Counting points on braid varieties and the Deligne--Simpson problem

1. Problem Statement

The paper addresses the Deligne–Simpson problem for a specific class of connections on the projective line $\mathbb{P}^1$ with two singular points ($0$ and $\infty$).

• Context: The problem asks for the existence of a $G$-connection (where $G$ is a complex reductive group) with prescribed singularities and monodromy.
• Specific Class: The authors focus on isoclinic connections. These are connections with:
  • An irregular singularity at $0$ with a regular semisimple Levelt–Turrittin leading term.
  • A regular singularity at $\infty$.
  • A slope $\nu = d/m$, where $m$ is a "regular number" for the Weyl group $W$ of $G$, and $\gcd(d, m) = 1$.
• The Core Question: For a given slope $\nu$, which unipotent conjugacy classes $C \subset G$ can appear as the monodromy at $\infty$?
• Previous Status:
  • For $G = GL_n$, the problem was solved using linear algebra (Sage et al.).
  • For classical groups, it was solved via non-abelian Hodge theory (Jakob–Yun).
  • Gap: The case for exceptional groups (types $E_6, E_7, E_8$) remained unresolved.

2. Methodology

The authors employ a geometric and combinatorial approach distinct from previous methods, relying on the Riemann–Hilbert correspondence and finite field counting.

A. Reduction to Braid Varieties

1. Riemann–Hilbert Correspondence: Using recent results by Hohl and Jakob, the authors establish an equivalence between the moduli stack of irregular connections ($Conn(A, C)$) and a braid stack $M(\beta, C)$.
2. Identification of Braids: They prove that the braid $\beta_A$ associated with an isoclinic connection of slope $\nu = d/m$ is the $d$-th power of a lift of an $m$-Springer element ($w_m$) in the positive braid monoid. Specifically, $\beta_A = (\tilde{w}_m)^d$.
3. Group-Theoretic Criterion: The non-emptiness of the connection moduli space is equivalent to the non-emptiness of the intersection between a product of Bruhat double cosets and the conjugacy class:
   $$(\tilde{w}_m B \tilde{w}_m^{-1} \dots) \cap C \neq \emptyset \iff B(\beta)^d \cap C \neq \emptyset$$
   where $B(\beta)$ is the product of double cosets associated with the braid.

B. Counting Points over Finite Fields

To determine when these intersections are non-empty for exceptional groups, the authors adapt Lusztig's strategy (originally used for minimal length elements in the Weyl group):

1. Finite Field Reduction: They count the number of $\mathbb{F}_q$-points on the braid varieties (or stacks) over a finite field.
2. Character Theory: The point count is expressed as a sum involving character values of the finite reductive group $G(\mathbb{F}_q)$ and the finite Hecke algebra $H_q$.
   $$|M((\tilde{w}_m)^d, C)^{\mathbb{F}_q}| = \frac{1}{|G^F|} \sum_{g \in C^F} \sum_{E \in \text{Irr}(W)} \text{tr}((T_w)^d, E_q) \text{tr}(g, \rho_E)$$
3. Computational Simplification:
   • Standard expansion of $(T_w)^d$ is computationally infeasible for exceptional types.
   • The authors utilize Lemma 8, which relates the trace of the power of the Hecke algebra element to the trace of the Weyl group element: $\text{tr}((T_w)^d, E_q) = q^{\nu c(E_q)} \text{tr}(w^d, E)$.
   • This reduces the problem to computing character values of $W$ and $G(\mathbb{F}_q)$, which are efficiently handled using the CHEVIE package (for Julia/GAP).

3. Key Contributions and Results

A. Solution to the Isoclinic Deligne–Simpson Problem

The authors prove Theorem 1, which completely solves the problem for exceptional groups:

• There exists a unique minimal unipotent conjugacy class $C_\nu$ for a given slope $\nu$.
• An isoclinic connection with slope $\nu$ and monodromy $C$ exists if and only if $C \geq C_\nu$ (in the partial order of unipotent classes).
• Specifics:
  • If $\nu > 1$, $C_\nu = \{1\}$ (trivial).
  • If $\nu < 1$, the classes $C_\nu$ are explicitly listed in the appendix for exceptional types.

B. New Examples of Rigid Connections

• The paper identifies new examples of physically rigid irregular $G$-connections.
• Physical Rigidity: A connection is physically rigid if its moduli space is a singleton (a single point).
• The authors prove that the cohomologically rigid isoclinic connections identified in previous literature are indeed physically rigid by showing their associated braid varieties have exactly one point over finite fields.

C. Explicit Realizations

• The authors establish Conjecture 5.7(2) of Sage (2024) regarding Coxeter $G$-connections.
• They show that the specific Coxeter connections constructed by Kamgarpour and Sage (using sums of root vectors) possess the minimal possible monodromy predicted by the theoretical classification.

4. Significance

• Completing the Program: This work finalizes the classification of isoclinic connections for all simple algebraic groups, bridging the gap left by previous work on classical groups.
• Methodological Innovation: It demonstrates the power of combining the Riemann–Hilbert correspondence with finite field point counting to solve existence problems in complex geometry. This approach bypasses the need for intricate linear algebra (used for $GL_n$) or non-abelian Hodge theory (used for classical groups).
• Connection to Lusztig's Work: The paper generalizes Lusztig's map from Weyl group conjugacy classes to unipotent classes. While Lusztig's original result applied to minimal length elements, this paper extends the logic to powers of Springer elements (which are generally not minimal length) and powers of Bruhat double cosets.
• Applications: The results provide a foundation for constructing $\ell$-adic analogues and Hecke eigensheaves for these connections, which are relevant to the Langlands program and mathematical physics (e.g., gauge theory and knot invariants).

5. Technical Appendix

The paper includes extensive appendices containing:

• Tables of minimal unipotent classes ($C_\nu$) for exceptional groups.
• Data on rigid connections.
• Sample computations and code snippets used to verify the character sums via the CHEVIE package.