The minimal orbit in a simple Lie algebra and its associated maximal ideal

Let 9 be a simple Lie algebra over C. If 9 is different from sl(n + 1) : n = 1, 2, ..., then g* admits a single non-trivial G-orbit (Po of minimal dimension.This orbit consists of nilpotent elements, contains (^ -{ 0 }, where (^ is the root subspace of the highest root (3, and is not polarizable.Through the study of a certain Heisenberg subalgebra of g associated with g 0 , it is shown that there exists a unique completely prime, two-sided ideal Jo ofU (9) whose characteristic variety coincides with OQ u { 0 }.Jo is shown to be a maximal ideal and that it cannot be induced from any proper subalgebra of 9.The construction of Jo is very explicit and its central character is computed.For sp ( 4), Jo coincides with an ideal constructed by Conze and Dixmier [8] (Ex.3).* Work supported by the C.N.R.S. ANNALES SCIENT1FIQUES DE L'^COLE NORMALE SUP^RIEURE r = {^et^ : (X, a,) ^ 0, for all a,e7i}.Note that P e ^.Given X e Of, set A^ = { a e A* : (X, a) = 0 }, ^ = n n A^ and n^ the complement of n^ in TT.