The minimal Morse components of translations on flag manifolds are normally hyperbolic

TL;DR

This paper proves that the minimal Morse components of translations on flag manifolds are normally hyperbolic, extending previous results from diagonalizable matrices to general elements of semisimple Lie groups.

Contribution

It generalizes the normal hyperbolicity of Morse components from diagonalizable matrices to arbitrary semisimple Lie group elements acting on flag manifolds.

Findings

Morse components are normally hyperbolic in this general setting

The result applies to various geometric contexts including projective spaces and Grassmannians

Lie theory tools are essential for solving this problem

Abstract

Consider the iteration of an invertible matrix on the projective space: are the Morse components normally hyperbolic? As far as we know, this was only stablished when the matrix is diagonalizable over the complex numbers. In this article we prove that this is true in the far more general context of an arbitrary element of a semisimple Lie group acting on its generalized flag manifolds: the so called translations on flag manifolds. This context encompasses the iteration of an invertible non-diagonazible matrix on the real or complex projective space, the classical flag manifolds of real or complex nested subspaces and also symplectic grassmanians. Without these tools from Lie theory we do not know how to solve this problem even for the projective space.