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

Published:2010-08-26
Printed: Mar 2011
• Wai-Shun Cheung,
Department of Mathematics and Statistics, Auburn University, AL 36849--5310, U.S.A.
• Tin-Yau Tam,
Department of Mathematics and Statistics, Auburn University, AL 36849--5310, U.S.A.
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Abstract

Given a complex semisimple Lie algebra $\mathfrak{g}=\mathfrak{k}+i\mathfrak{k}$ ($\mathfrak{k}$ is a compact real form of $\mathfrak{g}$), let $\pi\colon\mathfrak{g}\to \mathfrak{h}$ be the orthogonal projection (with respect to the Killing form) onto the Cartan subalgebra $\mathfrak{h}:=\mathfrak{t}+i\mathfrak{t}$, where $\mathfrak{t}$ is a maximal abelian subalgebra of $\mathfrak{k}$. Given $x\in \mathfrak{g}$, we consider $\pi(\mathop{\textrm{Ad}}(K) x)$, 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 $f(\mathop{\textrm{Ad}}(K)x)$, where $f$ is a linear functional on $\mathfrak{g}$. We establish the star-shapedness of $f(\mathop{\textrm{Ad}}(K)x)$ for simple Lie algebras of type $B$.
 MSC Classifications: 22E10 - General properties and structure of complex Lie groups [See also 32M05] 17B20 - Simple, semisimple, reductive (super)algebras