**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.

Read more about Springer Correspondence: Construction, Example, Applications

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**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 ...