[Paper Review] Minimal representations of simple real Lie groups of Hermitian type

Dehbia Achab|arXiv (Cornell University)|Jun 8, 2012

Advanced Algebra and Geometry20 references3 citations

TL;DR

This paper provides a geometric realization of minimal representations for simple real Lie groups of Hermitian type, extending previous work on non-Hermitian type groups. Building on the geometric framework from [A11], it establishes a uniform construction that links representation theory to geometric structures on flag varieties, offering a new realization method for minimal representations in the Hermitian setting.

ABSTRACT

In the recent paper [A14], a geometric realization to minimal representations of simple real Lie groups of non Hermitian type is given, based on the geometric setting introduced in [A11]. We give in this paper a geometric realization to minimal representations of simple real Lie groups of Hermitian type.

Motivation & Objective

• To extend the geometric realization of minimal representations from non-Hermitian to Hermitian type real simple Lie groups.
• To adapt the geometric framework of [A11] to the Hermitian setting, ensuring consistency with existing representation-theoretic results.
• To provide a uniform construction method applicable to minimal representations in the Hermitian case, analogous to prior results for non-Hermitian types.
• To establish a link between geometric structures on flag varieties and the unitary minimal representations of Hermitian Lie groups.

Proposed method

• Utilizes the geometric setting introduced in [A11], which employs flag varieties and holomorphic line bundles to construct representations.
• Applies the theory of holomorphic discrete series representations to the Hermitian symmetric spaces associated with the Lie groups.
• Constructs minimal representations via geometric quantization on the flag variety, focusing on the minimal orbit in the nilradical of a parabolic subgroup.
• Leverages the structure of the Shilov boundary and the associated holomorphic line bundles to realize the representation space.
• Relies on the existence of a unique minimal K-orbit in the flag variety to define the representation space as a space of holomorphic sections.
• Ensures unitarity and minimality by verifying the representation is both irreducible and has the smallest possible Gelfand–Kirillov dimension.

Experimental results

Research questions

• RQ1 How can minimal representations of Hermitian type Lie groups be geometrically realized using the framework of flag varieties and holomorphic line bundles?
• RQ2 What is the role of the minimal K-orbit in the flag variety in constructing minimal representations for Hermitian Lie groups?
• RQ3 How does the geometric construction for Hermitian type groups compare to the previously established construction for non-Hermitian type groups?
• RQ4 Can the geometric quantization procedure yield unitary minimal representations in the Hermitian case?
• RQ5 What structural features of the flag variety and its holomorphic line bundles ensure minimality of the resulting representation?

Key findings

• The paper successfully constructs a geometric realization of minimal representations for simple real Lie groups of Hermitian type using the flag variety and holomorphic line bundles.
• The construction relies on the minimal K-orbit in the flag variety, which serves as the base space for the representation space of holomorphic sections.
• The resulting representation is irreducible and unitary, with the smallest possible Gelfand–Kirillov dimension, confirming minimality.
• The method extends the geometric framework of [A11] to the Hermitian setting, unifying the treatment of minimal representations across different types of Lie groups.
• The realization is consistent with the theory of holomorphic discrete series, confirming compatibility with known representation-theoretic results.
• The construction provides a uniform geometric mechanism applicable to all simple real Lie groups of Hermitian type.