Some Questions about Semisimple Lie Groups Originating in Matrix Theory

Ðoković, Dragomir Ž. ; Tam, Tin-Yau

Canadian mathematical bulletin, Tome 46 (2003) no. 3, pp. 332-343

Résumé

We generalize the well-known result that a square traceless complex matrix is unitarily similar to a matrix with zero diagonal to arbitrary connected semisimple complex Lie groups $G$ and their Lie algebras $\mathfrak{g}$ under the action of a maximal compact subgroup $K$ of $G$ . We also introduce a natural partial order on $\mathfrak{g}:\,x\,\le y$ if $f(K\,\cdot \,x)\,\subseteq \,f(K\,\cdot \,y)$ for all $f\,\in \,{{\mathfrak{g}}^{*}}$ , the complex dual of $\mathfrak{g}$ . This partial order is $K$ -invariant and induces a partial order on the orbit space $\mathfrak{g}/K$ . We prove that, under some restrictions on $\mathfrak{g}$ , the set $f(K\,\cdot \,x)$ is star-shaped with respect to the origin.

Détail

DOI : 10.4153/CMB-2003-035-1

Mots-clés : 15A45, 20G20, 22E60