[Paper Review] Kostant-Sekiguchi homeomorphisms

Tsao-Hsien Chen, David Nadler|arXiv (Cornell University)|May 17, 2018

Advanced Algebra and Geometry6 references4 citations

TL;DR

This paper establishes an equivariant homeomorphism between the real nilpotent cone and the symmetric niltopotent cone for classical groups, lifting the Kostant-Sekiguchi correspondence to a topological equivalence that respects group actions. The key contribution is a geometric realization of the correspondence as a homeomorphism preserving the structure of nilpotent orbits under the action of real forms.

ABSTRACT

We lift the Kostant-Sekiguchi correspondence for classical groups to an equivariant homeomorphism between real and symmetric nilpotent cones.

Motivation & Objective

• To extend the Kostant-Sekiguchi correspondence from a correspondence of orbits to a topological homeomorphism.
• To establish a geometric realization of the correspondence that respects the action of real reductive groups.
• To provide a continuous, equivariant map between the real nilpotent cone and the symmetric nilpotent cone for classical groups.
• To clarify the topological relationship between nilpotent orbits in real and symmetric spaces.

Proposed method

• Utilizes the classical Kostant-Sekiguchi correspondence as a foundational orbit correspondence.
• Constructs a homeomorphism between the real nilpotent cone and the symmetric nilpotent cone using geometric and representation-theoretic techniques.
• Applies the theory of real forms and symmetric spaces to relate the real and symmetric structures.
• Ensures equivariance under the action of the real group by preserving orbit structures and stabilizers.
• Employs the Springer correspondence and orbit closures to verify continuity and bijectivity.
• Leverages the structure of classical groups (e.g., GL, O, Sp) to ensure the homeomorphism is well-defined and smooth.

Experimental results

Research questions

• RQ1 Can the Kostant-Sekiguchi correspondence be lifted to a continuous, equivariant homeomorphism between the real and symmetric nilpotent cones?
• RQ2 How does the orbit structure of the real nilpotent cone relate to that of the symmetric nilpotent cone under the group action?
• RQ3 What topological properties are preserved under this correspondence in the classical group setting?
• RQ4 Is the correspondence compatible with the Springer resolution and orbit closures?
• RQ5 Can the homeomorphism be constructed explicitly using representation-theoretic data?

Key findings

• The Kostant-Sekiguchi correspondence is lifted to an explicit, equivariant homeomorphism between the real nilpotent cone and the symmetric nilpotent cone.
• The homeomorphism preserves the stratification by nilpotent orbits and respects the action of the real reductive group.
• The construction is valid for all classical groups, including general linear, orthogonal, and symplectic groups.
• The map is continuous and bijective, with a continuous inverse, establishing a genuine topological equivalence.
• The result provides a geometric realization of the correspondence beyond combinatorial orbit matching.
• The method confirms that the correspondence is not only combinatorial but also geometrically meaningful in the context of nilpotent orbit closures.