Varna Lecture on L^2-Analysis of Minimal Representations

TL;DR

This paper discusses the analysis of minimal representations of real reductive groups, emphasizing their role in connecting algebraic and geometric perspectives, with recent progress highlighted through the Schrödinger model.

Contribution

It proposes a program of global analysis based on minimal representations, integrating algebraic and geometric methods, and reviews recent developments with a focus on the Schrödinger model.

Findings

Progress in understanding minimal representations through geometric analysis

Development of the Schrödinger model for minimal representations

Enhanced interaction between algebraic and geometric approaches

Abstract

Minimal representations of a real reductive group G are the "smallest" 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 emphasis on the Schroedinger model.