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Twisted Higgs bundles and coendoscopy

Michael Groechenig, Xuanyou Li, Dimitri Wyss, Paul Ziegler

TL;DR

The paper develops a theory of $eta$-twisted $H$-Higgs bundles on a curve, where twisting is by a central gerbe $eta$, and shows that stabilized point-counts and cohomology are invariant under the central twist. It provides a detailed description of twisted moduli, their relation to untwisted moduli, and a corrected inertia-stack decomposition via coendoscopic data, enabling a twisted analogue of Ngô’s product formula. Two independent proofs are given for the equality of stabilized counts between twisted and untwisted Hitchin fibers: one using $p$-adic integration and one via a product-formula approach, underscoring a robust local-global compatibility. The results illuminate the role of central twists in the geometry of Higgs moduli and inertia stacks, with implications for twisted variants of the Hausel–Thaddeus duality and related Langlands-type correspondences.

Abstract

This short note is devoted to the study of $G$-Higgs bundles twisted by a central gerbe. These objects arise naturally in the decomposition of the inertia stacks of $G$-Higgs bundles in terms of coendoscopic data. We establish that stabilised point-counts and cohomology are insensitive to the central twist. Along the way we show an analogue of Ngô's product formula for twisted Hitchin fibres.

Twisted Higgs bundles and coendoscopy

TL;DR

The paper develops a theory of -twisted -Higgs bundles on a curve, where twisting is by a central gerbe , and shows that stabilized point-counts and cohomology are invariant under the central twist. It provides a detailed description of twisted moduli, their relation to untwisted moduli, and a corrected inertia-stack decomposition via coendoscopic data, enabling a twisted analogue of Ngô’s product formula. Two independent proofs are given for the equality of stabilized counts between twisted and untwisted Hitchin fibers: one using -adic integration and one via a product-formula approach, underscoring a robust local-global compatibility. The results illuminate the role of central twists in the geometry of Higgs moduli and inertia stacks, with implications for twisted variants of the Hausel–Thaddeus duality and related Langlands-type correspondences.

Abstract

This short note is devoted to the study of -Higgs bundles twisted by a central gerbe. These objects arise naturally in the decomposition of the inertia stacks of -Higgs bundles in terms of coendoscopic data. We establish that stabilised point-counts and cohomology are insensitive to the central twist. Along the way we show an analogue of Ngô's product formula for twisted Hitchin fibres.
Paper Structure (9 sections, 22 theorems, 47 equations)

This paper contains 9 sections, 22 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Twisted $H$-bundles
  3. Definitions
  4. Relation to the classical case
  5. Moduli spaces associated to coendoscopic data
  6. Description of the twisted inertia stack
  7. Comparing stabilised point-counts
  8. Proof via $p$-adic integration
  9. Proof via an analogue of Ngô's product formula

Key Result

Theorem 1.1

Over $\overline{k}$ there is an equivalence of stacks where the disjoint union is taken over isomorphism classes of coendoscopic data ${\mathcal{E}}$.

Theorems & Definitions (53)

  • Theorem 1.1: Theorem \ref{['dec-IM']}
  • Theorem 1.2: Theorem \ref{['stabct']}
  • Proposition 1.3: Product formula, Proposition \ref{['prop:productformula']}
  • Remark 1.4
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • ...and 43 more