Minimal representations are building blocks of unitary representations, which are the smallest infinite dimensional unitary representations of reductive groups. The Weil representation for the metaplectic group, which plays a prominent role in number theory, is a classic example.
Minimal representations (viewed from groups) have ''maximal symmetries (viewed from representation spaces)''. The second viewpoint brings us to a rich study of geometric analysis on minimal representations. Highlighting minimal representations of the indefinite orthogonal group O(p,q), I plan to discuss conservative quantities of ultra-hyperbolic equations, a generalized Fourier transform for the isotropic cone, and its deformation theory.
© Toshiyuki Kobayashi