Springer Correspondence

Springer Correspondence

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, φ) consisting of a unipotent element u of G and an irreducible representation φ of A(u), one can associate either an irreducible representation of the Weyl group, or 0. The association

depends only on the conjugacy class of u and generates a correspondence between the irreducible representations of the Weyl group and the pairs (u, φ) modulo conjugation, called the Springer correspondence. It is known that every irreducible representation of W occurs exactly once in the correspondence, although φ may be a non-trivial representation. The Springer correspondence has been described explicitly in all cases by Lusztig, Spaltenstein and Shoji. The correspondence, along with its generalizations due to Lusztig, plays a key role in Lusztig's classification of the irreducible representations of finite groups of Lie type.