The Geometry of Double Affine Hecke Algebra Superpolynomials

We formulate a conjecture which interprets DAHA superpolynomials colored by fundamental weights to the Borel-Moore cohomology of Jacobian factors and their flagged and higher rank generalizations. To study the generalized Jacobian factors we use a certain stratification derived from the valuation semi-group of the singularity and in special cases show that the components of the stratification of the flagged Jacobian factors and higher rank Jacobian factors are affine spaces. Once these results are in place it is easy to check the main conjecture. In the last section, we are able to check the conjecture for more sophisticated examples where the stratification does not consist of affine cells and study for a particular example which types of varieties may occur as components in the stratification.