New realizations of the maximal Satake compactifications of Riemannian symmetric spaces of noncompact type (Q703993)

The compactification of symmetric Riemannian spaces \(X=G/K\) of noncompact type and their finite volume quotients have received much attention. One such compactification of \(X\) was constructed by \textit{I. Satake} [Ann. Math. (2) 71, 77--110 (1960; Zbl 0094.34603)] using faithful, irreducible representations of \(G\) and is called the maximal Satake compactification of \(X\). In the paper under review, the authors construct new \(G\)-equivariant realizations of the maximal Satake compactification. These compactifications are given by taking the closure of \(X\) under suitable embeddings. Briefly, we describe the embeddings. The first compactification is given by embedding \(X\) into the Grassmannian of \(m\)-dimensional subspaces of the corresponding Lie algebra \({\mathfrak g}\) of \(G\), where \(m\) is the dimension of \(K\). The second compactification relates the wonderful compactification of \textit{C. De Concini} and \textit{C. Procesi} [Lect. Notes Math. 996, 1-44 (1983; Zbl 0581.14041)] with the maximal Satake compactification, the former being a compactification of the associated Hermitian symmetric space \(X_\mathbb{C}\) of \(X\). The third compactification is obtained from an embedding of \(X\) into the Grassmannian of real \(n\)-dimensional subspaces of the complexified Lie algebra \({\mathfrak g} \otimes_{\mathbb{R}}\mathbb{C}\), where \(n\) is the dimension of \({\mathfrak g}\). Finally, the fourth compactification is obtained via an embedding of \(X\) into the orthogonal group of the compact real form \({\mathfrak u}\) of \({\mathfrak g}\) equipped with the Killing form \(B\). This well written paper serves as a piece in a program to study the behavior of a natural Poisson structure on the boundary of \(X\). The authors note that corollary 1.3 is a first step in establishing a connection between Poisson geometry and harmonic analysis on \(X\).