Simple linear compactifications of odd orthogonal groups (Q372673)

The article classifies simple linear compactifications of the group \(G=\text{SO}(2n+1)\). All linear compactifications are given by the following construction. Let \(\Pi\) be a finite subset of the set of dominant weights of \(G\). Then NEWLINE\[NEWLINEX_{\Pi} := \overline{(G\times G)[\text{Id}]} \subset \mathbb{P}\left(\bigoplus_{\lambda \in \Pi}\text{End}(V(\lambda))\right),NEWLINE\]NEWLINE where \(V(\lambda)\) is the simple representation of highest weight \(\lambda\), is a linear compactification.NEWLINENEWLINESuch a compactification is called simple if it contains a unique closed \(G\times G\) orbit. The problem reduces to two questions. Firstly, for which \(\Pi\) is the compactification \(X_{\Pi}\) simple? Secondly, for which \(\Pi\) and \(\Pi'\) is there an isomorphism \(X_{\Pi} \cong X_{\Pi'}\)?NEWLINENEWLINEBoth these questions can be completely answered in terms of combinatorics of the weight lattice. The question when \(X_{\Pi}\) is simple was previously answered by \textit{D. A. Timashev} [Sb. Math. 194, No. 4, 589--616 (2003); translation from Mat. Sb. 194, No. 4, 119--146 (2003; Zbl 1074.14043)]. The author restates this result in his notation before he continues to completely answer the second question. The combinatorics are complicated and it takes most of the article to figure them out.