Construction
Several approaches to Springer correspondence have been developed. T. A. Springer's original construction (1976) proceeded by defining an action of W on the top-dimensional l-adic cohomology groups of the algebraic variety Bu of the Borel subgroups of G containing a given unipotent element u of a semisimple algebraic group G over a finite field. This construction was generalized by Lusztig (1981), who also eliminated some technical assumptions. Springer later gave a different construction (1978), using the ordinary cohomology with rational coefficients and complex algebraic groups.
Kazhdan and Lusztig found a topological construction of Springer representations using the Steinberg variety and, allegedly, discovered Kazhdan–Lusztig polynomials in the process. Generalized Springer correspondence has been studied by Lusztig-Spaltenstein (1985) and by Lusztig in his work on character sheaves. Borho and MacPherson (1983) gave yet another construction of the Springer correspondence.
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