Zero weight spaces and the Springer correspondence
The irreducible representations of \({\text{SL}}(n,{\mathbb C})\) are parameterised by Young tableau. If the tableau has \(n\) boxes, then the representation has the property that zero is a weight but twice a root is not a weight. For any semisimple group, a representation with this property is called `small'. It may be checked for \({\text{SL}}(n,{\mathbb C})\), that the action of the symmetric group \(S_n\) (as the Weyl group of \({\text{SL}}(n,{\mathbb C})\)) on the zero weight space of a small representation is the representation of \(S_n\) given by the dual tableau. This article generalises this fact to the other simply laced groups, the \(D\) and \(E\) series. For this, we need to give another description of the action of \(S_n\) corresponding to a particular tableau. It is the Springer representation determined as follows. Choose a unipotent element \(u\) in \({\text{SL}}(n,{\mathbb C})\) with Jordan blocks of sizes determined by the tableau regarded as a partition of \(n\). Then the representation is realised as the top cohomology with complex coefficients of the fixed point set of \(u\) in the full flag variety of \({\text{SL}}(n,{\mathbb C})\). It is in this form that the statement for \({\text{SL}}(n,{\mathbb C})\) is generalised. It asserts the existence of a unipotent element \(u\in G\) so that exactly the same statement holds. There is a difference in that these Weyl group representations may be reducible. The proof is empirical, i.e. by determining all small representations and so on. For \(G\) that are not simply laced, this particular formulation fails and it is an interesting question whether there is a generally valid reformulation.
scientific article; zbMATH DE number 1251203