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On the isotropy stratification of a real representation of a compact Lie group

Perla Azzi, Rodrigue Desmorat, Julien Grivaux, Boris Kolev

TL;DR

This work analyzes isotropy stratification for real representations of compact Lie groups by transporting the problem to the complex setting via complexification and applying Luna's slice theory. The authors prove two main results: (i) the algebraicity theorem, identifying the real closure of an isotropy stratum with the real points of the corresponding complex closure, and (ii) the normalization theorem, showing that the real restriction map to invariants becomes a normalization after complexification. The approach unifies differential-geometric and invariant-theoretic methods, clarifying how complex-analytic techniques control the real orbit structure and offering explicit links between real and complex invariants through normalizers and slice models. The results provide a rigorous algebraic description of orbit-type closures and a principled mechanism to compute them via complex quotients, with implications for real algebraic geometry of group actions and representation theory.

Abstract

The aim of the present paper is to provide a comprehensive introduction to some algebraic and geometric aspects of real representations of compact Lie groups, as well as some results concerning isotropy strata and restriction of invariants.

On the isotropy stratification of a real representation of a compact Lie group

TL;DR

This work analyzes isotropy stratification for real representations of compact Lie groups by transporting the problem to the complex setting via complexification and applying Luna's slice theory. The authors prove two main results: (i) the algebraicity theorem, identifying the real closure of an isotropy stratum with the real points of the corresponding complex closure, and (ii) the normalization theorem, showing that the real restriction map to invariants becomes a normalization after complexification. The approach unifies differential-geometric and invariant-theoretic methods, clarifying how complex-analytic techniques control the real orbit structure and offering explicit links between real and complex invariants through normalizers and slice models. The results provide a rigorous algebraic description of orbit-type closures and a principled mechanism to compute them via complex quotients, with implications for real algebraic geometry of group actions and representation theory.

Abstract

The aim of the present paper is to provide a comprehensive introduction to some algebraic and geometric aspects of real representations of compact Lie groups, as well as some results concerning isotropy strata and restriction of invariants.
Paper Structure (16 sections, 36 theorems, 36 equations, 2 tables)

This paper contains 16 sections, 36 theorems, 36 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Differential geometry background
  3. Isotropy strata
  4. Slices
  5. Closures of isotropy strata
  6. Algebraic background
  7. Reductive groups and invariants
  8. Luna's slice theorem
  9. Isotropy strata in the complex case
  10. Complexification
  11. Compact Lie groups
  12. Real representations of compact Lie groups
  13. Normalizers of subgroups of compact Lie groups
  14. Proof of the main theorems
  15. The algebraicity theorem (theorem \ref{['thm:first-main-theorem']})
  16. ...and 1 more sections

Key Result

Theorem 1.1

Given a real representation $(W, \rho)$ of a compact Lie group, all orbits of $\rho$ are real algebraic subsets of $W$. Equivalently, polynomial invariants of $(W, \rho)$ separate the orbits of $\rho$.

Theorems & Definitions (88)

  • Theorem 1.1: AS1983, Ser1993
  • Example 1.2
  • Theorem 1.3
  • Example 1.4: Chevalley restriction theorem
  • Theorem 1.5
  • Example 1.6
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • ...and 78 more