Theorem.  Assume $k, n$ are coprime.  Let $X_{k,n}$ denote the quotient of the big positroid cell of $Gr(k, k+n)$ by the appropriate torus action; i.e., “the image of the X variety in the A variety”.  Then the cohomology of $X_{k,n}$ is all in even degrees, and the $2i$’th Betti number are given by counting the number of Young diagrams with $i$ boxes which fit below a line of slope $k/n$.

Proof.  It is shown in [STWZ], (although it could also be seen directly from the definitions) that $X_{k,n}$ is the “wild character variety”, i.e. moduli space of Stokes data,  for connections on $\mathbb{P}^1$ with a single irregular singularity, at infinity, of formal type the same as for the connection $d^n – z^k$.   By nonabelian Hodge theory, this space is diffeomorphic to a certain Hitchin system on $\mathbb{P}^1$ with a certain high order pole at infinity, which in turn contracts to its central fibre, which is the compactified Jacobian for the curve $y^n = z^k$.  The Betti numbers for this compactified Jacobian are known to be given by the above formula.  The nonabelian Hodge theorem required above is in the to-appear thesis of Laura Fredrickson.

Remark.  I record this fact here because I was recently informed by Thomas Lam and David Speyer that this was not previously known.  A more detailed discussion, especially in terms of the relationship to [GORS] and HOMFLY homology, will eventually appear.

Remark.  It is possible to show that the weight polynomial of $X_{k,n}$ is the $k/n$ rational slope  $q$-catalan number.   This follows, for instance, from Theorem 1.10 of [STZ], plus a formula of Jones for the HOMFLY polynomial of torus knots, see e.g. the calculations in section 5 of [OS].

Conjecture.  The bigraded formula recording both the weight and the cohomological degree is the rational slope $k/n$ $(q,t)$-catalan number.

Remark.  This conjecture is related to conjectures in [GORS], by the “P = W” conjecture (formulated for wild character varieties).