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Very stable regular $G$-Higgs bundles

Miguel González

Abstract

We give a classification of very stable $G$-Higgs bundles in the generically regular Higgs field case for $G$ an arbitrary connected semisimple complex group. This extends the classification for $G=\mathrm{GL}_n(\mathbb C)$ and fixed point type $(1,1,\dots,1)$ given by Hausel and Hitchin.

Very stable regular $G$-Higgs bundles

Abstract

We give a classification of very stable -Higgs bundles in the generically regular Higgs field case for an arbitrary connected semisimple complex group. This extends the classification for and fixed point type given by Hausel and Hitchin.

Paper Structure

This paper contains 12 sections, 24 theorems, 119 equations, 1 table.

Table of Contents

  1. Introduction
  2. Biał ynicki-Birula theory
  3. Moduli space of $G$-Higgs bundles
  4. Main definitions, stability and moduli spaces
  5. $\mathbb{C}^\times$-action on $\mathcal{M}(G)$ and very stable Higgs bundles
  6. Upward and downward flows on $\mathcal{M}(G)$
  7. The affine Grassmannian and Hecke transformations
  8. Definitions and properties
  9. Hecke transformations
  10. $\mathbb{C}^\times$-action on the affine Grassmannian
  11. Classification of very stable $G$-Higgs bundles of Borel type
  12. Virtual equivariant multiplicities

Key Result

Theorem 1

Let $(E,\varphi)$ be a smooth $\mathbb{C}^\times$-fixed point of Borel type with associated multiplicity divisor $\mu_{(E,\varphi)}$. Then $(E,\varphi)$ is very stable if and only if all the coefficients in $\mu_{(E,\varphi)}$ are minuscule.

Theorems & Definitions (83)

  • Theorem : Theorem \ref{['verystablecharact']}
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3: bialynicki-birula_theorems_1973, hausel_very_2022
  • Definition 2.4
  • Proposition 2.5
  • Proposition 2.6: hausel_very_2022
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • ...and 73 more