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An Example of Holomorphically Induced Representations of Exponential Solvable Lie Groups

Conference paper
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Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 290)

Abstract

We discuss a holomorphically induced representation \(\rho =\rho (f,\mathfrak h)\) of Boidol’s group (split oscillator group) G from a real linear form f of the Lie algebra \(\mathfrak g\) of G and a one-dimensional complex subalgebra \(\mathfrak h\) of \(\mathfrak g_\mathbb C\) given by (2.2) in Sect. 2.\(\rho \) is a subrepresentation of the regular representation of G with the Plancherel measure \(\nu \). For \(\nu \)-almost all irreducible representations \(\pi \) of G, the spaces of generalized vectors satisfying the semi-invariance associated with f and \(\mathfrak h\) are one-dimensional subspaces. On the other hand, according to the choice of f, there are two cases that (1) \(\rho \) vanishes, and (2) \(\rho \) is non-zero.

Keywords

Holomorphically induced representation Solvable Lie group Plancherel formula 

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Education Center, Organization for Educational Support and International AffairsTottori UniversityTottoriJapan

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