Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Proper affine actions: a sufficient criterion

Ilia Smilga

Abstract

For a semisimple real Lie group $G$ with an irreducible representation $ρ$ on a finite-dimensional real vector space $V$, we give a sufficient criterion on $ρ$ for existence of a group of affine transformations of $V$ whose linear part is Zariski-dense in $ρ(G)$ and that is free, nonabelian and acts properly discontinuously on $V$. This new criterion is more general than the one given in the author's previous paper "Proper affine actions in non-swinging representations" (submitted; available at arXiv:1605.03833), insofar as it also deals with "swinging" representations. We conjecture that it is actually a necessary and sufficient criterion, applicable to all representations.

Proper affine actions: a sufficient criterion

Abstract

For a semisimple real Lie group with an irreducible representation on a finite-dimensional real vector space , we give a sufficient criterion on for existence of a group of affine transformations of whose linear part is Zariski-dense in and that is free, nonabelian and acts properly discontinuously on . This new criterion is more general than the one given in the author's previous paper "Proper affine actions in non-swinging representations" (submitted; available at arXiv:1605.03833), insofar as it also deals with "swinging" representations. We conjecture that it is actually a necessary and sufficient criterion, applicable to all representations.

Paper Structure

This paper contains 38 sections, 59 theorems, 136 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Basic notations
  4. Statement of main result
  5. Strategy of the proof
  6. Plan of the paper
  7. Acknowledgments
  8. Preliminaries
  9. Eigenvalues in different representations
  10. Properties of restricted weights
  11. Parabolic subgroups and subalgebras
  12. Metric properties and estimates
  13. Proximal maps
  14. $X$-regular linear maps
  15. Definitions
  16. ...and 23 more sections

Key Result

Theorem 1.2

Assume that $G$ is split. Let $\mathfrak{g} = \mathfrak{g}_1 \oplus \cdots \oplus \mathfrak{g}_n$ be its decomposition into simple factors; let $\rho$ be an irreducible representation of $G$ with highest weight $\lambda$. Then:

Figures (3)

  • Figure 1: This picture represents the situation for $G = \mathop{\mathrm{SO}}\nolimits^+(3,2)$ acting on $\mathbb{R}^5$, and $X$ chosen such that $\Pi_X = \{\alpha_1\}$ (or $\{1\}$ with the usual abuse of notations). This choice of $X$ is not random: it satisfies the conditions that will be required starting from Section \ref{['sec:choice']}. (This also corresponds to Example 3.7.3 in Smi16.) The group $W_X$ is then generated by the single reflection $s_{\alpha_1}$. Proposition \ref{['jordan_additivity']} states that $\operatorname{Jd}(gh)$ lies in the shaded trapezoid. Corollary \ref{['jordan_additivity_reformulation']} states that it lies on the thick line segment. In any case it lies by definition in the dominant open Weyl chamber $\mathfrak{a}^{++}$ (the shaded sector).
  • Figure 2: Relative positions of subspaces involved in the proof of Proposition \ref{['invariant_additivity_only']}. This picture should be understood to represent some affine cross-section of the full picture. Lines represent (the corresponding cross-sections of) affine parabolic spaces (such as $A_g^{\mathsmaller{\gtrsim}}$), and dots represent (the corresponding cross-sections of) intersections of transverse pairs of them (such as $A_g^{\mathsmaller{\approx}}$). We use the notational convention $A^{{\mathsmaller{\approx}}}_{u, v} := A^{{\mathsmaller{\gtrsim}}}_{u} \cap A^{{\mathsmaller{\lesssim}}}_{v}$ for any two $\rho$-regular maps $u$ and $v$. Double-headed arrows represent contraction and expansion by the maps $g$ and $h$. We encourage the reader to watch this picture on a screen, so as to benefit from helpful (even if dispensable) color cues.
  • Figure 3:

Theorems & Definitions (208)

  • Remark 1.1
  • Theorem 1.2: LFlSm
  • Remark 1.3
  • Example 1.4
  • Remark 1.5
  • Proposition 2.1: Jordan decomposition
  • proof
  • Proposition 2.2: Cartan decomposition
  • proof
  • Definition 2.3
  • ...and 198 more