Minimal representations are the smallest infinite dimensional unitary representations of reductive groups, which may be thought of as a quantization of minimal nilpotent orbits. The Segal-Shale-Weil representation for the metaplectic group, which plays a prominent role in number theory, is a classic example.Geometric realizations of minimal representations offer abundant symmetries on function spaces, far more than other (usual) irreducible infinite dimensional representations by the “minimality”. Highlighting geometric analysis on minimal representations of indefinite orthogonal groups, I plan to discuss interaction with conformal geometry, conservative quantities of PDEs, an analogue of the Schrodinger model and the Fock model, and a generalized Fourier transform and its deformation theory.

© Toshiyuki Kobayashi