Diagonal coinvariants have an interpretation from https://arxiv.org/abs/math/0201148 in terms of the Hilbert scheme.
There are two recent papers https://arxiv.org/pdf/1801.09033.pdf and https://arxiv.org/pdf/1808.02278.pdf which both detail (different and seemingly unconnected) relations between diagonal coinvariants and affine Springer fibers (another paper that does this is https://arxiv.org/pdf/1203.5878.pdf). In the first paper diagonal coinvariants act on suitable (co)homology of affine Springer fibers and in the second paper diagonal coinvariants are (sometimes conjecturally) identified with certain (co)invariants in suitable (co)homology of certain affine Springer fibers. Why should there be such relations between diagonal coinvariants and affine Springer fibers?
Is there any relation of the Hilbert scheme interpretation of diagonal coinvariants to the affine Springer fiber interpretations of diagonal coinvariants?
Also, can one explain intuitively the connections of the above relations between diagonal coinvariants and affine Springer fibers with the results on the action of DAHA/Cherednik algebra on the suitable (co)homology of the affine Springer fibers in https://arxiv.org/abs/1407.5685 and https://arxiv.org/abs/0705.2691 ?