There is a paper of Hatta and Kashiwara which identifies the "geometric" constructions people have mentioned with an "algebraic" one which came from Harish-Chandra's study of characters (invariant eigendistributions is the term I think). This gives you an explicit description of the D-module Rf_*(O) associated to the Springer resolution, one for the D-modules on the whole Lie algebra associated to the Grothendieck resolution on the whole Lie algebra, and proves how the two are related via the Fourier transform.

For sl_n this sheaf decomposes into simple summands corresponding to the middle extension of the structure sheaf of each nilpotent orbit. At least that these modules are the only possibilities follows from the GL_n equivariance of the Springer resolution: the constituents therefore have to be GL_n equivariant, and the equivariant fundamental groups are trivial in this case. If you work with SL_n, then on the regular orbit the equivariant fundamental group is isomorphic to the centre of SL_n, and in some sense this case is responsible for all the nontrivial equivariant local systems (Lusztig's generalized Springer correspondence arose from trying to understand related things).

The upshot though is that the simple GL_n-equivariant D-modules on the nilcone are the simple summands of the Springer sheaf. Now one way of saying Springer theory is that the Springer sheaf's decomposition into irreducibles is governed by the action of W the Weyl group on it (i.e. by the isotypic components of this action). Thus understanding the W-action on this sheaf gives a way of trying to understand these simple D-modules which is different from simply trying to calculate middle extensions.

There is a paper of Hotta and Kashiwara which identifies the "geometric" constructions people have mentioned with an "algebraic" one which came from Harish-Chandra's study of characters (invariant eigendistributions is the term I think). This gives you an explicit description of the D-module Rf_*(O) associated to the Springer resolution, one for the D-modules on the whole Lie algebra associated to the Grothendieck resolution on the whole Lie algebra, and proves how the two are related via the Fourier transform.

For sl_n this sheaf decomposes into simple summands corresponding to the middle extension of the structure sheaf of each nilpotent orbit. At least that these modules are the only possibilities follows from the GL_n equivariance of the Springer resolution: the constituents therefore have to be GL_n equivariant, and the equivariant fundamental groups are trivial in this case. If you work with SL_n, then on the regular orbit the equivariant fundamental group is isomorphic to the centre of SL_n, and in some sense this case is responsible for all the nontrivial equivariant local systems (Lusztig's generalized Springer correspondence arose from trying to understand related things).

The upshot though is that the simple GL_n-equivariant D-modules on the nilcone are the simple summands of the Springer sheaf. Now one way of saying Springer theory is that the Springer sheaf's decomposition into irreducibles is governed by the action of W the Weyl group on it (i.e. by the isotypic components of this action). Thus understanding the W-action on this sheaf gives a way of trying to understand these simple D-modules which is different from simply trying to calculate middle extensions.