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Semigroups in semi-simple Lie groups: Flag type and estimation of cocycles

Adriano Da Silva, Luiz A. B. San Martin, Joao Victor Uzita

TL;DR

The paper develops a sharp criterion linking the flag type Theta(S) of a semigroup S in a noncompact semisimple Lie group G to lower bounds of K-invariant cocycles on flag manifolds. It adapts the Iwasawa decomposition to define cocycles rho_lambda and analyzes their behavior via cone and rank-one reductions, connecting root data to dynamical bounds. The main result states that, for x0 in the invariant core, rho_alpha(g,x0) remains positively bounded for alpha outside the flag type and vanishes in the flag directions, enabling recovery of Theta(S) from cocycle behavior. This provides a new dynamical criterion for flag type and has implications for random products in semisimple groups and spectral radius phenomena in associated operators.

Abstract

The flag type of a semigroup S of a noncompact semisimple Lie group is an algebraic tool related to the geometry of the invariant control set determined by S on the flag manifolds of G. In the present paper we show that it is possible to recover the flag type by studying the existence of lower bounds for cocycles on the maximal flag manifold

Semigroups in semi-simple Lie groups: Flag type and estimation of cocycles

TL;DR

The paper develops a sharp criterion linking the flag type Theta(S) of a semigroup S in a noncompact semisimple Lie group G to lower bounds of K-invariant cocycles on flag manifolds. It adapts the Iwasawa decomposition to define cocycles rho_lambda and analyzes their behavior via cone and rank-one reductions, connecting root data to dynamical bounds. The main result states that, for x0 in the invariant core, rho_alpha(g,x0) remains positively bounded for alpha outside the flag type and vanishes in the flag directions, enabling recovery of Theta(S) from cocycle behavior. This provides a new dynamical criterion for flag type and has implications for random products in semisimple groups and spectral radius phenomena in associated operators.

Abstract

The flag type of a semigroup S of a noncompact semisimple Lie group is an algebraic tool related to the geometry of the invariant control set determined by S on the flag manifolds of G. In the present paper we show that it is possible to recover the flag type by studying the existence of lower bounds for cocycles on the maximal flag manifold

Paper Structure

This paper contains 9 sections, 14 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Lower bounds of cocycles
  3. The main result
  4. Nonexistence of lower bound for $\alpha \in \Theta \left( S\right)$
  5. Simple root outside $\Theta \left( S\right)$
  6. Preliminaries
  7. Semisimple theory
  8. Semigroup actions on flag manifolds
  9. Cocycles on flag manifolds

Key Result

Theorem 1

Let $S\subset G$ be a semigroup with $\mathrm{int}S\neq \emptyset$ and write $\mathbb{F}_{\Theta \left( S\right) }$, $\Theta \left( S\right) \subset \Sigma$, for its flag type. Denote by $C$ the invariant control set of $S$ in the maximal flag manifold $\mathbb{F}$. If $x_{0}\in C_{0}$, then for any $\alpha \in \Sigma \setminus \Theta \left( S\right)$. Conversely, $\inf_{g\in S}\rho _{\alpha }\le

Theorems & Definitions (15)

  • Theorem
  • Lemma 2.1
  • Proposition 2.2
  • Remark 2.3
  • Lemma 2.4
  • Proposition 2.5
  • Corollary 2.6
  • Proposition 2.7
  • Theorem 3.1
  • Lemma 3.2
  • ...and 5 more