Projective representations of real reductive Lie groups and the gradient map

TL;DR

This paper investigates the projective representations of real reductive Lie groups using gradient map techniques, providing insights into their structure and properties within the context of symmetric endomorphisms and scalar products.

Contribution

It introduces a method to analyze projective representations of real reductive Lie groups via gradient maps, connecting group structure with geometric representation theory.

Findings

Characterization of the projective representation using gradient maps

Identification of conditions for the existence of a $K$-invariant scalar product

Analysis of the geometric structure of the representation space

Abstract

Let $G$ be a connected semisimple noncompact real Lie group and let $\rho: G \longrightarrow \mathrm{SL}(V)$ be a representation on a finite dimensional vector space $V$ over $\mathbb R$, with $\rho(G)$ closed in $\mathrm{SL}(V)$. Identifying $G$ with $\rho(G)$, we assume there exists a $K$-invariant scalar product $\mathtt g$ such that $G=K\exp(\mathfrak p)$, where $K=\mathrm{SO}(V,\mathtt g)\cap G$, $\mathfrak p=\mathrm{Sym}_o (V,\mathtt g)\cap \mathfrak g$ and $\mathfrak g$ denotes the Lie algebra of $G$. Here $\mathrm{Sym}_o (V,\mathtt g)$ denotes the set of symmetric endomorphisms with trace zero. Using the $G$-gradient map techniques we analyze the natural projective representation of $G$ on $\mathbb P(V)$.