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Springer developed his resolution of singularities for the unipotent (or nilpotent) variety of a semsimple algebraic group (or its Lie algebra) as part of a bigger program to construct Weyl group representations in a new way by making the Weyl group act on cohomology groups of fibers of the resolution. This is a multifaceted subject by now and can be approached in somewhat different ways. As Mike indicates in his answer, Ginzburg and others provide at least a partial pathway (mainly in characteristic 0) using modern algebraic geometry and emphasizing the best-behaved example $SL_n$. Some of the machinery used is necessarily rather sophisticated, as in Springer's original papers.