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A note on the admissibility of smooth simple $RG$-modules

Mihir Sheth

Abstract

Let $G$ be a $p$-adic reductive group and $R$ be a noetherian Jacobson $\mathbb{Z}[1/p]$-algebra. In this note, we show that every smooth irreducible $R$-linear representation of $G$ is admissible using the finiteness result of Dat, Helm, Kurinczuk and Moss for Hecke algebras over $R$.

A note on the admissibility of smooth simple $RG$-modules

Abstract

Let be a -adic reductive group and be a noetherian Jacobson -algebra. In this note, we show that every smooth irreducible -linear representation of is admissible using the finiteness result of Dat, Helm, Kurinczuk and Moss for Hecke algebras over .

Paper Structure

This paper contains 2 theorems, 2 equations.

Key Result

Theorem 1

For any noetherian $\mathbb{Z}[1/p]$-algebra $R$ and any compact open subgroup $K\subseteq G$, the Hecke algebra $H_{R}(G,K)$ is a finitely generated module over $Z_{R}(G,K)$ and $Z_{R}(G,K)$ is a finitely generated $R$-algebra.

Theorems & Definitions (6)

  • Theorem 1
  • Theorem 2
  • proof
  • proof : Proof of Theorem \ref{['mainthm']}
  • Remark 3
  • Remark 4