Abstract Let G be a complex reductive group and X r G denote the G -character variety of the free group of rank r . Using geometric methods, we prove that E ( X r S L n ) = E ( X r P G L n ) , for any n , r ∈ N , where E ( X ) denotes the Serre (also known as E -) polynomial of the complex quasi-projective variety X , settling a conjecture of Lawton-Munoz in Lawton and Munoz (2016).The proof involves the stratification by polystable type introduced in Florentino et al. (0000), and shows moreover that the equality of E -polynomials holds for every stratum and, in particular, for the irreducible stratum of X r S L n and X r P G L n . We also present explicit computations of these polynomials, and of the corresponding Euler characteristics, based on our previous results and on formulas of Mozgovoy-Reineke for G L n -character varieties over finite fields.

中文翻译：

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自由群的 SLn 和 PGLn 特征变体的 Serre 多项式

摘要 令 G 为复数约简群，X r G 表示秩为 r 的自由群的 G 特征簇。使用几何方法，我们证明 E ( X r SL n ) = E ( X r PGL n ) ，对于任何 n ， r ∈ N ，其中 E ( X ) 表示复数的 Serre（也称为 E -）多项式准射影变体 X ，解决了 Lawton and Munoz (2016) 中 Lawton-Munoz 的猜想。 证明涉及到 Florentino 等人引入的多稳定类型分层。(0000)，并且还表明 E 多项式的等式对于每个层都成立，特别是对于 X r SL n 和 X r PGL n 的不可约层。我们还提供了这些多项式的显式计算，以及相应的欧拉特征，