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

DOI:
https://doi.org/10.1090/S0002-9939-1994-1189550-8

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,
<!– MATH ${C^\infty }$ –> <IMG WIDTH="38" HEIGHT="18" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="${C^\infty }$"> and holomorphic functions,
quasi-affine variety,
reductive and unipotent groups

Article copyright:
© Copyright 1994
American Mathematical Society