Springer's Weyl group representations through characteristic classes of cone bundles
A nilpotent orbit \({\mathcal O}_ u\) in a complex semisimple Lie algebra gives rise to a collection of cone bundles on the flag variety, by taking the closed components of its preimage under Springer's resolution of singularities. Using the generalization of inverse Chern classes of vector bundles to Segre classes of cone bundles due to Fulton and the third author, we attach to each such cone bundle a characteristic class in the cohomology of the flag variety, which is interpreted as a harmonic polynomial on the Cartan subalgebra. Using the intersection homology approach to the study of nilpotent varieties as by \textit{W. Borho} and \textit{R. MacPherson} in Astérisque 101-102, 23-74 (1983; Zbl 0576.14046) and C. R. Acad. Sci., Paris, Sér. I 292, 707-710 (1981; Zbl 0467.20036) we show that this collection of polynomials transforms under the action of the Weyl group according to Springer's irreducible representation \(\rho_ u\) which is usually constructed from \({\mathcal O}_ u\) by quite different means.
scientific article; zbMATH DE number 4012802
scientific article