Bigraded Polynomials for the Cohomology of Wild Hitchin Systems

We introduce a bi-graded polynomial that encodes the cohomology groups of the wild Hitchin system of type~$A_{n-1}$, constructed using an irregular singularity (determined by an integer~$m$) and an arbitrary regular singularity~$f$. When the regular singularity is of the form~$f = [1, \ldots, 1]$, the bi-graded polynomial~$C_{m,n}(q,t)$ coincides with the bigraded rational parking function defined combinatorially, admitting a Schur expansion~$C_{m,n}(q,t) = \sum_\lambda f_\lambda(q,t) s_\lambda(x)$. For general~$f$, the polynomial takes the form~$C^f_{m,n}(q,t) = \sum_\lambda f_\lambda(q,t) K_{\lambda f}$, where~$K_{\lambda f}$ denotes the Kostka number. We conjecture that this bi-graded polynomial agrees with the one arising from the perverse filtration of the Hitchin fibration, or equivalently, from the weight filtration of the mixed Hodge structure from the character variety. We also give a description by using the geometry of affine Springer fiber.}