The following theorem is a celebrated result from [1]. A very good reference for this subject is also [2].
(Borel, Harish-Chandra)
Let \(\mathcal{G}\subseteq SL_n(\mathbb{C})\) be an algebraic group defined over \(\mathbb{Q}\). Then \(\mathcal{G}_{\mathbb{R}}/\mathcal{G}_{\mathbb{Z}}\) has a finite invariant measure if any only if \(X_{\mathbb{Q}}(\mathcal{G}^{0}) = \{ e\}\), where \(\mathcal{G}^{0} \subseteq \mathcal{G}\) is the connected component of identity in the Zariski topology. That is to say, there are no non-trivial \(\mathbb{Q}\)-characters of \(\mathcal{G}^{0}\).
If the interest is in finding whether or not \(\mathcal{G}_{\mathbb{R}}/\mathcal{G}_{\mathbb{Z}}\) is compact, BHC theorem also deals with this special case in the paper [1]. We have that \(\mathcal{G}_{\mathbb{R}}/\mathcal{G}_{\mathbb{Z}}\) is compact if and only if every unipotent element of \(\mathcal{G}_{\mathbb{Z}}\), or equivalently of \(\mathcal{G}_{\mathbb{Q}}\) belongs to the radical of \(\mathcal{G}\).
This condition also holds if and only if \(\mathcal{G}_{\mathbb{Q}}\) is anisotropic.
In particular, this implies the Dirichlet’s unit theorem when \(\mathcal{G}\) is the multiplicative group of unit norm element in a number field \(K\). This group is defined over \(\mathbb{Q}\), but not over \(K\).
This page was updated on December 8, 2021.
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