Counting points on generic character varieties

Technical Summary: Counting Points on Generic Character Varieties

Problem Statement
The paper addresses the enumeration of points on character varieties associated with punctured surface groups and regular semisimple conjugacy classes in arbitrary reductive groups $G$ over finite fields. While the geometry and topology of $GL_n$-character varieties are well-understood—thanks to foundational work by Hausel, Letellier, and Rodriguez-Villegas [HRV08, HLRV11]—there is limited understanding of character varieties for general reductive groups. Specifically, the authors seek to determine the number of $\mathbb{F}_q$-points for both multiplicative character varieties (associated with the fundamental group of a punctured surface) and their additive analogues (associated with the Lie algebra $\mathfrak{g}$). The study focuses on "generic" tuples of conjugacy classes, a condition ensuring the varieties are smooth and equidimensional.

Methodology
The authors employ a combination of representation theory of finite groups of Lie type, Deligne-Lusztig theory, and harmonic analysis on Lie algebras.

1. Multiplicative Case:

   • The starting point is the Frobenius Mass Formula, which expresses the number of points on the representation variety in terms of irreducible complex characters of the finite group $G(\mathbb{F}_q)$.
   • Using the assumption that the conjugacy classes are generic and contain strongly regular elements, the authors apply a theorem of Deligne and Lusztig [DL76]. This restricts the contributing characters to the principal series representations.
   • The sum over characters is reorganized by grouping representations into "types" $\tau = [L, \rho]$, where $L$ is a standard endoscopy group (a pseudo-Levi subgroup) and $\rho$ is an irreducible character of the Weyl group of $L$.
   • The character sums are evaluated using Möbius inversion on the poset of endoscopy groups. A crucial step involves analyzing the stabilizers of the Frobenius action on the dual torus, requiring a finite base change to ensure certain component groups split.

2. Additive Case:

   • The additive character variety is defined via the Lie algebra $\mathfrak{g}$ and adjoint orbits. The counting problem is approached via an additive analogue of the Frobenius Mass Formula, which involves the Fourier transforms of characteristic functions of adjoint orbits.
   • The authors utilize results by Kazhdan and Letellier [Let05] to evaluate these Fourier transforms. The non-vanishing of the transform is linked to "split" elements in the Lie algebra.
   • Similar to the multiplicative case, the sum is decomposed by "types" $\tau = [L, N]$, where $L$ is a Levi subgroup and $N$ is a nilpotent orbit.
   • The evaluation relies on the properties of Green functions and Springer fibers. The authors note that for groups with an $E_8$ factor, a quadratic base change may be required to ensure the point count is a polynomial.

Key Contributions and Results

1. Polynomial Counting for Multiplicative Varieties:

   • Theorem 2: The authors prove that for a generic character variety $X$ associated with a reductive group $G$, there exists a positive integer $r$ such that the number of points $|X(\mathbb{F}_{q^{rm}})|$ is given by a polynomial $f_X(q^{rm})$ for all $m \geq 1$.
   • Palindromic Property: The counting polynomial $f_X(t)$ is shown to be palindromic (symmetric coefficients). This suggests that the variety satisfies a "curious" Poincaré duality and a "curious" Hard Lefschetz property, supporting a $P=W$ conjecture for these varieties.
   • Connected Components: The number of connected components of $X$ is determined to be $|\pi_0(Z(\check{G}))|$, where $\check{G}$ is the Langlands dual group.
   • Independence: The polynomial $f_X$ depends only on the group $G$, the genus $g$, and the number of punctures $n$, but is independent of the specific generic conjugacy classes chosen.

2. Polynomial Counting for Additive Varieties:

   • Theorem 6: The authors establish that the additive character variety $Y$ is also polynomial count, specifically after a quadratic base change (i.e., $|Y(\mathbb{F}_{q^{2m}})| = f_Y(q^{2m})$).
   • Non-negative Coefficients: While a general proof is lacking due to the absence of a direct link to quiver varieties (which exists only for $GL_n$), computational evidence for groups of rank less than 7 suggests the coefficients of $f_Y$ are non-negative. This supports the conjecture that the cohomology of $Y$ is pure.
   • Connectedness: The additive variety $Y$ is proven to be connected.

3. Explicit Formulas and Examples:

   • The paper provides explicit rational function formulas for the counting polynomials $f_X$ and $f_Y$ in terms of root system data, Weyl group characters, and Green polynomials.
   • Section 7 presents detailed tables and explicit polynomials for low-rank groups ($GL_n, SO_5, G_2, B_3, C_3, D_4, F_4, E_6$) and specific values of genus and punctures.

Significance and Claims
The paper claims to generalize the seminal results of Hausel, Letellier, and Rodriguez-Villegas from type $A$ ($GL_n$) to arbitrary reductive groups.

• Structural Insight: The demonstration that generic character varieties are polynomial count and palindromic provides strong evidence for deep topological properties (Hodge-Tate structure, Poincaré duality, Hard Lefschetz) in the reductive setting, mirroring the known behavior for $GL_n$.
• Reductive Analogue of Kac Polynomials: The counting polynomials for the additive varieties are proposed as the reductive analogues of Kac polynomials for comet-shaped quivers. The non-negativity of their coefficients is conjectured, linking the geometry of additive character varieties to the representation theory of Borcherd algebras.
• Foundational for Future Conjectures: The authors note that their results provide the necessary data to formulate conjectures regarding mixed Hodge polynomials in the reductive setting, analogous to the Macdonald polynomial formulas known for $GL_n$. They emphasize that proving the non-negativity of coefficients for general $G$ remains a non-trivial open problem due to the lack of a direct quiver variety correspondence.

The work relies heavily on the assumption that the center $Z(G)$ is connected and the characteristic of the field is "very good" for $G$. The authors acknowledge that relaxing the connected center assumption would complicate the formulas due to disconnected centralizers in the dual group, and that the genericity assumption is crucial for the smoothness and equidimensionality of the varieties.