A topological approach to Springer's representations

Let u be a unipotent element in a complex semisimple group G and let 3,, be the variety of Bore1 subgroups of G which contain u. In [4, 51, Springer has defined a representation of W, the Weyl group of G, on the homology of 2:, and showed how to decompose the Wrepresentation in the top homology of .53’U so that all irreducible representations of W are obtained. His mehod used etale cohomology of algebraic varieties in characteristic p. In this paper, we shall give an elementary construction (independent of a he conjectured that the representation in the top homology of 2 is the two-sided regular representation of W and showed how this could be used to prove the completeness of the set of W-representation in the top homologies of z#,,.