Polystability of Stokes representations and differential Galois groups

TL;DR

This work extends Richardson's tameness-based polystability criterion to wild (Stokes) representations by linking polystability to the differential Galois group of the associated connection and by embedding the problem in wild character varieties. It develops both extrinsic (twisted) and intrinsic formalisms: extrinsically via twisted groups $G(\phi)$, fission spaces, and generalized reduction data, and intrinsically via Stokes local systems and graded local systems, with stability characterized by absence of invariant parabolics and polystability by linear reductivity of the Galois image. The paper proves several equivalent formulations of these criteria, applies the results to wild character varieties $\mathcal{M}_{\rm B}(\mathbf{\Sigma})$, and provides a framework that accommodates interior twists and outer automorphism actions. By tying algebraic group-theoretic conditions to the geometry of Stokes data, it connects differential Galois theory with wild nonabelian Hodge theory and braid/group actions on moduli spaces, offering a canonical description of wild moduli spaces and their deformations.

Abstract

Polystability of (twisted) Stokes representations (i.e. wild monodromy representations) will be characterised, in terms of the corresponding differential Galois group (generalising the Zariski closure of the monodromy group in the tame case). This extends some results of Richardson. Further, the intrinsic approach to such results will be established, in terms of reductions of Stokes local systems.

Theorems & Definitions (41)

• Theorem 1: Richardson
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5: Richardson
• Theorem 6: Richardson
• Theorem 7
• Remark 8
• Proposition 9
• Lemma 10