BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Yujie Xu (Columbia University) DTSTART:20240708T130000Z DTEND:20240708T140000Z DTSTAMP:20260421T154340Z UID:UEAPS/35 DESCRIPTION:Title: H ecke algebras for p-adic groups and explicit Local Langlands Correspondenc e\nby Yujie Xu (Columbia University) as part of ANTLR seminar\n\n\nAbs tract\nI will talk about several results on Hecke algebras attached to Ber nstein blocks of reductive p-adic groups\, where we construct a local Lang lands correspondence for these Bernstein blocks. Our techniques draw inspi rations from the foundational works of Deligne\, Kazhdan\, Langlands\, Lus ztig and Shahidi.\n\n\nAs an application\, we prove the Local Langlands Co njecture for G_2\, which is the first known case in literature of LLC for exceptional groups. Our correspondence satisfies an expected property on c uspidal support\, which is compatible with the generalized Springer corres pondence\, along with a list of characterizing properties including the st abilization of character sums\, formal degree property etc. In particular\ , we obtain (not necessarily unipotent) "mixed" L-packets containing "F-si ngular" supercuspidals and non-supercuspidals. I will give explicit exampl es of such mixed L-packets in terms of Deligne-Lusztig theory and Kazhdan- Lusztig parametrization.\n\nIf time permits\, I will explain how to pin do wn certain choices in the construction of the correspondence using stabili ty of L-packets\; one key input is a homogeneity result due to Waldspurger and DeBacker. Moreover\, I will mention how to adapt our general strategy to construct explicit LLC for other reductive groups\, such as GSp_4\, Sp _4\, etc\, generalizing the depth-zero L-packets of Lust-Stevens. Such exp licit description of the L-packets has been useful in number-theoretic app lications\, e.g. modularity lifting questions.\n\nVarious parts of this ta lk are based on joint works with various collaborators\, and attributions will be made clear throughout the talk.\n LOCATION:https://researchseminars.org/talk/UEAPS/35/ END:VEVENT END:VCALENDAR