Reduction theory using semistability. II. (English) Zbl 0616.20020

Grayson, Daniel R.

Comment. Math. Helv. 61, 661-676 (1986).

This paper extends the results of Part I [ibid. 59, 600-634 (1984; Zbl 0564.20027)] to the case of an arithmetic subgroup of any semisimple algebraic group G. The methods and results of the previous paper are assumed in the current one. A numerical notion of semistability of lattices endowed with metrics is used to construct distance functions to the cusps in the symmetric space X belonging to G, leading to a realization of the Borel-Serre completion of X as a proper subset of X. Of independent interest: (1) the description of X as a certain space of inner products H on the Lie algebra g of G; (2) a criterion for when an element of the canonical filtration of the \({\mathbb{Z}}\)-points of g (with respect to a given H) is a parabolic subalgebra of g, solely in terms of the slopes of the canonical filtration.

Reviewer: A.Ash

MSC:

20H05	Unimodular groups, congruence subgroups (group-theoretic aspects)
20G10	Cohomology theory for linear algebraic groups
11E04	Quadratic forms over general fields
20G15	Linear algebraic groups over arbitrary fields
11F06	Structure of modular groups and generalizations; arithmetic groups
11E57	Classical groups

Keywords:

arithmetic subgroup; semisimple algebraic group; semistability of lattices; distance functions; cusps; symmetric space; Borel-Serre completion; inner products; Lie algebra; parabolic subalgebra; filtration

Citations:

Zbl 0564.20027