Representations of $p$-adic groups over coefficient rings

TL;DR

This paper surveys recent advances in the representation theory of reductive p-adic groups over various coefficient rings, highlighting developments motivated by the Langlands program in the 21st century.

Contribution

It compiles and reviews key results and progress in the area of p-adic group representations over coefficient rings beyond complex numbers, emphasizing recent developments.

Findings

Summarizes foundational results in p-adic representation theory.

Highlights progress related to the Langlands program.

Provides an overview of recent research trends and open problems.

Abstract

Motivated by the Langlands program in representation theory, number theory and geometry, the theory of representations of a reductive $p$-adic group over a coefficient ring different from the field of complex numbers has been widely developped during the last two decades. This article provides a survey of basic results obtained in the 21st century.