On the Springer resolution of the minimal unipotent conjugacy class
Let \(G\) be a simple complex Lie group, \(U\) the subvariety of unipotent elements in \(G\), and \(\mathcal B\) the variety of all Borel subgroups of \(G\). \textit{T. Springer} [cf. Algebr. Geom., Bombay Colloq. 1968, 373--391 ( Zbl 0195.50803)] proved that the projection \(\pi: V=\{(u,B)\in U\times\mathcal B: u\in B\}\to U\) resolves the singularities of \(U\). The fibre \({\mathcal B}_ u=\pi^{-1}(u)\) depends (up to isomorphism) only on the conjugacy class of an element \(u\in U\). In general, one can associate a graph \(\Gamma_ u\) to any fibre \(\mathcal B_ u\) [\textit{D. Kazhdan} and \textit{G. Lusztig}, Invent. Math. 53, 165--184 (1979; Zbl 0499.20035)]. In the paper under review the authors compute the graph \(\Gamma_ u\), for a unipotent element u belonging to the minimal conjugacy class, the unique class whose closure does not contain any other conjugacy class, except the class \(\{\) 1\(\}\). If \(G\) is of type A, D, or E, then \(\Gamma_ u\) turns out to be 'dual' to the graph, \(\Gamma_ u'\), of the resolution of a subregular element. They show that for every \(G\) of type other than \(G_ 2\) (resp. of type A,D,E) the graph, \(\Gamma_ u\), of a subregular element \(u\) (resp. of an element \(u\) in the minimal class) can be found among the KL-graphs. Moreover if \(G\) is of type A,D or E, then the KL-graph corresponding to a unipotent element from the minimal conjugacy class is the 'dual' to the KL-graph corresponding to a subregular element.
scientific article; zbMATH DE number 3867623
scientific article