In physics, the Dirac equation has the invariance under the representation of $SL(2,\mathbb{C})$ (see, wiki). So I expect the following to happen, in general mathematics:
Let $M$ be an oriented compact Riemannian manifold together with the Riemannian metric $g$, let $E$ be a Dirac bundle equipped with the metric $h$ and let $D:\Gamma(E)\rightarrow \Gamma(E)$ be its Dirac operator on smooth sections $\Gamma(E)$. Then there is some (infinite-dimensional, compact?) Lie group $G$ and exists its representation on $L^2(E)$ such that the eigenspace $\text{ker}(D-m)$ for the eigenvalue $m$ is invariant under the representation.
Here the Hilbert space $L^2(E)$ is the completion of the pre-Hilbert space $\Gamma(E)$ with respect to the inner product defined by $$<\phi, \psi>_{L^2}=\int_M h(\phi, \psi)v_g,$$ where $v_g$ is the canonical volume form of $M$.
It seems difficult in general Dirac bundle $E$, but some special case (for example, the spinor bundle for a spin manifold) looks promising, I think. In a view point of physics, I guess one of candidates is the universal covering group of the connected component of the subgroup $\lbrace f\in \text{Diff}(M):\text{$f$ preserves the metric and orientation.} \rbrace$ of the diffeomorphism group, but there is no proof. Please any information.