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Maximal Weinstein neighborhoods of symmetric R-spaces and their symplectic capacities

Johanna Bimmermann

TL;DR

The paper addresses the problem of constructing maximal Weinstein neighborhoods for symmetric $R$-spaces and computing their symplectic capacities. It develops a systematic, equivariant embedding of fiberwise convex neighborhoods $U_rN$ into the Hermitian complexification $N_{\mathbb C}$ by analyzing orbits at infinity, building an invariant Liouville field, and completing the symplectic structure to identify with $TN$. Leveraging a Hamiltonian circle action and known results for coadjoint orbits, it derives explicit formulas for the Gromov width $c_G$ and Hofer–Zehnder capacity $c_{HZ}$ of $U_rN$, depending on the rank data $\rk(N)$ and $\rk(N_{\mathbb C})$, and relates disc bundles $D_rN$ to the geodesic systole ${\rm sys}$ in simply connected cases. The results extend prior work on spheres and Hermitian symmetric spaces, providing a broad framework for symplectic capacities of tangent/cotangent disc bundles of symmetric $R$-spaces and linking geometric representation theory to symplectic invariants.

Abstract

Symmetric R-spaces can be characterized as real forms of Hermitian symmetric spaces, and as such, they are all embedded as Lagrangian submanifolds. We show that their maximal Weinstein tubular neighborhoods are dense and use this property to compute both the Gromov width and the Hofer--Zehnder capacity of the corresponding disc (co)tangent bundles of the symmetric R-spaces.

Maximal Weinstein neighborhoods of symmetric R-spaces and their symplectic capacities

TL;DR

The paper addresses the problem of constructing maximal Weinstein neighborhoods for symmetric -spaces and computing their symplectic capacities. It develops a systematic, equivariant embedding of fiberwise convex neighborhoods into the Hermitian complexification by analyzing orbits at infinity, building an invariant Liouville field, and completing the symplectic structure to identify with . Leveraging a Hamiltonian circle action and known results for coadjoint orbits, it derives explicit formulas for the Gromov width and Hofer–Zehnder capacity of , depending on the rank data and , and relates disc bundles to the geodesic systole in simply connected cases. The results extend prior work on spheres and Hermitian symmetric spaces, providing a broad framework for symplectic capacities of tangent/cotangent disc bundles of symmetric -spaces and linking geometric representation theory to symplectic invariants.

Abstract

Symmetric R-spaces can be characterized as real forms of Hermitian symmetric spaces, and as such, they are all embedded as Lagrangian submanifolds. We show that their maximal Weinstein tubular neighborhoods are dense and use this property to compute both the Gromov width and the Hofer--Zehnder capacity of the corresponding disc (co)tangent bundles of the symmetric R-spaces.
Paper Structure (22 sections, 22 theorems, 94 equations)

This paper contains 22 sections, 22 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Symmetric $R$-spaces
  3. Symmetric spaces
  4. Hermitian symmetric spaces
  5. Symmetric $R$-spaces
  6. Table with groups
  7. Geometric interpretation
  8. Symplectic embedding
  9. The Orbits at Infinity
  10. An Invariant Liouville Vector Field
  11. The symplectic completion of $N_{\mathbb C}\setminus \Delta$
  12. Comparing momentum map images
  13. Computing capacities
  14. Embedding balls via Hamiltonian circle yctions
  15. Capacities of $U_rN$
  16. ...and 7 more sections

Key Result

Theorem 1

For $r \leq \frac{\mathrm{rk}(N_{\mathbb{C}})}{\mathrm{rk}(N)}$, there exists a $K$-equivariant symplectic embedding which is open-dense for $r = \frac{\mathrm{rk}(N_{\mathbb{C}})}{\mathrm{rk}(N)}$.

Theorems & Definitions (35)

  • Theorem 1: Corollary \ref{['symplectomorphism']}
  • Theorem 2
  • Corollary 3
  • Theorem 4
  • Theorem 5: Tak84
  • Lemma 6
  • proof
  • Example 7
  • Example 8
  • Proposition 9
  • ...and 25 more