Higher $P$-symmetric Ekeland-Hofer capacities

TL;DR

This paper develops higher P-symmetric Ekeland-Hofer capacities in symplectic geometry, explores their relationships with existing capacities, and provides computational examples, extending the theory of symmetric capacities.

Contribution

It introduces new higher P-symmetric capacities, analyzes their relationships with other capacities, and defines real symmetric capacities as a complement to recent studies.

Findings

Established relationships between P-symmetric capacities and other symplectic capacities.

Provided explicit computation examples of the new capacities.

Extended the framework of symmetric capacities to real symmetric cases.

Abstract

This paper is devoted to the construction of analogues of higher Ekeland-Hofer symplectic capacities for $P$-symmetric subsets in the standard symplectic space $(\mathbb{R}^{2n},\omega_0)$, which is motivated by Long and Dong's study $P$-symmetric closed characteristics on $P$-symmetric convex bodies. We study the relationship between these capacities and other capacities, and give some computation examples. Moreover, we also define higher real symmetric Ekeland-Hofer capacities as a complement of Jin and the second named author's recent study of the real symmetric analogue about the first Ekeland-Hofer capacity.