Some topics for unitary representations of solvable Lie groups
Fujiwara, Hidenori Some topics for unitary representations of solvable Lie groups. In: 表現論シンポジウム, 2005/11/15-18.
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November 2005: 表現論シンポジウム
Introduction In this talk I shall explain some topics for unitary representations of solvable Lie groups, their present state and problems for futur development. At the beginning of 1970’s Auslander-Kostant succeeded in the framework of the orbit method to construct the unitary dual for a connected and simply connected type I solvable Lie group, and then their results were extended to non type I solvable Lie groups by Pukanzsky. These works are landmarks in the representation theory of solvable Lie groups. If, however, we try to study the holomorphically induced representation and its application in detail, it remains until now to be difficult. Concerning induced representations or restricted representations, we would like to decompose them, construct intertwining operators or study some related algebra of invariant differential operators. Then, we know little even for exponential Lie groups. We have more tools in hand only for nilpotent Lie groups. The theory of representations is developed in a rather different fashion between semi-simple and solvable Lie groups. The algebraic structure of semi-simple Lie groups is so rich that it offers us many ingredients. As for solvable Lie groups, the poor structure obliges us to use the main method of induction. In any way it’s incontestable that the orbit method is very fruitful in the unitary representation theory of solvable Lie groups. The innovatory idea of Kirillov to associate a coadjoint orbit to an irreducible unitary representation seems to be proud of its worthy results. It’s a nice application of Mackey’s theory to solvable Lie groups. Once this frame is opted for, we can study many objects in analysis by means of algebraic and geometric properties of coadjoint orbits. The aim of this talk is to invite young people into the research of this domain, where many problems are waiting them.