Measures, Integration and Quadratic Model

Abstract

We shall give various formulas related to measures on GL_n and its subgroups. We also compute the volume of a fundamental domain, a computation which was originally carried out by Minkowski. Essentially we follow Siegel’s proof [Sie 45]. We note historically that people used to integrate over fundamental domains, until Weil pointed out the existence of a Haar (invariant) measure on homogeneous spaces with respect to unimodular subgroups in his book [We 40], and observed that Siegel’s arguments could be cast in the formalism of this measure [We 46].