Minimal representations, geometric quantization, and unitarity

Area of Science:

• Mathematics
• Representation Theory
• Geometric Quantization

Background:

• Geometric quantization is a method for constructing representations of Lie groups.
• Minimal representations are the smallest non-trivial representations of a group.
• Understanding minimal representations is crucial for classifying group representations.

Purpose of the Study:

• To explicitly construct unitary minimal representations for a class of Lie groups.
• To establish algebraic and analytic properties of these minimal representations.
• To connect the geometry of minimal nilpotent orbits to representation theory.

Main Methods:

• Utilizing the framework of geometric quantization.
• Analyzing algebraic and symplectic geometry of minimal nilpotent orbits.
• Quantizing geometric results to obtain representations.

Main Results:

• A uniform construction of unitary minimal representations (pio) for simply-connected real Lie groups with specific properties.
• Derivation of algebraic and analytic results concerning these representations.
• Demonstration of how geometric properties of nilpotent orbits lead to representation construction.

Conclusions:

• The study provides a unified approach to constructing minimal representations for a significant class of Lie groups.
• The interplay between geometry and representation theory is highlighted.
• The results offer a foundation for further investigation into minimal representations and their applications.