Minimal representations are the smallest infinite dimensional unitary representations. The Weil representation for the metaplectic group, which plays a prominent role in number theory, is a classic example. Our viewpoint of minimal representations is that they shoud have ''maximal symmetries'' on representation spaces. We then initiate a new line of investigations to use minimal representations as a guiding principle to find interactions with other fields of mathematics.
Highlighting geometric analysis on minimal representations of generalized Lorentz group O(p,q), I plan to discuss conservative quantities of ultrahyperbolic equations, the generalization of the Fourier-Hankel transform on the L2-model, and its deformation.
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© Toshiyuki Kobayashi