Study on decomposition of unitary representations of groups and harmonic functions on branching graphs

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We intend to develop harmonic analysis on a large group, which plays a role in clarifying complex phenomena involving a huge number of factors. Our project aims at establishing a concrete framework by which one clearly understands decomposition of representations of a large group. For wreath products of a compact group, including the infinite symmetric group and infinite complex reflection groups, we took interactive approach from probability theory, group representations and potential theory, and obtained concrete and detailed results on such objects as group character formulas and their characterization, boundaries of the branching graph and minimal harmonic functions, and description of ergodic probabilities on the path space of the graph.