Minimal representations are the smallest infinite dimensional unitary representations of reductive groups. About ten years ago, I suggested a program of ''geometric analysis” with minimal representations as a motif. We have found various geometric realizations of minimal representations that interact with conformal geometry, conservative quantities of PDEs, holomorphic model (e.g. Fock-type model), L2-model (Schrödinger-type model), and Dolbeault cohomology models. I plan to discuss some of these models based on works with my collaborators, Hilgert, Mano, Möllers, and Ørsted among others. From the viewpoint of the orbit philosoply by Kirillov-Kostant, minimal representations may be thought of as a quantization of minimal nilpotent orbits. In certain setting, we give a ''geometric quantization” of minimal representations by using certain Lagrangean manifolds. Our construction includes the Schrödinger model of the Segal-Shale-Weil representation of the metaplectic group, and the commutative model of the complementary series representations of O(n,1) due to A. M. Vershik and M. I. Graev.
© Toshiyuki Kobayashi