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

Author: Juan Tirao
Journal: Proc. Amer. Math. Soc. 121 (1994), 1257-1266
MSC: Primary 22E15; Secondary 17B35, 22E60
MathSciNet review: 1189550
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Abstract: Let $ {G_0}$ be a connected Lie group with Lie algebra $ {g_0}$ and let h be a Lie subalgebra of the complexification g of $ {g_0}$ . Let $ {C^\infty }{({G_0})^h}$ be the annihilator of h in $ {C^\infty }({G_0})$ and let $ \mathcal{A} = \mathcal{A}({C^\infty }{({G_0})^h})$ be the annihilator of $ {C^\infty }{({G_0})^h}$ in the universal enveloping algebra $ \mathcal{U}(g)$ of g. If h is the complexification of the Lie algebra $ {h_0}$ of a Lie subgroup $ {H_0}$ of $ {G_0}$ then $ \mathcal{A} = \mathcal{U}(g)h$ whenever $ {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^\infty }$, the other for holomorphic functions. In the first part the Lie algebra $ {\bar h_0}$ of the closure of $ {H_0}$ is characterized as the annihilator in $ {g_0}$ of $ {C^\infty }{({G_0})^h}$, and it is proved that $ {h_0}$ is an ideal in $ {\bar h_0}$ and that $ {\bar h_0} = {h_0} \oplus v$ where v is an abelian subalgebra of $ {\bar h_0}$. In the second part we consider a complexification G of $ {G_0}$ and assume that h is the Lie algebra of a closed connected subgroup H of G. Then we establish that $ \mathcal{A}(\mathcal{O}{(G)^h}) = \mathcal{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 $ \mathcal{A}({C^\infty }{({G_0})^h}) = \mathcal{U}(g)h$.

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Keywords: Lie subgroup, closure, annihilator, complexification, $ {C^\infty }$ and holomorphic functions, quasi-affine variety, reductive and unipotent groups
Article copyright: © Copyright 1994 American Mathematical Society

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