Arch. Math., gol. 56, 491-496 (1991) 0003-889X/91/5605-0491 $ 2.70/0 9 1991 Birkh/iuser Verlag, Basel Actions of semisimple Lie groups and orbits of Cartan subgroups By Yu. G. ZARHIN Let G be a real semisimple Lie group of rank r, acting analytically and effectively on an n-dimensional (connected) real analytic manifold M. E.E. Shnol [9] proved that always r < n, and r < n/2 in the case of symplectic M and the Hamiltonian G-action. He deduced the general case from the symplectic one, making use of the existence of canon- ical symplectic structure on the cotangent bundle. In this paper we give another proof of the theorem of Shnol, based on the following idea. Let H be a Cartan subgroup of G. We prove the existence of a point m ~ M, whose stabilizer H min H is a discrete subgroup. In the case of symplectic M and Hamiltonian G-action we prove that each H-orbit is isotropic. Notice that if G is compact then the existence of a point, whose stabilizer in H is trivial, is an immediate corollary of Mostov's finiteness theorem for conjugacy classes of stabi- lizers [7]. The existence of such a point in the case of an algebraic action of a reductive algebraic group is also known [5]. I am grateful to E. E. Shnol, V. L. Popov, M. Gromov and G. L. Litvinov for useful discussions. The proof of Theorem 1 was inspired by some ideas of theory of/-adic representations [8]. 1. General case. Let K be either the field P~ of real numbers or the field C of complex numbers or an ultrametric complete non-discrete commutative field of characteristic zero [4] (e.g. the field Qp of p-adic numbers). Let G be a finite-dimensional K-Lie group, L(G) its Lie algebra, and Ad: G -~Aut L(G) the adjoint representation of G [3]. We assume that L(G) is a semisimple Lie algebra. Since the set of ideals of L(G) is finite, elementary properties of the exponential map [3] imply the existence of a finite (if K = P~ or K = C) or countable (in the ultrametric case) subset A of G such that: 1) A does not contain the unit e of G, 2) if X is a Lie subgroup of G and the Lie algebra of X contains a non-zero ideal of L(G) then the intersection A • X is non-empty. Let V be L(G) viewed as a finite-dimensional K-vector space, and 7: G -~Aut (V) be the composition of Ad and the natural embedding Aut L(G) c Aut (V). The Lie algebra