| Budget Amount *help |
¥2,200,000 (Direct Cost: ¥2,200,000)
Fiscal Year 2005: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2004: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2003: ¥800,000 (Direct Cost: ¥800,000)
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| Research Abstract |
1.According to the orbit method, an irreducible unitary representation of an exponential solvable Lie group is realized as an induced representation from a unitary character of a connected subgroup on a space of L^2-functions of the homogeneous space diffeomorphic with R^n. When the group is nilpotent, the space of C^∞ vectors coincides with the space of Schwartz functions of R^n, and it fits in with a description of the Fourier transform for the group. However, when the group is general exponential, the space of C^∞ vectors does not have such simple descriptions. As a collaboration with J.Ludwig (University of Metz, France), we investigated C^∞ vectors for irreducible representations of exponential groups G, and obtained the following results : Taking a nilpotent ideal which includes the derived ideal of the Lie algebra of G, and choosing a real polarization adapted to the ideal, we realize the representation on a space of L^2-functions on the homogeneous space. We treat a subspace consisting of functions with some rapidly decreasing property associated with the nilpotent ideal, and show that it can be embedded as the space of C^∞ vectors for an irreducible representation of a group containing G. Using the space, we also describe a certain space of functions included in the image of Fourier transforms of L^1-functions on G of finite ranks. Furthermore, specific descriptions of the space of C^∞ vectors for some special classes of representations are obtained.
2.I investigated generalized holomorphically induced representations, and described associated semiinvariant vectors for some examples of low dimensional solvable Lie groups. Their descriptions depend on particular group structures.
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