Infinitesimally tight Lagrangian submanifolds in adjoint orbits: A classification of real forms

Jhoan Báez; Luiz A. B. San Martin

doi:10.56754/0719-0646.2703.523

DOI:

https://doi.org/10.56754/0719-0646.2703.523

Abstract

In this paper, we study the geometry of real flag manifolds within complex flag manifolds, focusing on their Lagrangian properties. We prove that the natural immersion of real flag manifolds into their corresponding complex flag manifolds can be characterized as infinitesimally tight Lagrangian submanifolds with respect to the Kirillov-Kostant-Souriau (KKS) symplectic form. This property of tightness provides a significant geometric constraint, indicating that the submanifolds are locally minimal and cannot be deformed infinitesimally to reduce their volume further in the ambient space. We further provide a comprehensive classification of these immersions, detailing the conditions under which such submanifolds exist across various symmetric pairs. This classification elucidates the relationship between the structure of the real flags and the associated complex flags, contributing to a deeper understanding of the interplay between symplectic geometry and representation theory.

Keywords

Flag manifolds , homogeneous space , Lagrangian submanifolds , infinitesimally tight

Mathematics Subject Classification:

14M15 , 22F30 , 53D12

Jhoan Báez Beauchef 851, Edificio Norte – Office 610, Santiago, Chile. https://orcid.org/0000-0002-2227-7532
Luiz A. B. San Martin Imecc - Unicamp, Departamento de Matemática. Rua Sérgio Buarque de Holanda, 651, Cidade Universitária Zeferino Vaz. 13083-859 Campinas - SP, Brasil. https://orcid.org/0000-0003-4340-4809

Pages: 523–551
Date Published: 2025-11-21
In Press

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