Analysis of minimal representations and branching laws of infinite-dimensional representations

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Minimal representations are one of building blocks of unitary representations. Classic examples are the Weil representation, and intensive algebraic studies have been made since 1990s by many experts. In contrast, I proposed yet another geometric approach to minimal representations, by which we could expect a fruitful theory on global analysis by maximal symmetries. It includes a theory of unitary inversion operator on the L^2-model that generalizes the Euclidean Fourier transform with G. Mano ([Memoirs of AMS, 1000, (2011)]), a deformation theory of the Fourier transform in [Compositio Math. 2012], a theory of new "special functions" satisfying a certain ODE of order four with G. Mano, Hilgert, and Moellers in [Ramanujan J. 2011], and a generalization of the Schroodinger/Fock model in the framework of the Jordan algebra among others.