The goal is to construct irreducible representation of Weyl group in a geometric way rather than combinatorial. Recall the Springer resolution of the nilpotent cone where can also be identified with the cotangent bundle of the flag variety .
The nilpotent cone are acted on by where acts by dilation. If we let be the stablizer of a nilpotent element , then acts on the Springer fiber . If we consider the induced action on the cohomology , then the action of factors through the component group . Since surjects onto with kernel , the component group surjects onto , so we get a representation of .
Let be the Weyl group of . Springer discovered that even though doesn’t act directly on the Springer fiber , there is a natural -action on the cohomology . Moreover, this action commute with the above action by . For every and every irrep of , the -isotypic component will exhaust all the irrep of the Weyl group . See Thm. 1.5.1 of Yun’s note.
For example, if , then the Springer representation can be identified with the induced action on from the right action of on . Using Borel’s presentation of the cohomology ring of the flag and Chevalley’s restriction theorem, we can further identify this representation with the regular representation of .
One construction is by the theory of perverse sheaves. The key theorem is Thm. 1.5.7, which says the complex is a perverse sheaf on whose endomorphism ring is canonically isomorphic to the group ring . In particular, acts on the stalk of , i.e. it acts on for all . The idea is that by the dimension formula for the Springer fiber, one sees that the Springer resolution is semismall and the Grothendieck resolution is small, which is a technical condition that guarantees is perverse and it is the intermediate extension of its restriction to any open dense subset of . In particular over the regular semisimple locus is a -torsor and is a -local system placed in dimension By the functoriality of intermediate extension we get an action of on .
There is another proof using the Steinberg variety . Using this Kazhdan-Lusztig give a topological construction in this paper. It is the union of the conormal bundles to all -orbits in , which are indexed by the Weyl group , and the irreducible components of are the closures of . See Ginzburg’s book Corollary 3.3.5. As in the above proof a key role is the following dimension identity: Let be the conjugacy class of . Then we have . See Corollary 3.3.24. The proof uses some symplectic geometry and it still to be understood.
Lie algebraic version of the story: The exponential map is -equivariant and identifies with for large.
Reducedness of Springer fiber:
For , the fiber is non-reduced. We can compute this by parametrizing Borels by , and the Lie algebra of the corresponding Borel can be identified with subsets of that fixed the line defined by . Then the equation for is . On the other hand, the equation for is reduced, as its equation becomes .
Example:
For , the Springer correspondence is supposed to give us a bijection between irreps of indexed by Young Tableaux (the corresponding representation has basis in bijection with the set of standard Young tableaux of shape ) and nilpotent conjugacy classes indexed by partitions of . Say the top cohomology of the Springer fiber realizes this representation for in the nilpotent conjugacy class . The nilpotent Steinberg variety has irreducible components indexed by the Weyl group . On the other hand, we can also describe the cohomology as direct sum of where range over nilpotent orbits (to show this we first define a filtration of by ; then we show is a two-sided ideal in . The key is to show is semisimple.). Each is isomorphic to . Finally, we deduce . Since each has a vector space basis that can be identified with the set of standard Young tableaux of shape , we get a bijection between the set of pairs of standard Young tableaux of the same shape and .
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