Constructions and decompositions of induced representations of solvable Lie groups and their applications

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Holomorphically induced representations of a Lie group are usually constructed starting from a real linear from f of the Lie algebra and a complex polarization at f. In this research, I investigated holomorphically-induced representations of solvable Lie groups from weak polarizations or general complex subalgebra n.First of all, let G be a connected and simply connected Lie group whose Lie algebra is a normal j-algebra. When f belongs to an open coadjoint G-orbit and n is a positive weak polarization at f, the holomorphically-induced representation of G is non-zero if some term o defined by the modular function is suitably chosen. It decomposes into a direct sum of irreducible representations, which is described by the orbit method. In the course of this research, I reviewed and checked again the term o above and the construction of intertwining operators using algebraic structures of normal j-algebras. I revised the paper of the results above, and it has been published.I investigated some cases for low-dimensional exponential groups G and weak polarizations or complex subalgebras n which are isotropic (not necessarily maximally isotropic) for f. In some cases, I actually obtained non-zero representations and decompositions of them. The descriptions of semi-invariant vectors, which are used in computations, essentially depend on each algebraic structure of Lie algebras. I will try to find better descriptions suitable for treating a general setting in further study.For irreducible representations of exponential groups, I also treated another problem to find "good" operators or "good" subspaces of representation spaces which are compatible with the Fourier transforms. I have tried to characterize "good" subspaces by using "smooth operators" introduced by Ludwig, and I plan to proceed with it in further research.