  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.
