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Covariant representations of algebraic group actions and applications

Yvann Gaudillot-Estrada

Abstract

If $G$ is an algebraic affine group acting on an affine variety $X$, there is a natural notion of covariant representation for the pair $(G,X)$. In this paper, we classify the irreducible covariant representations for any such pair by adapting the Mackey machine to this algebraic setting. Next, we give applications for continuous representations of motion groups on Banach spaces and other related examples.

Covariant representations of algebraic group actions and applications

Abstract

If is an algebraic affine group acting on an affine variety , there is a natural notion of covariant representation for the pair . In this paper, we classify the irreducible covariant representations for any such pair by adapting the Mackey machine to this algebraic setting. Next, we give applications for continuous representations of motion groups on Banach spaces and other related examples.
Paper Structure (15 sections, 18 theorems, 19 equations)

This paper contains 15 sections, 18 theorems, 19 equations.

Table of Contents

  1. Preliminaries
  2. Modules over the ring of regular functions
  3. Actions of algebraic groups
  4. Induction for representations of affine groups
  5. Irreducible covariant representations
  6. The classification
  7. The transitive case
  8. Application to motion groups
  9. Compact Lie groups and complex reductive groups
  10. General representation theory of motion groups
  11. Irreducible admissible representations of general motion groups
  12. The case of Cartan motion groups
  13. Quantum analogues of Cartan motion groups and their irreducible representations
  14. Background from the theory of symmetric spaces of compact type
  15. A restriction theorem and a parametrization of the closed orbits

Key Result

Lemma 2

Let $M$ be a finitely generated $k[X]$-module. If the map $x \mapsto \mathrm{dim}_k M|_x$ is locally constant on $X$ then $\mathrm{Sh}M$ is locally free.

Theorems & Definitions (33)

  • Definition 1
  • Lemma 2
  • proof
  • Theorem 3
  • proof : Proof, based on BBHM
  • Lemma 4
  • proof
  • Theorem 5
  • proof : Proof of the theorem assuming that is is true in the transitive case.
  • Theorem 6
  • ...and 23 more