Spinor representation and Clifford modules

Question

Let $V$ be an even-dimensional real inner product space. We denote the Clifford algebra of $V$ by $C(V)$ and the spinor representation by $S$.

For a finite-dimensional $\mathbb Z_2$-graded complex Clifford module $E$ the following facts are known.

Denote by $W$ the trivial Clifford module $\mathrm{Hom}_{C(V)}(S,E)$.

1. $E$ is isomorphic to $W \otimes S$ as a Clifford module.
2. $\mathrm{End}(W)$ is isomorphic to $\mathrm{End}_{C(V)}(E)$.

Question: Why do the above statements hold?

The statements can be found in [BGV,§3.2] and [R,4.12] without any details. According to [BGV], the isomorphism $W \otimes S \to E$ is given by the evaluation. This is obviously a homomorphism of Clifford modules, but it is not obvious that the evaluation map is bijective.

References

• [BGV] Berline-Getzler-Vergne, Heat Kernels and Dirac operators
• [R] Roe, Elliptic operators, topology and asymptotic methods

Answer 1

A good reference for that is the lecture notes of Nicolaescu "Notes on the Atiyah-Singer Index Theorem", proposition 2.2.6