The Restricted Schatten-class Grassmannian $\mathrm{Gr}_{\mathrm{res}, p}(\mathcal{H})$ as affine coadjoint orbit

TL;DR

This paper demonstrates that the restricted p-Schatten class Grassmannian for 1≤p≤2 can be viewed as an affine coadjoint orbit of an infinite-dimensional unitary group and admits natural symplectic structures.

Contribution

It establishes the geometric structure of the restricted p-Schatten class Grassmannian as an affine coadjoint orbit with symplectic forms for 1≤p≤2.

Findings

Identifies the Grassmannian as an affine coadjoint orbit of an infinite-dimensional group.

Shows the existence of natural weak symplectic structures on these Grassmannians.

Proves the Lie algebra admits a non-trivial 2-cocycle.

Abstract

In this paper, we consider the restricted $p$-Schatten class Grassmannian $\mathrm {Gr}_{{\rm res}, p}(\mathcal{H})$ consisting of infinite-dimensional and infinite codimensional subspaces $W$ of a polarized complex separable Hilbert space $\mathcal{H} = \mathcal{H}_+\oplus \mathcal{H}_-$ such that the orthogonal projection from $W$ onto $\mathcal{H}_+$ is Fredholm and the orthogonal projection from $W$ onto $\mathcal{H}_-$ is in the Schatten ideal $L_p$, $p\geq 1$. The aim of this paper is to show that, for $1\leq p\leq 2$, the restricted $p$-Schatten class Grassmannian $\mathrm {Gr}_{{\rm res}, p}(\mathcal{H})$ is an affine (co-)adjoint orbit of an infinite-dimensional restricted unitary group $\operatorname{U}_{{\rm res}, p}(\mathcal{H})$, and that it admits natural weak symplectic structures. These results follow from the fact that the Lie algebra of the restricted $p$-Schatten class unitary group $\operatorname{U}_{{\rm res}, p}(\mathcal{H})$ admits a non-trivial $2$-cocycle.