ON THE LEFT IDEAL IN THE UNIVERSAL ENVELOPING ALGEBRA OF A LIE GROUP GENERATED BY A COMPLEX LIE SUBALGEBRA

Let G0 be a connected Lie group with Lie algebra g0 and let h be a Lie subalgebra of the complexification g of g0. Let C(infinity)(G0)h be the annihilator of h in C(infinity)(G0) and let A = A(C(infinity)(G0)h) be the annihilator of C(infinity)(G0)h in the universal enveloping algebra U(g) of g. If h is the complexification of the Lie algebra h0 of a Lie subgroup H0 of G0 then A = U(g)h whenever H0 is closed, is a known result and the point of this paper is to prove the converse assertion. The paper has two distinct parts, one for C(infinity), the other for holomorphic functions. In the first part the Lie algebra ho of the closure of H0 is characterized as the annihilator in go of C(infinity)(G0)h, and it is proved that h0 is an ideal in h0BAR and that h0BAR = h0 + v where v is an abelian subalgebra of h0BAR. In the second part we consider a complexification G of G0 and assume that h is the Lie algebra of a closed connected subgroup H of G. Then we establish that A(O(G)h) = U(g)h if and only if G/H has many holomorphic functions. This is the case if G/H is a quasi-affine variety. From this we get that if H is a unipotent subgroup of G or if G and H are reductive groups then A(C(infinity)(G0)h) = U(g)h.