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

Published online by Cambridge University Press:  20 November 2018

Wai-Shun Cheung  and
Tin-Yau Tam
Wai-Shun Cheung
Affiliation:
Department of Mathematics and Statistics, Auburn University, AL 36849–5310, U.S.A.e-mail: cheungwaishun@gmail.comtamtiny@auburn.edu
Tin-Yau Tam
Affiliation:
Department of Mathematics and Statistics, Auburn University, AL 36849–5310, U.S.A.e-mail: cheungwaishun@gmail.comtamtiny@auburn.edu
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Abstract

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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 $B$.

Keywords

22E1017B20
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 54 , Issue 1 , 01 March 2011 , pp. 44 - 55
Copyright
Copyright © Canadian Mathematical Society 2011

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