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I would like to know why people are interested in transverse slices. In particular, I would like to know why people study the Slodowy slice to orbits in the nilpotent cone. In Chriss and Ginzburg's book, Slodowy slices are defined in Proposition 3.7.15, and it is mentioned that Springer fibers are homotopy retracts of the preimage of Slodowy slices under the Springer resolutions. However, it seems there are many other reasons for the interest in Slodowy slices.

In addition, people also study transverse slices to stratum closures in stratified spaces. Are there some general theorems about how these transverse slices relate to intersection cohomology of these stratified spaces?

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My motivation for learning them is for the role they play in the quantum Hamiltonian reduction construction of W-algebras as outlined in the following sources:

https://arxiv.org/abs/1502.07025

https://arxiv.org/abs/1605.00138

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