Remarks on the wonderful compactification of semisimple algebraic groups (Q1306808)

De Concini and Procesi constructed and studied ``wonderful compactifications'' of symmetric varieties [cf. \textit{C. De Concini} and \textit{C. Procesi}, in: Invariant theory, Proc. 1st 1982 Sess. CIME, Montecatini, Lect. Notes Math. 996, 1-44 (1983; Zbl 0581.14041)]. More precisely, if \(G\) is a semisimple algebraic group of adjoint type over a field of complex numbers, \(H\) is the subgroup of all fixed points of an involution \(\sigma\) of \(G\) that is induced by an involution \({\widehat{\sigma}}\) of the simply connected covering \({\widehat G}\) of \(G\), then they have constructed a complete embedding \(\overline{G/H}\) of the homogeneous space \(G/H\), with boundary being a union of normal crossing divisors. In particular, one gets such a compactification \(\overline G\) for the group \(G\) (\(G\) being considered as \((G{\times}G)/{\triangle(G)}\)). In the paper under review the author considers the subgroup \(H\) as above and views \(\overline G\) as a \(H\) variety (\(H\) acting on the right). He also \(H\) linearises ample line bundles \(L\), and one can therefore take the G.I.T. quotients \({\overline G}^{ss}(L)//H\) of \(\overline G\). This way also one obtains a compactification of \(G/H\). A natural question is to get an explicit relationship between these compactifications and the ``wonderful compactifications''. The aim of the reviewed paper is to prove the following result: (a) There is a \(G\)-linearised ample line bundle \(L\) on \(\overline G\) such that \({\overline G}^{ss}(L)//H\) is isomorphic to \(\overline{G/H}\). The author also obtains: (b) a natural functorial property of the wonderful compactification \(\overline G\) of \(G\) [cf. theorem 4.7 of the paper for a precise statement].