Abstract
LetG o be a non compact real semisimple Lie group with finite center, and letU U(g)K denote the centralizer inU U(g) of a maximal compact subgroupK o ofG o. To study the algebraU U(g)K, B. Kostant suggested to consider the projection mapP:U U(g)→U(k)⊗U(a), associated to an Iwasawa decompositionG o=K o A o N o ofG o, adapted toK o. WhenP is restricted toU U(g)K J. Lepowsky showed thatP becomes an injective anti-homomorphism ofU U(g)K intoU(k)M⊗U(a). HereU(k)M denotes the centralizer ofM o inU(k),M o being the centralizer ofA o inK o. To pursue this idea further it is necessary to have a good characterization of the image ofU U(g)K inU(k)M×U(a). In this paper we describe such image whenG o=SO(n,1)e or SU(n,1). This is acomplished by establishing a (minimal) set of equations satisfied by the elements in the image ofU U(g)K, and then proving that they are enough to characterize such image. These equations are derived on one hand from the intertwining relations among the principal series representations ofG o given by the Kunze-Stein interwining operators, and on the other hand from certain imbeddings among Verma modules. This approach should prove to be useful to attack the general case.
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Tirao, J.A. On the centralizer ofK in the universal enveloping algebra of SO(n,1) and SU(n,1). Manuscripta Math 85, 119–139 (1994). https://doi.org/10.1007/BF02568189
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DOI: https://doi.org/10.1007/BF02568189