High Quality Content by WIKIPEDIA articles! In mathematics, the Springer representations are certain representations of the Weyl group W associated to unipotent conjugacy classes of a semisimple algebraic group G. There is another parameter involved, a representation of a certain finite group A(u) canonically determined by the unipotent conjugacy class. To each pair (u, ?) consisting of a unipotent element u of G and an irreducible representation ? of A(u), one can associate either an irreducible representation of the Weyl group, or 0. 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.

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