Satake diagrams and real structures on spherical varieties (Q2788771)

Let \(G\) be a complex Lie group and \(\sigma\) be a real structure on \(G\). A real structure \(\mu\) on a homogeneous space \(X=G/H\) is said to be \textit{\(\sigma\)-equivariant} if \(\mu(gx)=\sigma(g)\mu(x)\), \(\forall g\in G\), \(x\in X\). It is shown that such a real structure \(\mu\) exists and is unique if \(H\) is \(G\)-conjugate to \(\sigma(H)\) (a necessary condition for existence) and self-normalizing (a sufficient condition for uniqueness). If \(G\) is connected and semisimple, then \(\sigma\) is determined by a Satake diagram [\textit{I. Satake}, Ann. Math. (2) 71, 77--110 (1960; Zbl 0094.34603); \textit{A. L. Onishchik}, Lectures on real semisimple Lie algebras and their representations. Zürich: European Mathematical Society Publishing House (2004; Zbl 1080.17001)]; in particular, there is a Satake involution \(\epsilon_{\sigma}\) of the Dynkin diagram of \(G\). The main result of the paper is that, if \(\epsilon_{\sigma}=\text{id}\) and \(H\) is a self-normalizing \textit{spherical} algebraic subgroup of \(G\) (i.e., \(X\) contains an open orbit of a Borel subgroup of \(G\)), then there is a unique \(\sigma\)-equivariant real structure \(\mu\) on \(X\). Furthermore, in this case \(X\) admits a wonderful compactification \(\overline{X}\) [\textit{F. Knop}, J. Am. Math. Soc. 9, No. 1, 153--174 (1996; Zbl 0862.14034)] and \(\mu\) extends to \(\overline{X}\). These results were previously proved by the author and S.~Cupit-Foutou for \(\sigma\) defining a split real form of \(G\) [\textit{D. Akhiezer} and \textit{S. Cupit-Foutou}, J. Reine Angew. Math. 693, 231--244 (2014; Zbl 1314.14101)].NEWLINENEWLINEThe idea of the proof is to look at the action of \(\sigma\) on the Luna-Vust invariants of \(X\) [\textit{F. Knop}, in: Proceedings of the Hyderabad conference on algebraic groups held at the School of Mathematics and Computer/Information Sciences of the University of Hyderabad, India, December 1989. Madras: Manoj Prakashan. 225--249 (1991; Zbl 0812.20023); \textit{D. A. Timashev}, Homogeneous spaces and equivariant embeddings. Berlin: Springer (2011; Zbl 1237.14057)]. It comes from the action of \(\sigma\) on the weight lattice of the Borel subgroup \(B\subset G\) corresponding to the simple roots on the Satake diagram. The latter action is given by \(\lambda\mapsto\lambda^{\sigma}\), \(\lambda^{\sigma}(b)=\overline{\lambda(g^{-1}\sigma(b)g)}\), \(\forall b\in B\), where \(\sigma(B)=gBg^{-1}\). It turns out that this action of \(\sigma\) on weights coincides with the natural action of \(\epsilon_{\sigma}\), hence it is trivial whenever \(\epsilon_{\sigma}=\text{id}\). Then it is easy to deduce that the Luna-Vust invariants of \(G/H\) and \(G/\sigma(H)\) are the same. By \textit{I. V. Losev}'s theorem [Duke Math. J. 147, No. 2, 315--343 (2009; Zbl 1175.14035)], \(H\) and \(\sigma(H)\) are \(G\)-conjugate, which yields a unique \(\sigma\)-equivariant real structure \(\mu\) on \(X=G/H\). The extension of \(\mu\) to \(\overline{X}\) comes from the uniqueness of the wonderful compactification (which is also described in terms of the Luna-Vust invariants).