Star-Shapedness and K-Orbits in Complex Semisimple Lie Algebras

Cheung, Wai-Shun ; Tam, Tin-Yau

Canadian mathematical bulletin, Tome 54 (2011) no. 1, pp. 44-55

Résumé

Given a complex semisimple Lie algebra $\mathfrak{g}=\mathfrak{k}+i\mathfrak{k}$ ( $\mathfrak{t}$ is a compact real form of $\mathfrak{g}$ ), let $\text{ }\pi \text{ }\text{:}\mathfrak{g}\to \mathfrak{h}$ be the orthogonal projection (with respect to the Killing form) onto the Cartan subalgebra $\mathfrak{h}:=\mathfrak{t}\text{+}i\mathfrak{t}$ , where $\mathfrak{t}$ is a maximal abelian subalgebra of $\mathfrak{k}$ . Given $x\,\in \,\mathfrak{g}$ , we consider $\text{ }\!\!\pi\!\!\text{ (Ad(}K\text{)}x)$ , where $K$ is the analytic subgroup $G$ corresponding to $\mathfrak{k}$ , and show that it is star-shaped. The result extends a result of Tsing. We also consider the generalized numerical range $f(\text{Ad(}K\text{)}x)$ , where $f$ is a linear functional on $\mathfrak{g}$ . We establish the star-shapedness of $f(\text{Ad(}K\text{)}x)$ for simple Lie algebras of type $B$ .

Détail

DOI : 10.4153/CMB-2010-097-7

Mots-clés : 22E10, 17B20