BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Linus Hamann (Princeton) DTSTART:20211104T210000Z DTEND:20211104T220000Z DTSTAMP:20260423T022807Z UID:UCSD_NTS/46 DESCRIPTION:Title: Compatibility of the Fargues-Scholze and Gan-Takeda local Langlands\ nby Linus Hamann (Princeton) as part of UCSD number theory seminar\n\nLect ure held in APM 6402 and online.\n\nAbstract\nGiven a prime $p$\, a finite extension $L/\\mathbb{Q}_{p}$\, a connected $p$-adic reductive group $G/L $\, and a smooth irreducible representation $\\pi$ of $G(L)$\, Fargues-Sch olze recently attached a semisimple Weil parameter to such $\\pi$\, giving a general candidate for the local Langlands correspondence. It is natural to ask whether this construction is compatible with known instances of th e correspondence after semisimplification. For $G = GL_{n}$ and its inner forms\, Fargues-Scholze and Hansen-Kaletha-Weinstein show that the corres pondence is compatible with the correspondence of Harris-Taylor/Henniart. We verify a similar compatibility for $G = GSp_{4}$ and its unique non-spl it inner form $G = GU_{2}(D)$\, where $D$ is the quaternion division algeb ra over $L$\, assuming that $L/\\mathbb{Q}_{p}$ is unramified and $p > 2$. In this case\, the local Langlands correspondence has been constructed by Gan-Takeda and Gan-Tantono. Analogous to the case of $GL_{n}$ and its inn er forms\, this compatibility is proven by describing the Weil group actio n on the cohomology of a local Shimura variety associated to $GSp_{4}$\, u sing basic uniformization of abelian type Shimura varieties due to Shen\, combined with various global results of Kret-Shin and Sorensen on Galois r epresentations in the cohomology of global Shimura varieties associated to inner forms of $GSp_{4}$ over a totally real field. After showing the par ameters are the same\, we apply some ideas from the geometry of the Fargue s-Scholze construction explored recently by Hansen\, to give a more precis e description of the cohomology of this local Shimura variety\, verifying a strong form of the Kottwitz conjecture in the process.\n\npre-talk at 1: 20pm.\n\nThe talk will be given via Zoom\, but we will meet in the lecture hall as usual.\n LOCATION:https://researchseminars.org/talk/UCSD_NTS/46/ END:VEVENT END:VCALENDAR