Algebraic Analysis of Minimal Representations

Abstract

Small representations of a group yield large symmetries in the representation space. Analysis of minimal representations utilizes large symmetries in their geometric models, and serves as a driving force in creating new interesting problems that interact with other branches of mathematics. This article discusses the following three topics that arise from minimal representations of the indefinite orthogonal group:

1. construction of conservative quantities for ultra-hyperbolic equations,
2. quantitative discrete branching laws,
3. deformation of the Fourier transform,

with emphasis on the prominent role of Sato's ideas in algebraic analysis.