Varna Lecture on L2-Analysis of Minimal Representations Toshiyuki Kobayashi

Abstract Minimal representations of a real reductive group G are the ‘small-est ’ irreducible unitary representations of G. The author suggests a program of global analysis built on minimal representations from the philosophy: small representation of a group = large symmetries in a representation space. This viewpoint serves as a driving force to interact algebraic representation theory with geometric analysis of minimal representations, yielding a rapid progress on the program. We give a brief guidance to recent works with em-phasis on the Schrödinger model. 1 What are Minimal Representations? Minimal representations of reductive groups G are the ‘smallest ’ infinite di-mensional irreducible unitary representations. The Weil (metaplectic, oscillator, the Segal–Shale–Weil, harmonic) rep-resentation, known by a prominent role in number theory, consists of two minimal representations of the metaplectic group Mp(n,R). The minimal representation of a conformal group SO(4, 2) arises on the Hilbert space of bound states of the Hydrogen atom. Minimal representations are distinguished among other (continuously many) irreducible unitary representations of G by the following properties that I state loosely. • ‘Smallest ’ infinite dimensional representations of G. • One of the ‘building blocks ’ of unitary representations of Lie groups