The authors study the real Grassmannian \(G^\mathbb{R}_{n,n}\) of \(n\)-planes in \(\mathbb{R}^{2n}\), with \(n\ge 3\), and its reduced space. The latter is the irreducible symmetric space \(\bar G^\mathbb{R}_{n,n}\), which is the quotient of the space \(G^\mathbb{R}_{n,n}\) under the action of its isometry which sends a \(n\)-plane} into its orthogonal complement. One of the main results of this monograph asserts that the irreducible symmetric space \(\bar G^\mathbb{R}_{3,3}\) possesses non-trivial infinitesimal isospectral deformations; it provides the first example of an irreducible reduced symmetric space which admits such deformations. The authors also give a criterion for the exactness of a form of degree one on \(\bar G^\mathbb{R}_{n,n}\) in terms of a Radon transform.

A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.

Readership

Graduate students and research mathematicians interested in analysis.

Table of Contents

• Introduction
• Symmetric spaces of compact type and the Guillemin condition
• Invariant symmetric forms on symmetric spaces
• The real Grassmannians
• The Stiefel manifolds and the real Grassmannians
• Functions and forms of degree one on the real Grassmannians
• Isospectral deformations of the real Grassmannian of 3-planes in \(\mathbb{R}^6\)
• Forms of degree one
• A family of polynomials
• Exactness of the forms of degree one
• Branching laws
• The special Lagrangian Grassmannian \(SU(4)/SO(4)\)
• The complex quadric of dimension three
• Bibliography