Ryshkov domains of reductive algebraic groups

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.k/-invariant subset of the adele group G.‫./ށ‬ We show that a fundamental domain for Q.k/nR yields a fundamental domain for G.k/nG.‫./ށ‬ We also see that any local maximum of m Q is attained on the boundary of . MSC2010: primary 11H55; secondary 11F06, 22E40.