ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

7. Lie Groups and Lie Algebras

1991 MS Classification: 22E46, 17B35 Leandro Cagliero and Juan Tirao, National University of Cordoba (K, M)-spherical submodules of the universal enveloping algebra of a real rank one semisimple Lie group G_{∘} Let G_{∘} be a connected, non-compact semisimple Lie group with finite center and let K_{∘} be a maximal compact subgroup of G_{∘}. If 𝔨 ⊂ 𝔤 denote the complexifications of the Lie algebras of K_{∘} and G_{∘} respectively, U(𝔤) will be the universal enveloping algebra of 𝔤 and U(𝔤)^{K} will denote the centralizer 𝔨 in U(𝔤). In order to study the algebra U(𝔤)^{K} B. Kostant suggested to study the injective homomorphism of algebras [1]

P : U(𝔤)^{K} → U(𝔨)^{M} ⊗ U(𝔞)

associated with an Iwasawa decomposition G = KAN. Here M is the centralizer of 𝔞 in K. It is well known that P is not surjective. In this work we present some transversality properties for the (K, M)-spherical submodules of U(𝔨) which allows to complete the characterization of P (U(𝔤)^{K}) for a real rank one Lie group. In addition we characterize the dominant weights of the (K, M)-spherical modules for the rank one case without using the Cartan classification. Reference:

[1] J. Lepowsky, Algebraic results on representations of semisimple Lie groups, Trans. Amer. Math. Soc. 176 (1973), 1-44