Non-commutative Harmonic Analysis, Branching Problems, and Discontinuous Groups. 7th Tunisian-Japanese Conference: Geometric and Harmonic Analysis on Homogeneous Spaces and Applications in Honor of Professor Toshiyuki Kobayashi. Monastir, Tunisia, 31 October-4 November 2023.
In this talk, I plan to discuss two different disciplines, infinite-dimensional representation theory and geometry of discontinuous groups.

First, I discuss some fundamental questions in representation theory, giving a nice framework for induction (analysis on homogeneous spaces) and restriction (branching problems).

I then discuss some fundamental questions of discontinuous groups such as properness criterion, its quantification, and deformation theory.

These apparently very different themes are interacted not only philosophically but also the former gives a new clue to the global analysis of locally homogeneous spaces with indefinite metric.

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