On the shape of fundamental domains in \(\text{GL}(n,{\mathbb{R}})/\text{O}(n)\)
A further study is made of Grenier's fundamental domain for \(\text{GL}(n,\mathbb{Z})\) acting on positive \(n \times n\) real matrices of determinant 1. See \textit{D. Grenier} [Pac. J. Math. 132, 293-317 (1988; Zbl 0699.10045)]. This fundamental domain has the advantage of being very close to that for \(n = 2\). It is thus more useful for the theory of automorphic forms for \(\text{GL}(2,\mathbb{Z})\) than that of Minkowski. The paper makes comparisons of the Grenier fundamental domain with Siegel sets. These results are needed for work on lower bounds of the eigenvalues of the Laplacian on these fundamental domains for \(\text{GL}(n,\mathbb{Z})\). See \textit{J. Huntley} [Ph. D. Thesis, Stanford, 1987]. In particular, it is proved that one needs 10 copies of the fundamental domain for \(\text{GL}(3,\mathbb{Z})\) to cover a certain Siegel set. This result was used by Huntley to show that in this case a lower bound for the absolute value of an eigenvalue of \(\Delta\) is \(3 \pi^ 2/2\). The problem of reduction of quadratic forms is connected to this. See the reviewer [Harmonic analysis on symmetric spaces and applications. II (1988; Zbl 0668.10033)] for some of the history and motivation. A Satake compactification of the fundamental domain is also considered. This is analogous to that originally considered by Satake for the Siegel modular group and generalized by Satake and others to other algebraic groups. The compactification considered here is the maximal Satake compactification unlike that of the previous paper of Grenier [loc. cit.].
fundamental domain for \(\text{GL}(n,\mathbb{Z})\)
scientific article; zbMATH DE number 572465