The central sheaf of the category of smooth mod-$p$ representations of $SL_2(\mathbb{Q}_p)$

Abstract: This is work in progress with Peter Schneider. The Bernstein centre of a $p$-adic reductive group plays a fundamental role in the classical local Langlands correspondence. In the mod-p local Langlands program, the naive analogue of the Bernstein centre, namely the centre of the category $Mod(G)$ of all smooth mod-$p$ representations, turns out to be too small: it is for example trivial whenever the group in question has trivial centre. Instead, we consider the centres $Z(Mod(G)/\mathcal{L})$ of the quotient categories $Mod(G)/\mathcal{L}$, as $\mathcal{L}$ runs over all localizing subcategories of $Mod(G)$. We show that provided one restricts to the localizing subcategories that are $stable$ $under$ $injective$ $envelopes$, this defines a sheaf with 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 the structure of the pro-$p$ Iwahori $Ext$-algebra to construct a certain projective variety $X$ having the property that every Zariski open subset $U$ of $X$ gives rise to a stable localizing subcategory $\mathcal{L}_U$ of $Mod(G)$. Both connected components of $X$ are certain chains of projective lines, and $X$ and is closely related to the recent work of Dotto, Emerton and Gee.

Mathematics

Audience: researchers in the topic

( paper )

ANTLR seminar

Series comments: This is the Algebra, Number Theory, Logic and Representation theory seminar.