We prove several results on weak symplectic fillings of contact $3$-manifolds, including: (1) Every weak filling of any planar contact manifold can be deformed to a blow up of a Stein filling. (2) Contact manifolds that have fully separating planar torsion are not weakly fillable—this gives many new examples of contact manifolds without Giroux torsion that have no weak fillings. (3) Weak fillability is preserved under splicing of contact manifolds along symplectic pre-Lagrangian tori—this gives many...

Let $G$ be the semidirect product $V⋊K$ where $K$ is a semisimple compact connected Lie group acting linearly on a finite-dimensional real vector space $V$. Let $𝒪$ be a coadjoint orbit of $G$ associated by the Kirillov-Kostant method of orbits with a unitary irreducible representation $\pi$ of $G$. We consider the case when the corresponding little group $H$ is the centralizer of a torus of $K$. By dequantizing a suitable realization of $\pi$ on a Hilbert space of functions on ${ℂ}^n$ where $n=dim(K/H)$, we construct a symplectomorphism between...