Fundamental domains of arithmetic quotients of reductive groups over number fields

Lee Tim Weng

Vol. 290 (2017), No. 1, 139–168

DOI: 10.2140/pjm.2017.290.139

Abstract

For a connected reductive algebraic group $G$ over a number field $\mathbb{𝕜}$, we investigate the Ryshkov domain $R_Q$ associated to a maximal $\mathbb{𝕜}$-parabolic subgroup $Q$ of $G$. By considering the arithmetic quotients $G(\mathbb{𝕜})\setminus G{(\mathbb{𝔸})}^1∕K$ and ${\Gamma }_i\setminus G(\mathbb{𝕜})∕K_{\infty }$, with $K$ a maximal compact subgroup of the adele group $G(\mathbb{𝔸})$ and the ${\Gamma }_i$ arithmetic subgroups of $G(\mathbb{𝕜})$, we present a method of constructing fundamental domains for $Q(\mathbb{𝕜})\setminus R_Q$ and ${\Gamma }_i\setminus G{({\mathbb{𝕜}}_{\infty })}^1$. We also study the particular case when $G={\mathrm{GL}<!--FUNCTION\; APPLICATION-->}_n$, and subsequently construct fundamental domains for $P_n$, the cone of positive definite Humbert forms over $\mathbb{𝕜}$, with respect to the subgroups ${\Gamma }_i$.

Keywords

fundamental domain, arithmetic quotient, Ryshkov domain, Humbert form, Voronoi reduction

Mathematical Subject Classification 2010

Primary: 11H55

Secondary: 11F06

Milestones

Received: 31 May 2015

Revised: 23 January 2017

Accepted: 23 January 2017

Published: 7 July 2017