On the conjugacy of maximal unipotent subgroups of real semisimple Lie groups (Q2905197)

Let \(G\) be a connected real reductive group, identified with its \(\mathbb{R}\)-points. Let \(G=KAN\) be an Iwasawa decomposition. The aim of this article is to prove the following theorem: any unipotent subgroup \(U\) of \(G\) is conjugate by \(G\) to a subgroup of \(N\). As a consequence, all maximal unipotent subgroups of \(G\) are conjugate.NEWLINENEWLINEAlthough this is known among experts, the authors give a simple and geometric proof. The crucial ingredient is the existence of closed orbits under the action of a real algebraic group; the complex case is well-known, and the real case follows rather easily. The assertion then follows by considering the action of \(U\) on \(G/H\), where \(H\) is (the \(\mathbb{R}\)-points) of the Zariski closure of \(AN\) in \(G\).NEWLINENEWLINEThe authors also mentioned that their arguments prove the following result of Mostow and Vinberg: embed \(G\) into some \(\text{GL}(V)\) for some real vector space \(V\). Let \(S\) be a connected solvable subgroup of \(G\) whose elements have all real eigenvalues as linear transformations on \(V\), then \(S\) is conjugate to a subgroup of \(AN\).