Representations of $p$-adic groups over commutative rings

Abstract

Motivated by the Langlands program in representation theory, number theory, and geometry, the theory of representations of a reductive $p$-adic group, originally in complex vector spaces, has been widely developed in modules over a commutative ring during the last two decades. This article surveys basic results obtained during this period, assuming some familiarity with the representation theory connected to the Langlands program. Addressed to a broader audience, the 2022 ICM Noether Lecture should be accessible without prerequisites and convey intuition on the most striking results.