Indice et décomposition de Cartan d'une algèbre de Lie semi-simple réelle

Abstract

The Iwasawa decomposition g = k 0 ⊕ a ˆ 0 ⊕ n 0 of the real semisimple Lie algebra g 0 comes from its Cartan decomposition g 0 = k 0 ⊕ p 0 . Then we get g 0 = k 0 ⊕ b 0 where b 0 = a ˆ 0 ⊕ n 0 . In this note, we establish an explicite formula for the index, ind b , of b , where b is the complexification of b 0 . More precisely, we show the following result: ind b = rg g − rg k , where g and k are respectively the complexifications of g 0 and k 0 . In particular, this answers positively a question by Raïs in [M. Raïs, Notes sur l'indice des algèbres de Lie, preprint, 2004]: is the index additive for the following decomposition: g 0 = k 0 ⊕ b 0 ? In the proof, we use the Kostant construction and the Cayley transforms. We also give a characterization of the semisimple real Lie algebra g 0 whose subalgebra b 0 has a stable form.