Mod $p$ Iwasawa algebras of pro-$p$ Iwahori subgroups

TL;DR

This paper analyzes the structure of mod p Iwasawa algebras associated with pro-p Iwahori subgroups of certain p-adic groups, revealing their graded structure, maximal commutative quotients, and implications for smooth mod p representations.

Contribution

It determines the graded mod p Iwasawa algebra structure, its maximal commutative quotient, and connects these to Gelfand--Kirillov dimensions of smooth mod p representations.

Findings

Explicit description of the graded mod p Iwasawa algebra structure.

Identification of the maximal commutative quotient.

Relation to Gelfand--Kirillov dimensions of representations.

Abstract

Suppose $F$ is a finite unramified extension of $\mathbb{Q}_p$, and $G$ is the group of $F$-points of a split, connected, reductive group over $F$. Under a natural restriction on $p$, we determine the structure of the graded mod $p$ Iwasawa algebra $\textrm{gr}_{\mathfrak{m}}(\mathbb{F}_p [\![ I]\!])$, where $I$ is a pro-$p$ Iwahori subgroup of $G$. We also determine its maximal commutative quotient, and relate these results to Gelfand--Kirillov dimensions of smooth mod $p$ representations of $G$.