Type: Article
Publication Date: 2021-01-01
Citations: 13
DOI: https://doi.org/10.1515/math-2021-0038
Abstract Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>F</m:mi> </m:math> F be a finite group and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>X</m:mi> </m:math> X be a complex quasi-projective <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>F</m:mi> </m:math> F -variety. For <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>r</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant="double-struck">N</m:mi> </m:math> r\in {\mathbb{N}} , we consider the mixed Hodge-Deligne polynomials of quotients <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>X</m:mi> </m:mrow> <m:mrow> <m:mi>r</m:mi> </m:mrow> </m:msup> <m:mspace width="-0.15em" /> <m:mtext>/</m:mtext> <m:mspace width="-0.08em" /> <m:mi>F</m:mi> </m:math> {X}^{r}\hspace{-0.15em}\text{/}\hspace{-0.08em}F , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>F</m:mi> </m:math> F acts diagonally, and compute them for certain classes of varieties <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>X</m:mi> </m:math> X with simple mixed Hodge structures (MHSs). A particularly interesting case is when <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>X</m:mi> </m:math> X is the maximal torus of an affine reductive group <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>G</m:mi> </m:math> G , and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>F</m:mi> </m:math> F is its Weyl group. As an application, we obtain explicit formulas for the Hodge-Deligne and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>E</m:mi> </m:math> E -polynomials of (the distinguished component of) <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>G</m:mi> </m:math> G -character varieties of free abelian groups. In the cases <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>G</m:mi> <m:mo>=</m:mo> <m:mi>G</m:mi> <m:mi>L</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>,</m:mo> <m:mi mathvariant="double-struck">C</m:mi> <m:mspace width="-0.1em" /> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> G=GL\left(n,{\mathbb{C}}\hspace{-0.1em}) and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>S</m:mi> <m:mi>L</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>,</m:mo> <m:mi mathvariant="double-struck">C</m:mi> <m:mspace width="-0.1em" /> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> SL\left(n,{\mathbb{C}}\hspace{-0.1em}) , we get even more concrete expressions for these polynomials, using the combinatorics of partitions.