Root Systems for Levi Factors and Borel-de Siebenthal Theory

Let $\frak{m}$ be a Levi factor of a proper parabolic subalgebra $\frak{q}$ of a complex semisimple Lie algebra $\frak{g}$. Let $\frak{t} = cent \frak{m}$. A nonzero element $\nu \in \frak{t}^*$ is called a $\frak {t}$-root if the corresponding adjoint weight space $\frak{g}_{nu}$ is not zero. If $\nu$ is a $\frak{t}$-root, some time ago we proved that $\frak{g}_{\nu}$ is $ad \frak{m}$ irreducible. Based on this result we develop in the present paper a theory of $\frak{t}$-roots which replicates much of the structure of classical root theory (case where $\frak{t}$ is a Cartan subalgebra). The results are applied to obtain new reults about the structure of the nilradical $\frak{n}$ of $\frak{q}$. Also applications in the case where $dim \frak{t}=1$ are used in Borel-de Siebenthal theory to determine irreducibility theorems for certain equal rank subalgebras of $\frak{g}$. In fact the irreducibility results readily yield a proof of the main assertions of the Borel-de Siebenthal theory.