Let $V$ be an euclidean vector space, $\mathrm{Cl}$ its Clifford algebra, $S$ the spinor module and $E$ some Clifford module. The space $\mathrm{Hom}_{\mathrm{Cl}}(S,E)$ appears both in proposition $3.27$ of [G] and remark $4.11$ in [R]. How is it defined? Do its elements super-commute or simply commute with the Clifford action? Please also justify your answer.
Edit: As pointed out in the comments, the most obvious interpretation would be that this as a special case of the notation $\mathrm{Hom}(M,N)$, where $M$ and $N$ are modules over the same ring (and hence that elements of $\mathrm{Hom}_{\mathrm{Cl}}(S,E)$ simply commute with the Clifford multiplication). My confusion arises from the following literature:
From page 104 of [G]:
From page 107 of Nicolaescu's notes:
References
- [G] Berline-Getzler-Vergne, Heat Kernels and Dirac operators
- [R] Roe, Elliptic operators, topology and asymptotic methods