T. A. Springer's original construction[1] 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.
Springer later gave a different construction,[4] 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[5] and, allegedly, discovered Kazhdan–Lusztig polynomials in the process.
Its irreducible representations over a field of characteristic zero are also parametrized by the partitions of n. The Springer correspondence in this case is a bijection, and in the standard parametrizations, it is given by transposition of the partitions (so that the trivial representation of the Weyl group corresponds to the regular unipotent class, and the sign representation corresponds to the identity element of G).
Springer correspondence turned out to be closely related to the classification of primitive ideals in the universal enveloping algebra of a complex semisimple Lie algebra, both as a general principle and as a technical tool.