The Restricted Schatten-class Grassmannian $\mathrm{Gr}_{\mathrm{res}, p}(\mathcal{H})$ as affine coadjoint orbit
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arXiv
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| Natura: | Preprint |
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2026
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| _version_ | 1866917520097673216 |
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| author | Tahiri, Amin Tumpach, Alice Barbora |
| author_facet | Tahiri, Amin Tumpach, Alice Barbora |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_22512 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Restricted Schatten-class Grassmannian $\mathrm{Gr}_{\mathrm{res}, p}(\mathcal{H})$ as affine coadjoint orbit Tahiri, Amin Tumpach, Alice Barbora Functional Analysis Mathematical Physics Differential Geometry Operator Algebras 58B25, 22E65, 46T05, 53D17 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. |
| title | The Restricted Schatten-class Grassmannian $\mathrm{Gr}_{\mathrm{res}, p}(\mathcal{H})$ as affine coadjoint orbit |
| topic | Functional Analysis Mathematical Physics Differential Geometry Operator Algebras 58B25, 22E65, 46T05, 53D17 |
| url | https://arxiv.org/abs/2605.22512 |