I think that $\Lambda$ should be a $k$-algebra and the tensor product should be over $k$. Then $R_p\otimes_k \Lambda$ makes sense as a representation of $Q$ over $\Lambda$, or equivalently as a $\Lambda Q$-module.

Namely, the Grassmannian $\mathrm{Gr}(R_p,\alpha)$ doesn't make sense as a scheme, but only as a $k$-scheme, determined by a functor from $k$-algebras to sets. In this case the functor sends a $k$-algebra $\Lambda$ to the set of summands of $R_p \otimes_k \Lambda$ of rank $\alpha$ (i.e., localizing at any maximal ideal of $\Lambda$, the projective $\Lambda$-module corresponding to a vertex $v$ should give a free module over the localization of rank $\alpha(v)$).

I think that $\Lambda$ should be a $k$-algebra and the tensor product should be over $k$. Then $R_p\otimes_k \Lambda$ makes sense as a representation of $Q$ over $\Lambda$, or equivalently as a $\Lambda Q$-module.

Namely, the Grassmannian $\mathcal{Gr}(R_p,\alpha)$ doesn't make sense as a scheme, but only as a $k$-scheme, determined by a functor from $k$-algebras to sets. In this case the functor sends a $k$-algebra $\Lambda$ to the set of summands of $R_p \otimes_k \Lambda$ of rank $\alpha$ (i.e., localizing at any maximal ideal of $\Lambda$, the projective $\Lambda$-module corresponding to a vertex $v$ should give a free module over the localization of rank $\alpha(v)$).