ON THE SHAPE OF FUNDAMENTAL DOMAINS IN GL(N, R)/O(N)

We investigate parameters for the symmetric space H = G/K, G = GL(n, R) , K = O(n), in the sense of positive definite quadratic forms. This leads to a description for the fundamental domain H/GAMMA where GAMMA is an arithmetic subgroup of G . We also see interesting relations with the Siegel sets. This enables us to explicitly describe Satake compactifcations of H/GAMMA. We will also consider the behavior at the ''bottom'' of the fundamental domains.