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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
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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  • [1] A. Bialynicki-Birula, G. Hochschild, and G. D. Mostow, Extensions of representations of algebraic linear groups, Amer. J. Math. 85 (1963), 131-144. MR 0155938 (27:5871)
  • [2] C. Chevalley, Theory of Lie groups, Vol. 1, Princeton Univ. Press, Princeton, NJ, 1946.
  • [3] J. Dixmier, Algèbres enveloppantes, Gauthier-Villars, Paris, 1974. MR 0498737 (58:16803a)
  • [4] M. Gotô, Faithful representations of Lie groups. I, Math. Japon. 1 (1948), 107-119. MR 0029919 (10:681a)
  • [5] S. Helgason, Differential geometry and symmetric spaces, Academic Press, New York and London, 1962. MR 0145455 (26:2986)
  • [6] G. Hochschild and G. D. Mostow, Affine embeddings of complex analytic homogeneous spaces, Amer. J. Math. 87 (1965), 807-839. MR 0191900 (33:127)
  • [7] -, Representations and representative functions of Lie groups, Ann. of Math. (2) 66 (1957), 495-542. MR 0098796 (20:5248)
  • [8] J. Humphreys, Linear algebraic groups, Springer-Verlag, Berlin, New York, and Heidelberg, 1975. MR 0396773 (53:633)
  • [9] A. T. Huckleberry and E. Oeljeklaus, Homogeneous spaces from a complex analytic viewpoint, Manifolds and Lie Groups, Progr. Math., vol. 14, Birkhäuser, Basel and Boston, MA, 1981, pp. 159-186. MR 642856 (84i:32045)
  • [10] G. D. Mostow, The extensibility of local Lie groups of transformations and groups on surfaces, Ann. of Math. (2) 52 (1950), 606-636. MR 0048464 (14:18d)
  • [11] M. Sugiura, The Tannaka duality theorem for semisimple Lie groups and the unitarian trick, Manifolds and Lie Groups (Notre Dame, Indiana, 1980), Progr. Math., vol. 14, Birkhäuser, Basel and Boston, MA, 1981, pp. 405-428. MR 642870 (83d:22011)
  • [12] G. Warner, Harmonic analysis on semi-simple Lie groups. I, Springer-Verlag, Berlin and New York, 1972. MR 0498999 (58:16979)

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Additional Information

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