Sur la classification des idéaux primitifs dans l’algèbre enveloppante d’une algèbre de Lie semi-simple


Let $\mathfrak{g}$ be a semi-simple complex Lie algebra. We denote by $U(\mathfrak{g})$ its enveloping algebra. A (two-sided) ideal $I$ of $U(\mathfrak{g})$ is called primitive if it is the kernel of an irreducible representation of $U(\mathfrak{g})$. For several reasons classifying such ideals is interesting. For instance, it is a well known result of Harish-Chandra that the quasi-sipmle irreducible representaitons of a real Lie group whose Lie algebra is a real form of $\mathfrak{g}$ define irreducible representations of $U(\mathfrak{g})$, and thus, primitive ideals in $U(\mathfrak{g})$. The main result of this paper is the following.

Fix a Borel subalgebra $\mathfrak{p}$ of $\mathfrak{g}$. Let $I$ be a primitive ideal of $U(\mathfrak{g})$. Then there exists an irreducible $U(\mathfrak(g)$- module with highest weight with respect to $\mathfrak{p}$ whose kernel is $I$.

This implies in particular that ${}\;^\wedge tI = I$, where $u\to {}\;^\wedge tu$ is an involution of $U(\mathfrak{g})$ which is the identity on some Cartan subalgebra of $\mathfrak{g}$.

The proof uses a result on this composition series of principal series of representations of complex semi-simple Lie groups announced by T. Hiraï in [7]. This paper contains a proof of this result.


Michel Duflo