Indice et decomposition de Cartan d'une algebre de Lie semi-simple reelle

Details Indice et decomposition de Cartan d'une algebre de Lie semi-simple reelle Indice et decomposition de Cartan d'une algebre de Lie semi-simple reelle

2005-06-10

arxiv.org/abs/math/0506206v3

Journal of Algebra 303 (1) (2006) 382-406

Mathematics

Representation Theory

21 pages en francais

Scientific paper

The Iwasawa decomposition $\mathfrak{g}=\mathfrak{k}_0 \oplus \hat{\mathfrak{a}}_0 \oplus \mathfrak{n}_0$ of the real semisimple Lie algebra $\mathfrak{g}_0$ comes from its Cartan decomposition $\mathfrak{g}_0=\mathfrak{k}_0 \oplus \mathfrak{p}_0$. Then we get $\mathfrak{g}_0=\mathfrak{k}_0 \oplus \mathfrak{b}_0$ where $\mathfrak{b}_0=\hat{\mathfrak{a}}_0 \oplus \mathfrak{n}_0$. The question of knowing if the index were additive in the decomposition $\mathfrak{g}_0=\mathfrak{k}_0 \oplus \mathfrak{b}_0$ goes back M. Ra\"{i}s \cite{Rais}. In \cite{Moreau3}, I wrote that the index always is additive for this decomposition. Precisly, I claim that the index of $\mathfrak{b}$ is given by the following formula : ${\rm ind} \mathfrak{b} = {\rm rk \} \mathfrak{g} - {\rm rk \} \mathfrak{k}$, where $\mathfrak{b}$ is the complexification of $\mathfrak{b}_0$. This result is false in general. We actually have an inequality : ${\rm ind} \mathfrak{b} \geq {\rm rg \} \mathfrak{g} - {\rm rg \} \mathfrak{k}$. The goal of this paper is to correct this mistake. We resume the approach of \cite{Moreau3} to obtain this time the previous inequality. Then we give in more a characterization of the semisimple real Lie algebra $\mathfrak{g}_0$ for which the index is additive in the decomposition $\mathfrak{g}_0=\mathfrak{k}_0 \oplus \mathfrak{b}_0$. Moreover, we study in this paper the quasi-reductive character of some subalgebras of $\mathfrak{g}$. This is a new part in comparison with \cite{Moreau3}.

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