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Journal of Lie Theory 33 (2023), No. 3, 845--873
Copyright Heldermann Verlag 2023

A Fundamental Domain for the General Linear Group by Means of Successive Minima

Masami Fujimori
Kanagawa Institute of Technology, Atsugi, Kanagawa, Japan

We present a variant of the method of construction of Watanabe of a fundamental domain for the translation action by the subgroup of rational points on the whole group of adélic points of the general linear group over an arbitrary number field. The obtained domain is a generalization of Grenier's one, not Minkowski's. For several fields, we profit our presentation to show that the fundamental domain is in a so-called {Siegel} set. As an example, inequalities bounding our fundamental domain are explicitly determined in the case of degree 3 over the imaginary quadratic field with discriminant -3, which will be interesting to some people.

Keywords: Fundamental domain, Ryshkov domain, general linear group.

MSC: 11H55, 11H50, 20G35.

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