Harmonic Analysis for 4-Dimensional Real Frobenius Lie Algebras
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A real Frobenius Lie algebra is characterized as the Lie algebra of a real Lie group admitting open coadjoint orbits. In this paper, we study irreducible unitary representations corresponding to open coadjoint orbits for each of 4-dimensional Frobenius Lie algebras. We show that such unitary representations are square-integrable, and their Duflo–Moore operators are closely related to the Pfaffian of the Frobenius Lie algebra.
KeywordsFrobenius Lie algebras Square-integrable representations Duflo-Moore operators
MSC Classification22E45 22D10 43A32
The present authors are very grateful to Professor Michel Duflo for suggesting that we study Frobenius Lie algebras. They thank Professor Hidenori Fujiwara for his interest to this work, and thank the referee for helpful comments. And they express their gratitude to Professors Ali Baklouti and Takaaki Nomura for invitation to the fifth Tunisian-Japanese conference and its wonderful hospitality. The first author would like to thank The Research, Technology Directorate General of Higher Education (RISTEK-DIKTI), Ministry of Research, Technology and Higher Education of Indonesia. This research is partially supported by KAKENHI 16K05174.
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