On the sum of the index of a parabolic subalgebra and of its nilpotent radical

Details On the sum of the index of a parabolic subalgebra and of its nilpotent radical On the sum of the index of a parabolic subalgebra and of its nilpotent radical

2006-06-12

arxiv.org/abs/math/0606268v2

Proceedings of the American Mathematical Society 136 (2008) 1515-1522

Mathematics

Representation Theory

Author's affiliation added in the second version

Scientific paper

In this short note, we investigate the following question of Panyushev : ``Is the sum of the index of a parabolic subalgebra of a semisimple Lie algebra $\mathfrak{g}$ and the index of its nilpotent radical always greater than or equal to the rank of $\mathfrak{g}$?''. Using the formula for the index of parabolic subalgebras conjectured by Tauvel and the author, and proved by Millet-Fauquant and Joseph, we give a positive answer to this question. Moreover, we also obtain a necessary and sufficient condition for this sum to be equal to the rank of $\mathfrak{g}$. This provides new examples of direct sum decomposition of a semisimple Lie algebra verifying the ``index additivity condition'' as stated by Ra{\"\i}s.

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