$G$-equivariant embedding theorems for CR manifolds of high codimension

Let $(X,T^{1,0}X)$ be a $(2n+1+d)$-dimensional compact CR manifold with codimension $d+1$, $d\geq1$, and let $G$ be a $d$-dimensional compact Lie group with CR action on $X$ and $T$ be a globally defined vector field on $X$ such that $\mathbb C TX=T^{1,0}X\oplus T^{0,1}X\oplus\mathbb C T\oplus\mathbb C\underline{\mathfrak{g}}$, where $\underline{\mathfrak{g}}$ is the space of vector fields on $X$ induced by the Lie algebra of $G$. In this work, we show that if $X$ is strongly pseudoconvex in the direction of $T$ and $n\geq 2$, then there exists a $G$-equivariant CR embedding of $X$ into $\mathbb C^N$, for some $N\in\mathbb N$. We also establish a CR orbifold version of Boutet de Monvel's embedding theorem.