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On integral images of Curtis homomorphisms for $\mathrm{GL}_n$ and $\mathrm{U}_n$

Tzu-Jan Li

Abstract

For $G = \mathrm{GL}_n$ or $\mathrm{U}_n$ defined over a finite field of characteristic $p$, we refine a result of Bonnafé and Kessar on the saturatedness of the Curtis homomorphism $\mathrm{Cur}^G$ by describing the image of $\mathrm{Cur}^G$ over $\overline{\mathbb{Z}}[1/p]$ via a system of linear conditions.

On integral images of Curtis homomorphisms for $\mathrm{GL}_n$ and $\mathrm{U}_n$

Abstract

For or defined over a finite field of characteristic , we refine a result of Bonnafé and Kessar on the saturatedness of the Curtis homomorphism by describing the image of over via a system of linear conditions.
Paper Structure (4 sections, 3 theorems, 20 equations)

This paper contains 4 sections, 3 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. A "Fourier transform"
  3. Translation of Theorem \ref{['mainthm']} into the dual side
  4. Proof of Theorems \ref{['mainthm']}, \ref{['K-thm0']} and \ref{['K-thm']}

Key Result

Theorem 1.2

Let $G$ be $\mathrm{GL}_n$ or $\mathrm{U}_n$ (defined over $\mathbb{F}_q$) with $n\in\mathbb{Z}_{>0}$, and let $\Lambda$ be a subring of $\overline{\mathbb{Q}}$ containing $\overline{\mathbb{Z}}[1/p]$. Then where $\Omega$ is the set of the elements $(f_S)_{S\in\mathscr{T}_G}$ of $\prod_{S\in\mathscr{T}_G}\Lambda[S^F]$ such that for every $F$-stable Levi subgroup $L$ of $G$ and every $s\in Z(L)^F

Theorems & Definitions (3)

  • Theorem 1.2
  • Theorem 3.2
  • Theorem 3.3