Algebraic analysis of minimal representations

TL;DR

This paper explores the algebraic structure and applications of minimal representations of the indefinite orthogonal group, highlighting their role in constructing conserved quantities, branching laws, and deformations of Fourier transforms.

Contribution

It introduces new methods for analyzing minimal representations, connecting algebraic analysis with geometric models and symmetry properties.

Findings

Construction of conservative quantities for ultra-hyperbolic equations

Quantitative analysis of discrete branching laws

Deformation of Fourier transform using algebraic analysis concepts

Abstract

Small representations of a group bring us to large symmetries in a representation space. Analysis on minimal representations utilises 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. quantative discrete branching laws, 3. deformation of the Fourier transform with emphasis on the prominent roles of Sato's idea on algebraic analysis.