Holomorphic bundles on diagonal Hopf manifolds

We show that every stable holomorphic bundle on the Hopf manifold $M=(\mathbb C^n\setminus0)/\langle A\rangle$ with $\dim M\geqslant 3$, where $A\in\operatorname{GL}(n,\mathbb C)$ is a diagonal linear operator with all eigenvalues satisfying $|\alpha_i|<1$, can be lifted to a $\widetilde G_F$-equivariant coherent sheaf on $\mathbb C^n$, where $\widetilde G_F\cong(\mathbb C^*)^l$ is a commutative Lie group acting on $\mathbb C^n$ and containing $A$. This is used to show that all bundles on $M$ are filtrable, that is, admit a filtration by a sequence $F_i$ of coherent sheaves with all subquotients $F_i/F_{i-1}$ of rank $1$.