|Budget Amount *help
¥1,500,000 (Direct Cost : ¥1,500,000)
Fiscal Year 1999 : ¥700,000 (Direct Cost : ¥700,000)
Fiscal Year 1998 : ¥800,000 (Direct Cost : ¥800,000)
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.