We prove an explicit formula for the Poincar\'e polynomials of parabolic character varieties of Riemann surfaces with semisimple local monodromies, which was conjectured by Hausel, Letellier and Rodriguez-Villegas. Using an approach of Mozgovoy and Schiffmann the problem is reduced to counting pairs of a parabolic vector bundles and a nilpotent endomorphism of prescribed generic type. The generating function counting these pairs is shown to be a product of Macdonald polynomials and the function counting pairs without parabolic structure. The modified Macdonald polynomial $\tilde H_\lambda[X;q,t]$ is interpreted as a weighted count of points of the affine Springer fiber over the constant nilpotent matrix of type $\lambda$.

中文翻译：

---

字符变体的 Poincaré 多项式、Macdonald 多项式和仿射 Springer 纤维

我们证明了 Hausel、Letellier 和 Rodriguez-Villegas 推测的具有半单局部单向性的黎曼曲面抛物线特征变体的 Poincar\'e 多项式的显式公式。使用 Mozgovoy 和 Schiffmann 的方法，问题被简化为对抛物线向量丛和规定泛型的幂零自同态的对进行计数。计算这些对的生成函数被证明是麦克唐纳多项式和没有抛物线结构的计算对的函数的乘积。修正的 Macdonald 多项式 $\tilde H_\lambda[X;q,t]$ 被解释为仿射 Springer 纤维在 $\lambda$ 类型的常数幂零矩阵上的加权点数。