Ryshkov domains of reductive algebraic groups

Takao Watanabe

Vol. 270 (2014), No. 1, 237–255

DOI: 10.2140/pjm.2014.270.237

Abstract

Let $G$ be a connected reductive algebraic group defined over a number field $k$. In this paper, we introduce the Ryshkov domain $R$ for the arithmetical minimum function $m_Q$ defined from a height function associated to a maximal $k$-parabolic subgroup $Q$ of $G$. The domain $R$ is a $Q\left(k\right)$-invariant subset of the adele group $G\left(\mathbb{A}\right)$. We show that a fundamental domain $\Omega$ for $Q\left(k\right)\setminus R$ yields a fundamental domain for $G\left(k\right)\setminus G\left(\mathbb{A}\right)$. We also see that any local maximum of $m_Q$ is attained on the boundary of $\Omega$.

Dedicated to Professor Ichiro Satake on his 85th birthday

Keywords

reduction theory, fundamental domain, Hermite constant

Mathematical Subject Classification 2010

Primary: 11H55

Secondary: 11F06, 22E40

Milestones

Received: 29 March 2013

Revised: 30 July 2013

Accepted: 5 August 2013

Published: 2 August 2014