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

TL;DR

The paper determines the structure of the graded mod $p$ Iwasawa algebra for a pro-$p$ Iwahori subgroup of a split reductive group over an unramified $p$-adic field, under the assumption $p>h+1$. It constructs an explicit graded Lie algebra $\mathrm{gr}(I)$ via Lazard $p$-valuations, describes its $P$-operator, and computes the universal enveloping algebra, yielding a precise presentation and a maximal commutative quotient. A key contribution is the comparison between the $\omega$-filtration and the $\mathfrak{m}$-adic filtration, establishing an isomorphism of the corresponding graded algebras up to rescaling, and thereby providing an explicit description of $\mathrm{gr}_{\mathfrak{m}}(\mathbb{F}_p[[I]])$ in terms of the Lazard data. The results feed into the study of Gelfand–Kirillov dimensions for smooth mod $p$ representations of $G$, showing that in many cases one must work with noncommutative quotients of the graded algebra to align with global expectations from completed cohomology. Overall, the work extends known $\mathrm{gr}$-algebra analyses from $\mathrm{GL}_2$ to arbitrary split reductive groups, with implications for mod $p$ Langlands and representation-theoretic size questions.

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 $G$.

Theorems & Definitions (33)

• Theorem 1.1: Proposition \ref{['prop:overline-gr(I)-pres']}
• Theorem 1.2: Proposition \ref{['prop: ideal-comparison']}
• Theorem 2.1
• Corollary 2.2
• Lemma 2.3
• proof
• Proposition 3.1
• proof
• Proposition 3.2
• proof