On the shape of fundamental domains in \(\text{GL}(n,{\mathbb{R}})/\text{O}(n)\)
type: Item
instance of: scholarly article
title: On the shape of fundamental domains in \(\text{GL}(n,{\mathbb{R}})/\text{O}(n)\)
zbMATH Open document ID: 0814.11025
author: Douglas Grenier
published in: Pacific Journal of Mathematics
publication date: 1995-06-18T00:00:00Z
review text: 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.].
Mathematics Subject Classification ID: 11F06
Mathematics Subject Classification ID: 11H55
zbMATH DE Number: 572465
zbMATH Keywords: fundamental domain for \(\text{GL}(n,\mathbb{Z})\)
zbMATH Keywords: reduction of quadratic forms
zbMATH Keywords: maximal Satake compactification
reviewed by: Audrey A. Terras
MaRDI profile type: Publication
full work available at URL: pjm.1993.160.53
OpenAlex ID: W1999064304
DOI: 10.2140/PJM.1993.160.53
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description: scientific article; zbMATH DE number 572465