Fundamental domains of arithmetic quotients of reductive groups over number fields

Abstract

For a connected reductive algebraic group G over a number field k, we investigate the Ryshkov domain R Q associated to a maximal k-parabolic subgroup Q of G. By considering the arithmetic quotients G(k)\G(‫)ށ‬ 1 /K and i \G(k)/K ∞ , with K a maximal compact subgroup of the adele group G(‫)ށ‬ and the i arithmetic subgroups of G(k), we present a method of constructing fundamental domains for Q(k)\R Q and i \G(k ∞ ) 1 . We also study the particular case when G = GL n , and subsequently construct fundamental domains for P n , the cone of positive definite Humbert forms over k, with respect to the subgroups i .