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SUMMARY:Konstantin Ardakov (Oxford)
DTSTART:20230328T130000Z
DTEND:20230328T140000Z
DTSTAMP:20260421T154240Z
UID:UEAPS/14
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/UEAPS/14/">T
 he central sheaf of the category of smooth mod-$p$ representations of $SL_
 2(\\mathbb{Q}_p)$</a>\nby Konstantin Ardakov (Oxford) as part of ANTLR sem
 inar\n\nLecture held in SCI 3.05.\n\nAbstract\nThis is work in progress wi
 th Peter Schneider. The Bernstein centre of a $p$-adic reductive group pla
 ys a fundamental role in the classical local Langlands correspondence. In 
 the mod-p local Langlands program\, the naive analogue of the Bernstein ce
 ntre\, namely the centre of the category $Mod(G)$ of all smooth mod-$p$ re
 presentations\, turns out to be too small: it is for example trivial whene
 ver the group in question has trivial centre. Instead\, we consider the ce
 ntres $Z(Mod(G)/\\mathcal{L})$ of the quotient categories $Mod(G)/\\mathca
 l{L}$\, as $\\mathcal{L}$ runs over all localizing subcategories of $Mod(G
 )$. We show that provided one restricts to the localizing subcategories th
 at are $stable$ $under$ $injective$ $envelopes$\, this defines a sheaf wit
 h respect to finite coverings. In the case where $G = SL_2(\\mathbb{Q}_p)$
  and $p \\neq 2\,3$\, we use recent results by Ollivier and Schneider on t
 he structure of the pro-$p$ Iwahori $Ext$-algebra to construct a certain p
 rojective variety $X$ having the property that every Zariski open subset $
 U$ of $X$ gives rise to a stable localizing subcategory $\\mathcal{L}_U$ o
 f $Mod(G)$. Both connected components of $X$ are certain chains of project
 ive lines\, and $X$ and is closely related to the recent work of Dotto\, E
 merton and Gee.\n
LOCATION:https://researchseminars.org/talk/UEAPS/14/
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