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Infinite-dimensional Siegel disc as symplectic and Kaehler quotient

Alice Barbora Tumpach

TL;DR

This work constructs the restricted infinite-dimensional Siegel disc $\mathfrak{D}_{\rm res}(\mathcal{H})$ as both a Marsden–Weinstein symplectic reduction and a Kaehler quotient of a weak Kaehler space. It introduces the Banach-Lie group $\operatorname{Sp}_{1,2}(\mathcal{V},\Omega)$ acting on the disc and analyzes a right action of $\operatorname{O}_{1,2}(\mathcal{V})$ with a momentum map $\mu_{\operatorname{O}}$, showing that the zero level set modulo $\operatorname{U}_{1}(\mathcal{H}_+)$ yields the disc and endowing it with an explicit reduced symplectic form. The paper proves transitivity, computes the reduced symplectic structure, and connects the resulting geometry to the Kirillov–Kostant–Souriau form on an affine coadjoint orbit of the restricted symplectic group (and its universal central extension), thereby linking infinite-dimensional dual-pair structures to coadjoint-orbit theory. This provides a rigorous Banach-space realization of the restricted Siegel disc within a dual-p pair framework and has potential implications for Teichmüller theory and Gaussian process geometry in infinite dimensions.

Abstract

In this paper, we construct the restricted infinite-dimensional Siegel disc as a Marsden-Weinstein symplectic reduced space and as Kaehler quotient of a weak Kaehler manifold. The obtained symplectic form is invariant with respect to the left action of the infinite-dimensional restricted symplectic group and coincides with the Kirillov-Kostant-Souriau symplectic form of the restricted Siegel disc obtained via the identification with an affine coadjoint orbit of the restricted symplectic group, or equivalently with a coadjoint orbit of the universal central extension of the restricted symplectic group.

Infinite-dimensional Siegel disc as symplectic and Kaehler quotient

TL;DR

This work constructs the restricted infinite-dimensional Siegel disc as both a Marsden–Weinstein symplectic reduction and a Kaehler quotient of a weak Kaehler space. It introduces the Banach-Lie group acting on the disc and analyzes a right action of with a momentum map , showing that the zero level set modulo yields the disc and endowing it with an explicit reduced symplectic form. The paper proves transitivity, computes the reduced symplectic structure, and connects the resulting geometry to the Kirillov–Kostant–Souriau form on an affine coadjoint orbit of the restricted symplectic group (and its universal central extension), thereby linking infinite-dimensional dual-pair structures to coadjoint-orbit theory. This provides a rigorous Banach-space realization of the restricted Siegel disc within a dual-p pair framework and has potential implications for Teichmüller theory and Gaussian process geometry in infinite dimensions.

Abstract

In this paper, we construct the restricted infinite-dimensional Siegel disc as a Marsden-Weinstein symplectic reduced space and as Kaehler quotient of a weak Kaehler manifold. The obtained symplectic form is invariant with respect to the left action of the infinite-dimensional restricted symplectic group and coincides with the Kirillov-Kostant-Souriau symplectic form of the restricted Siegel disc obtained via the identification with an affine coadjoint orbit of the restricted symplectic group, or equivalently with a coadjoint orbit of the universal central extension of the restricted symplectic group.
Paper Structure (19 sections, 9 theorems, 38 equations)

This paper contains 19 sections, 9 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Finite-dimensional picture.
  3. Infinite-dimensional picture
  4. Contributions
  5. Mathematical background
  6. The general linear group $\operatorname{GL}(\mathcal{V})\subset \operatorname{GL}(\mathcal{H})$.
  7. The general linear group $\operatorname{GL}_{1,2}(\mathcal{V})\subset \operatorname{GL}(\mathcal{V})$.
  8. The symplectic linear group $\operatorname{Sp}(\mathcal{V},\Omega)\subset \operatorname{GL}(\mathcal{V})$.
  9. The restricted symplectic linear group ${\rm Sp}_{\rm res}(\mathcal{V},\Omega)$.
  10. The symplectic linear group $\operatorname{Sp}_{1,2}(\mathcal{V},\Omega)$.
  11. The orthogonal group $\operatorname{O}(\mathcal{V})$.
  12. The orthogonal group $\operatorname{O}_{1,2}(\mathcal{V})$.
  13. The infinite-dimensional restricted Siegel disc $\mathfrak{D}_{\rm{res}}(\mathcal{H})$.
  14. $\mathfrak{D}_{\rm res}(\mathcal{H})$ as Homogeneous space under $\operatorname{Sp}_{1,2}(\mathcal{V},\Omega)$.
  15. The restricted Siegel Disc as Kähler quotient
  16. ...and 4 more sections

Key Result

theorem thmcountertheorem

The map $\rho: \operatorname{Sp}_{1,2}(\mathcal{V},\Omega) \times\mathfrak{D}_{\rm res}(\mathcal{H})\longrightarrow \mathfrak{D}_{\rm res}(\mathcal{H}),$ is a well-defined transitive action of the symplectic group $\operatorname{Sp}_{1,2}(\mathcal{V},\Omega)$ onto the restricted Siegel disc. The isotropy group of $0$ is isomorphic to the unitary group Therefore, the orbit map induces a diffeomorp

Theorems & Definitions (21)

  • remark thmcounterremark
  • theorem thmcountertheorem
  • proof
  • proposition thmcounterproposition
  • proof
  • corollary thmcountercorollary
  • remark thmcounterremark
  • theorem thmcountertheorem
  • proof
  • theorem thmcountertheorem
  • ...and 11 more