I'm studding Hitchin's Generalized Calabi-Yau Manifolds https://arxiv.org/abs/math/0209099 and I've stuck here:
Suppose that $V$ is a vector space and denote its dual by $V^*$. Now we know that the $\bigwedge^\bullet(V^*)$ that is the exterior algebra over the dual space is a representation for Clifford algebra $CL(V \oplus V^*)$ by the action $$(v,\xi).\varphi=i_v\varphi+\xi\wedge\varphi , (v,\xi)\in CL(V \oplus V^*)$$
we are mainly interested in those representations of Spin group $\operatorname{Spin}(V \oplus V^*)$ that is not a representation of $SO(V \oplus V^*)$ and we call them Spinor representation. I need to find out if the restriction of this representation to the subgroup $\operatorname{Spin}(V \oplus V^*)$ of $CL(V \oplus V^*)$ is one of these representations.
I see that if we take
$$\rho:CL(V \oplus V^*)\rightarrow \operatorname{End}(\bigwedge^\bullet(V^*))$$ by restricting the representation we have
$$\rho:\operatorname{Spin}(V \oplus V^*)\rightarrow GL(\bigwedge^\bullet(V^*))$$
and we have $$\rho(-1)=-\operatorname{id}$$ so why it cant be a representation of $SO(V \oplus V^*)$? I know that $\operatorname{Spin}(V \oplus V^*)$ is a double cover of $SO(V \oplus V^*)$ but can't see how its relevant.
That would be perfect if after figuring this out I get to understand how tensoring $\bigwedge^\bullet(V^*)$ in the space of top forms of $V$ $$\bigwedge^\bullet(V^*)\otimes (\bigwedge^n V)^\frac{1}{2}$$ will contruct another Spinor representation and in what aspects this will arise more useful constructions than $\bigwedge^\bullet(V^*)$ so Hitchin prefered this one.
Any help would be a lot appreciated.
To be able to get a well defined representation of the quotient you would want $\rho(-1)=id$, since if the representation passed to the quotient elements of the same equivalence class would have to give the same transformation.