Classification des structures $CR$ invariantes pour les groupes de Lie compacts

Jean-Yves Charbonnel et Hella Ounaïes Khalgui

Abstract: Let $G_0$ be a compact Lie group of dimension $N$ whose Lie algebra is ${\goth g}_0$. The notion of $CR$ structure on a C$^{\infty }$ manifold is known a long time ago. In this note we are interested by the $CR$ stuctures on $G_0$ which are invariant by the left action of the group on the tangent bundle and which are of maximal rank. Such a structure is defined by its fibre ${\goth h}$ at the neutral element which is a subalgebra of the complexification ${\goth g}$ of ${\goth g}_0$ whose dimension is the entire part $[N/2]$ of $N/2$ and whose intersection with ${\goth g}_0$ is equal to $\{0\}$. Up to conjugation by the adjoint group of ${\goth g}_0$, these subalgebras are classified. When $N$ is even, there is only one type, type $CR0$. When $N$ is odd, there are two types, type $CR0$ and type $CRI$. These types are given in terms of Cartan subalgebras and root systems. In any case, these subalgebras are solvable. Following Baouendi, M. S., L. P. Rothschild and F. Treves [Inventiones Mathematicae 82 (1985), 359--396], we introduce the notion of $CR$ structures which are $G_0$-invariant and invariant by the transverse action of a $G_0$-invariant Lie subgroup. When this group is commutative, we get the notion of $G_0$-rigidity. We then prove, when $N$ is odd, that a $G_0$-invariant $CR$ structure, of maximal rank, is $G_0$-rigid if and only if the fibre of the $CR$ structure at the neutral element, is of type $CR0$. Following H. Jacobowitz [Pacific Journal of Mathematics 127 (1987), 91--101], we introduce the canonical fibre bundle $K$ of a $G_0$-invariant $CR$ structure, of maximal rank, when $N$ is odd. We prove that $K$ contains a closed $G_0$-invariant form if and only if the fibre of the $CR$ structure at the neutral element, is of type $CR0$ or type $CRII$. As for type $CR0$ and $CRI$, the $CRII$ type is defined up to conjugation by the adjoint group of $\g_0$ in terms of Cartan subalgebras an root systems. In fact, every subalgebra of type $CRII$ is of type $CRI$.

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