Root Systems for Levi Factors and Borel-de Siebenthal Theory

2007/11/18 by Kostant, Bertram · 1 citation
#20CXX #20Gxx #22E10 #22E25 #22E46 #22EXX #FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.0711.2809

Abstract

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

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