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Stability and disconnected groups

Andres Fernandez Herrero, Andrés Ibáñez Núñez

Abstract

We study the notion of semistability for principal bundles over curves with possibly disconnected reductive structure group. We establish a new characterization of the behavior of semistability under change of group, novel even in the connected case, and prove that all existing notions of semistability are equivalent, thus settling a question by Biswas-Gomez. The key ingredients for our results include a study of cocharacters and characters of disconnected linear algebraic groups, and an extension of the recursive description of Kirwan stratifications in Geometric Invariant Theory to the case of disconnected groups.

Stability and disconnected groups

Abstract

We study the notion of semistability for principal bundles over curves with possibly disconnected reductive structure group. We establish a new characterization of the behavior of semistability under change of group, novel even in the connected case, and prove that all existing notions of semistability are equivalent, thus settling a question by Biswas-Gomez. The key ingredients for our results include a study of cocharacters and characters of disconnected linear algebraic groups, and an extension of the recursive description of Kirwan stratifications in Geometric Invariant Theory to the case of disconnected groups.

Paper Structure

This paper contains 14 sections, 17 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Weyl groups, cocharacters and rational characters
  3. Weyl groups
  4. Cocharacters
  5. Rational characters
  6. Kirwan stratifications for disconnected groups
  7. Semistable locus and the Hilbert--Mumford criterion
  8. Fixed points and attractors
  9. Kirwan stratification
  10. Semistability for principal bundles
  11. Rational degree
  12. Hilbert-Mumford semistability
  13. Semistability under change of group
  14. Comparison of existing notions of semistability

Key Result

Theorem 1.1.1

Suppose that the algebraically closed ground field $k$ has characteristic $0$. Let $f: G \to H$ be a homomorphism of linearly reductive affine algebraic groups over $k$. Let $E$ be a $G$-bundle on $C$. Then, the following hold: $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (30)

  • Theorem 1.1.1: ($=$ \ref{['thm: semistability under change of group']})
  • Remark 1.1.2
  • Theorem 1.1.3
  • Theorem 1.1.4: ($=$ \ref{['thm: description of centres']})
  • Proposition 2.1.3
  • Proposition 2.2.1
  • Proposition 2.3.2
  • Definition 3.2.3: Semistable locus
  • Theorem 3.2.5: Hilbert--Mumford criterion
  • Remark 3.3.4: Intrinsic definition
  • ...and 20 more