Induced action by an involution on spinor bundle and Dirac operator

Question

Let $M$ be a $4n$-dimensional spin manifold with a fixed Riemannian metric $g$. Let $S$ be a spinor bundle over $M$ and fix the Riemannian connection on it. There is a decomposition $S=S^+\oplus S^-$, where $S^\pm$ is the $\pm$1-eigenbundle with respect to the Clifford multiplication by $\omega=e_1\cdot ...\cdot e_{4n}$, where $\{e_1,...,e_{4n}\}$ is a positively oriented orthonormal basis of $T_xM$ at every $x\in M$.

Now let $t:M\rightarrow M$ be an orientation and spin structure preserving involution (fixed-point free) which is an isometry w.r.t. the metric $g$. This involution lifts to an action $T:S\rightarrow S$ on the spinor bundle.

Question 1: How does one show that $T$ preserves the above $\mathbb{Z}_2$-grading and that it commutes with the Dirac operator $D:\Gamma(S)\rightarrow \Gamma(S)$ defined by $D\sigma=\sum e_j\cdot \nabla_{e_j}\sigma$, where $\nabla$ is the covariant derivative associated to the Riemannian connection?

Question 2: How does the above Dirac operator induce a Dirac operator on the quotient manifold $M/t$?

Answer 1

The action $T\colon S\to S$ is compatible with the Clifford-multiplication, hence the decomposition into $\pm1$ eigenspaces is preserved. Moreover, the spin connection is also preserved, and by the definition of the Dirac operator this should solve question 1.

Answer 2

For question 2, the original manifold M will be a double cover of the quotient space since it's an involution, and if you have an operator that commutes with the involution on the covering space, it should descend to the quotient space. If there are "nice" fixed points one can get an orbifold or a stratified space, where you can still make sense of a Dirac operator and all the structure, possibly after resolving the space. For instance Hartmann, Lesch, and Vertman - On the domain of Dirac and Laplace type operators on stratified spaces and Albin and Gell-Redman - The index formula for families of Dirac type operators on pseudomanifolds.