## *Analysis of Minimal Representatinons-''Geometric
Quantization” of Minimal Nilpotent Orbits*.
(2 lectures). Analytic
Representation Theory of Lie Groups. Kavli IPMU, the University of Tokyo,
Japan, 1-4 July 2015.

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), *L*^{2}-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