The canonical combinatorial sheaf associated to an involution of a semisimple adjoint group. Case \(A_ n\)
Let \(G\) be a semisimple adjoint group over \(\mathbb{C}\) with involution \(\sigma\) and consider an associated restricted root system with basis of simple roots \(\Sigma\). The author defines a combinatorial sheaf of rings \({\mathcal F}_ G^ \sigma\) on the finite simplicial complex \(({\mathcal V},{\mathcal S})= (\Sigma,\{\text{subsets in }\Sigma\})\). As a consequence of some results of \textit{E. Bifet}, \textit{C. de Concini} and \textit{C. Procesi} [Adv. Math. 82, No. 1, 1-34 (1990; Zbl 0743.14018)] such a sheaf actually determines the rational \(G\)-equivariant cohomology ring of any good compactification of the symmetric variety \(G/G^ \sigma\). The author describes explicitly \({\mathcal F}_ G^ \sigma\) for the case \(G=\text{PGl}(n,\mathbb{C})\).
semisimple adjoint group over \(\mathbb{C}\)
rational \(G\)-equivariant cohomology ring
scientific article; zbMATH DE number 549750