622
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