Hecke algebras for p-adic groups, the explicit Local Langlands Correspondence and stability | Vidéo | Carmin.tv

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Hecke algebras for p-adic groups, the explicit Local Langlands Correspondence and stability

Appears in collection : Automorphic forms, endoscopy and trace formulas / Formes automorphes, endoscopie et formule des traces

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.

Information about the video

Date of recording 18/09/2023
Date of publication 12/10/2023
Institution CIRM
Licence CC BY NC ND
Language English
Audience Researchers, Graduate Students
Director(s) Guillaume Hennenfent
Format MP4

Citation data

DOI 10.24350/CIRM.V.20094203
Cite this video Xu, Yujie (18/09/2023). Hecke algebras for p-adic groups, the explicit Local Langlands Correspondence and stability. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.20094203
URL https://dx.doi.org/10.24350/CIRM.V.20094203

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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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