Induced representations of solvable Lie groups and their applications

1999 Fiscal Year Final Research Report Summary

• Project/Area Number: 10640177
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Tottori University
• Principal Investigator: INOUE Junko Tottori University, Faculty of Education and Regional Sciences, Associate Professor, 教育地域科学部, 助教授 (40243886)
• Project Period (FY): 1998 – 1999
• Keywords: solvable Lie group / induced representation / coadjoint orbit / polarization / holomorphically induced representation

Research Abstract

I investigated holomorphically induced representations of solvable Lie groups G from real linear forms f of their Lie algebras and weak polarizations at f. A slightly modified holomorphic induction ρ from f and a positive weak polarization at f is non-zero when G is a connected and simply connected Lie group whose Lie algebra is a normal j-algebra, and f belongs to an open coadjoint orbit. In this case, the decomposition or ρ into irreducible representations can be described in terms of the orbit method, and a distributional Frobenius reciprocity holds. I tried to generalize this results of myself : I studied examples in low dimensional (general) exponential Lie groups. I also reviewed the proof of my previous result mentioned above, and modified some technical parts. I expect that for general exponential groups, holomorphic inductions from positive weak polarizations are described similarly in terms of the orbit method. I was also concerned with holomorphic inductions from complex subalgebras h which are isotropic (not necessarily maximally isotropic) for the bilinear form defined by f when G is nilpotent : I studied low dimensional nilpotent Lie groups. General cases are too complicated to treat, but when h+g(f)c, where g(f) is the Lie algebra of the stabilizer of f, is maximally isotropic, and representations appearing in the decomposition are corresponding to flat orbits, I could describe the decomposition using the orbit method. I will try to generalize it for some class of holomorphic inductions.
For problems of "smooth operators" of a Hilbert space where an irreducible representation is realized, I mainly investigated examples in low dimensional exponential groups. For non-unimodular groups, we need to modify the definition of "smooth operators". I plan to find a good definition in order to use it in the theory of Fourier transforms in further research.