Stability of Lattices and the Partition of Arithmetic Quotients

Abstract

The sets $\gamma H_{Y}$, with $Y$ fixed, as $\gamma$ ranges over $\Gamma$ are disjoint (when not identical), and their union is stable under $\Gamma$ as is its complement in $H$. The quotient of this complement by $\Gamma$ is compact. As far as I know, it was Jim Arthur who first generalized this result explicitly to arbitrary arithmetical quotients (in 1977), although I think it's fair to say that this generalization was already implicit in Satake's work on compactifications of arithmetic quotients. In Arthur's generalization the subsets of the partition are parametrized by $\Gamma$-conjugacy classes of rational parabolic subgroups, which is also how Satake's rational boundary components are parametrized. Of course Arthur did this work with the intention of using it in dealing with his extension of the Selberg trace formula, but subsequently it has also been useful in other contexts. In this note, which is largely expositoray, I will explain Arthur's partition for $GL_{n}(\bb{Z})$, applying ideas almost entirely due to Harder, Stuhler, and Grayson, and including a self-contained account of their work.