If $G$ is the structure group of a manifold $M$ it is shown how a certain ideal in the character ring of $G$ corresponds to the set of geometric elliptic operators on $M$. This provides a simple method to construct these operators. For classical structure groups like $G = O(n)$ (Riemannian manifolds), $G = SO(n)$ (oriented Riemannian manifolds), $G = U(m)$ (almost complex manifolds), $G = Spin(n)$ (spin manifolds), or $G = Spin^c(n)$ (spin$^c$ manifolds) this yields well known classical operators like the Euler-deRham operator, signature operator, Cauchy-Riemann operator, or the Dirac operator. For some less well studied structure groups like $Spin^h(n)$ or $Sp(q)Sp(1)$ we can determine the corresponding operators. As applications, we obtain integrality results for such manifolds by applying the Atiyah-Singer Index Theorem to these operators. Finally, we explain how immersions yield interesting structure groups to which one can apply this method. This yields lower bounds on the codimension of immersions in terms of topological data of the manifolds involved.