M-spherical K-modules of a rank one semisimple Lie group
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Let G ○ =K ○ A ○ N ○ be an Iwasawa decomposition of a connected, noncompact real semisimple Lie group with finite center and let M ○ be the centralizer of A ○ in K ○ . B. Kostant proved that for every irreducible M-spherical K-module V there exists a unique d (the Kostant degree of V) such that V can be realized as a submodule of the space of all -harmonic homogeneous polynomials of degree d on . Here is a Cartan decomposition of the complexification of the Lie algebra of G ○ .In this paper we give an algorithm to obtain a highest weight vector from any M-invariant vector in an irreducible M-spherical K-module. This algorithm allows us to compute a sharp bound for the Kostant degree d(v) of any M-invariant vector v in a locally finite M-spherical K-module V. The method computes d(v) effectively for any V if G ○ is locally isomorphic to SO(n,1) and for if G ○ is locally isomorphic to SU(n,1).
KeywordsHigh Weight Weight Vector Homogeneous Polynomial High Weight Vector Iwasawa Decomposition
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