Spinor representation for $\operatorname{Spin}(V \oplus V^*)$

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 know that restriction of this representation to the subgroup $$\operatorname{Spin}(V \oplus V^*)$$ of $$CL(V \oplus V^*)$$ is one of these representations. but I don't 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.

• I guess it is not written explicitly in Hitchin’s paper, but this description of the spin representation is as modules over the connected component of the identity of the natural $GL(V)$ subgroup of $SO(V \oplus V^*)$. This is spelled out in Chapter 2 of Marco Gualtieri’s thesis. – José Figueroa-O'Farrill Nov 9 '18 at 11:17