##
**Introduction aux groupes arithmétiques.**
*(French)*
Zbl 0186.33202

Actualités Scientifiques et Industrielles. 1341. Paris: Hermann & Cie. 125 p. (1969).

This book is a nice introduction to the so called reduction theory in a real algebraic group \(G_R\) relative to an arithmetic subgroup \(\Gamma\). First (§1–§6), the author discusses the classical case which is the starting part of the general theory. Second (§7–§9), he introduces the general theory of linear algebraic groups and arithmetic subgroups which are used later. Third (§10–§16), he constructs fundamental domains of the reductive groups relative to arithmetic subgroups and deduces main properties of them corresponding to the theorems of Minkowski, Siegel and Hermite which are discussed in the previous parts. Comparing with the examples and proofs of the Iwasawa decompositions, Bruhat decompositions and the construction of Siegel domains in the classical groups, given in the first and second parts, the reader can clearly see in the third part how the classical theory has been generalized to the general algebraic groups. He who is familiar with the theory of algebraic groups and who wishes to know the theory rapidly can read directly the third part with §1.

Let \(G\) be an algebraic group in \(\mathrm{GL}(V)\) defined over \(\mathbb Q\), \(V\) being a vector space over \(\mathbb C\) of dimension \(n\). Let \(L\) be a lattice in \(V_{\mathbb Q}\), and let \(G_L= \{g\in G_{\mathbb Q};\ g(L) = L\}\) be the group of \(L\)-units of \(G\). A subgroup \(\Gamma\) of \(G_{\mathbb Q}\) is called an arithmetic subgroup if there exists a lattice of \(V_{\mathbb Q}\) such that \(\Gamma\) is commensurable to \(G_L\). A subset \(\Omega\) of \(G_R\) is called fundamental for \(\Gamma\), if it satisfies: (F\(_0\)) \(K\cdot \Omega = \Omega\) for a maximal compact subgroup \(K\) of \(G_R\), (F\(_1\)) \(\Omega\cdot\Gamma = G_R\), and (F\(_2\)) (Siegel’s property) for any element \(b\) of \(G_{\mathbb Q}\), the set of elements \(\gamma\) of \(\Gamma\) such that \(\Omega b\cap\Omega\cdot\gamma\ne\emptyset\) if finite. In particular, if \(G=\mathrm{GL}(n,\mathbb R)\), \(\Gamma =\mathrm{GL}(n,\mathbb Z)\) and \(K=O(n)\), then \(X=K\backslash G_R\) is the space of the positive definite, non-degenerate quadratic forms on \(\mathbb R^n\) and this gives rise to the reduction theory of the quadratic forms.

Let \(G\) be a connected reductive group defined over \(\mathbb Q\). The Siegel domain of \(G\) over \(\mathbb Q\) is defined as a natural extension of the classical case, manipulating a maximal compact subgroup \(K\), a minimal \(\mathbb Q\)-parabolic subgroup \(P\), a maximal \(\mathbb Q\)-trivial torus \(S\) and the \(\mathbb Q\)-root system with respect to \(S\). Then the fundamental domain of \(G_R\) for \(\Gamma\) is constructed as a finite union of the right translation by elements of \(G_R\) of a Siegel domain.

He gives two different methods of construction of fundamental domains. One constitutes a generalization of the method of Hermite in the classical case given in §5. Here, the compactness criterion of \(G_R/\Gamma\) and the finiteness of \(P_{\mathbb Q}\backslash G_{\mathbb Q}/\Gamma\) are used essentially. The other is analogous to those used by Godement and Weil in the case of adélic groups [Sém. Bourbaki 15 (1962/63), Exp. No. 257, 25 p. (1964)].

Finally, he deals with the space \(X=K\backslash G_R/\Gamma\) in a particular case which is diffeomorphic to the interior of a compact variety with boundary.

Let \(G\) be an algebraic group in \(\mathrm{GL}(V)\) defined over \(\mathbb Q\), \(V\) being a vector space over \(\mathbb C\) of dimension \(n\). Let \(L\) be a lattice in \(V_{\mathbb Q}\), and let \(G_L= \{g\in G_{\mathbb Q};\ g(L) = L\}\) be the group of \(L\)-units of \(G\). A subgroup \(\Gamma\) of \(G_{\mathbb Q}\) is called an arithmetic subgroup if there exists a lattice of \(V_{\mathbb Q}\) such that \(\Gamma\) is commensurable to \(G_L\). A subset \(\Omega\) of \(G_R\) is called fundamental for \(\Gamma\), if it satisfies: (F\(_0\)) \(K\cdot \Omega = \Omega\) for a maximal compact subgroup \(K\) of \(G_R\), (F\(_1\)) \(\Omega\cdot\Gamma = G_R\), and (F\(_2\)) (Siegel’s property) for any element \(b\) of \(G_{\mathbb Q}\), the set of elements \(\gamma\) of \(\Gamma\) such that \(\Omega b\cap\Omega\cdot\gamma\ne\emptyset\) if finite. In particular, if \(G=\mathrm{GL}(n,\mathbb R)\), \(\Gamma =\mathrm{GL}(n,\mathbb Z)\) and \(K=O(n)\), then \(X=K\backslash G_R\) is the space of the positive definite, non-degenerate quadratic forms on \(\mathbb R^n\) and this gives rise to the reduction theory of the quadratic forms.

Let \(G\) be a connected reductive group defined over \(\mathbb Q\). The Siegel domain of \(G\) over \(\mathbb Q\) is defined as a natural extension of the classical case, manipulating a maximal compact subgroup \(K\), a minimal \(\mathbb Q\)-parabolic subgroup \(P\), a maximal \(\mathbb Q\)-trivial torus \(S\) and the \(\mathbb Q\)-root system with respect to \(S\). Then the fundamental domain of \(G_R\) for \(\Gamma\) is constructed as a finite union of the right translation by elements of \(G_R\) of a Siegel domain.

He gives two different methods of construction of fundamental domains. One constitutes a generalization of the method of Hermite in the classical case given in §5. Here, the compactness criterion of \(G_R/\Gamma\) and the finiteness of \(P_{\mathbb Q}\backslash G_{\mathbb Q}/\Gamma\) are used essentially. The other is analogous to those used by Godement and Weil in the case of adélic groups [Sém. Bourbaki 15 (1962/63), Exp. No. 257, 25 p. (1964)].

Finally, he deals with the space \(X=K\backslash G_R/\Gamma\) in a particular case which is diffeomorphic to the interior of a compact variety with boundary.

Reviewer: Eiichi Abe (Ibaraki)

### MSC:

22-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to topological groups |

22E40 | Discrete subgroups of Lie groups |