Group actions on $p$-adic symplectic manifolds

TL;DR

This paper develops the theory of symplectic actions of p-adic Lie groups on p-adic symplectic manifolds, establishing momentum maps and conditions for Hamiltonian actions, and introduces p-adic symplectic toric manifolds.

Contribution

It introduces the concept of p-adic symplectic actions, proves the existence of momentum maps, and characterizes Hamiltonian actions in the p-adic setting, extending symplectic geometry.

Findings

Existence of momentum maps for p-adic symplectic actions

Proper p-adic symplectic actions are Hamiltonian iff orbits are isotropic

Definition of p-adic symplectic toric manifolds

Abstract

Let $p$ be a prime number. We introduce symplectic actions of $p$-adic analytic Lie groups on $p$-adic symplectic manifolds. Then we show that any $p$-adic symplectic action $G\times(M,\omega)\to(M,\omega)$ has a momentum map $\mu:M\to\mathfrak{g}^*$, and that a proper $p$-adic symplectic action is Hamiltonian if and only if every orbit is isotropic. We conclude by defining $p$-adic symplectic toric manifolds, by analogy with the real case.