On the left ideal in the universal enveloping algebra of a Lie group generated by a complex Lie subalgebra

Abstract

Let G 0 {G_0} be a connected Lie group with Lie algebra g 0 {g_0} and let h be a Lie subalgebra of the complexification g of g 0 {g_0} . Let C ∞ ( G 0 ) h {C^\infty }{({G_0})^h} be the annihilator of h in C ∞ ( G 0 ) {C^\infty }({G_0}) and let A = A ( C ∞ ( G 0 ) h ) \mathcal {A} = \mathcal {A}({C^\infty }{({G_0})^h}) be the annihilator of C ∞ ( G 0 ) h {C^\infty }{({G_0})^h} in the universal enveloping algebra U ( g ) \mathcal {U}(g) of g. If h is the complexification of the Lie algebra h 0 {h_0} of a Lie subgroup H 0 {H_0} of G 0 {G_0} then A = U ( g ) h \mathcal {A} = \mathcal {U}(g)h whenever H 0 {H_0} 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 ∞ {C^\infty } , the other for holomorphic functions. In the first part the Lie algebra h ¯ 0 {\bar h_0} of the closure of H 0 {H_0} is characterized as the annihilator in g 0 {g_0} of C ∞ ( G 0 ) h {C^\infty }{({G_0})^h} , and it is proved that h 0 {h_0} is an ideal in h ¯ 0 {\bar h_0} and that h ¯ 0 = h 0 ⊕ v {\bar h_0} = {h_0} \oplus v where v is an abelian subalgebra of h ¯ 0 {\bar h_0} . In the second part we consider a complexification G of G 0 {G_0} 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 \mathcal {A}(\mathcal {O}{(G)^h}) = \mathcal {U}(g)h if and only if G / H G/H has many holomorphic functions. This is the case if G / H 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 ∞ ( G 0 ) h ) = U ( g ) h \mathcal {A}({C^\infty }{({G_0})^h}) = \mathcal {U}(g)h .