The Restricted Schatten-class Grassmannian ${\mathrm{G}\mathrm{r}}_{\mathrm{r}\mathrm{e}\mathrm{s},p}(\mathcal{H})$ as affine coadjoint orbit

In this paper, we consider the restricted $p$-Schatten class Grassmannian ${\mathrm{G}\mathrm{r}}_{\mathrm{r}\mathrm{e}\mathrm{s},p}(\mathcal{H})$ 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 $L_p$, $p\ge 1$. The aim of this paper is to show that, for $1\le p\le 2$, the restricted $p$-Schatten class Grassmannian ${\mathrm{G}\mathrm{r}}_{\mathrm{r}\mathrm{e}\mathrm{s},p}(\mathcal{H})$ is an affine (co-)adjoint orbit of an infinite-dimensional restricted unitary group ${\mathrm{U}}_{\mathrm{r}\mathrm{e}\mathrm{s},p}(\mathcal{H})$, 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 ${\mathrm{U}}_{\mathrm{r}\mathrm{e}\mathrm{s},p}(\mathcal{H})$ admits a non-trivial $2$-cocycle.