Mod p Bernstein centres of p-adic groups

Abstract: The centre of the category of smooth mod $p$ representations of a $p$-adic reductive group does not distinguish the blocks of finite length representations, in contrast with Bernstein's theory in characteristic zero. Motivated by this observation and the known connections between the Bernstein centre and the local Langlands correspondence in families, we consider the case of $\mathrm{GL}_2(\mathbb{Q}_p)$ and we prove that its category of representations extends to a stack on the Zariski site of a simple geometric object: a chain $X$ of projective lines, whose points are in bijection with Paskunas's blocks. Taking the centre over each open subset we obtain a sheaf of rings on $X$, and we expect the resulting space to be closely related to the Emerton-Gee stack for $2$-dimensional representations of the absolute Galois group of $\mathbb{Q}_p$. Joint work in progress with Matthew Emerton and Toby Gee.

algebraic geometrynumber theoryrepresentation theory

Audience: researchers in the topic

Comments: Zoom ID: 650 3772 0269

Password: 585279

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POINTS - Peking Online International Number Theory Seminar

Series comments: Description: Seminar on number theory and related topics

This seminar series is sponsored by the Beijing International Center of Mathematical Research (BICMR) and the School of Mathematical Sciences of Peking University.

The conference number and password are available on the external website. See also the announcements on bicmr.pku.edu.cn

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