Chevalley groups over \Z: A representation-theoretic approach

2024/08/29 by Abid Ali, Ali, Abid, Lisa Carbone +3
Mathematics · #17B10 #17B67 #FOS: Mathematics #Finite Group Theory Research #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2408.16895

openalex publication_date 2024/08/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let G(ℚ) be a simply connected Chevalley group over ℚ corresponding to a simple Lie algebra \mathfrak g over ℂ. Let V be a finite dimensional faithful highest weight \mathfrak g-module and let V_ℤ be a Chevalley ℤ-form of V. Let Γ(ℤ) be the subgroup of G(ℚ) that preserves V_ℤ and let G(ℤ) be the group of ℤ-points of G(ℚ). Then G(ℚ) is integral if G(ℤ)=Γ(ℤ). Chevalley's original work constructs a scheme-theoretic integral form of G(ℚ) which equals Γ(ℤ). Here we give a representation-theoretic proof of integrality of G(ℚ) using only the action of G(ℚ) on V, rather than the language of group schemes. We discuss the challenges and open problems that arise in trying to extend this to a proof of integrality for Kac-Moody groups over ℚ.

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