an ind-model counterpart in [@BF14c], but there are no implications between these and the two proofs are totally different.

\[fLB\] For each $x \in W_{\mathrm{af}}$, every ${\widetilde{{\mathbf{I}}}}$-equivariant line bundle over the scheme ${\mathbf{Q}_{G}}( x )$ is isomorphic to some ${\mathcal{O}}_{{\mathbf{Q}_{G}}(x)}( \lambda )$ up to character twist.

Although $R_G$ itself is highly infinite-dimensional (it is [*not*]{} even finitely generated), it admits a grading such that it is almost like an Artin algebra in a graded sense. Moreover, Proposition \[fLB\] supplies “graded indecomposable projectives” of $R_G$. These two facts, combined with Theorem \[fBWB\], assert that the category of ${\widetilde{{\mathbf{I}}}}$-equivariant sheaves on ${\mathbf{Q}_{G}}$ (and on ${\mathbf{Q}_{G}^{\mathrm{rat}}}$) behaves almost like the category of coherent sheaves on an affine scheme.

This series of observations enables us to define a reasonable variant of an equivariant $K$-group $K^{\prime}_{{\widetilde{{\mathbf{I}}}}}({\mathbf{Q}_{G}})$ of ${\mathbf{Q}_{G}}$ (and $K_{{\widetilde{{\mathbf{I}}}}}({\mathbf{Q}_{G}^{\mathrm{rat}}})$ of ${\mathbf{Q}_{G}^{\mathrm{rat}}}$) with respect to $\tilde{{\mathbf{I}}}$; see Section \[sec:K-SiFl\] for details. They are rather involved, partly because we need to specify a class of formal power series that is large enough to afford the Pieri-Chevalley rule, and at the same time is small enough so that the Euler character map is injective. Nevertheless, we can prove that $K^{\prime}_{{\widetilde{{\mathbf{I}}}}}({\mathbf{Q}_{G}})$ contains (the classes of) the sheaves $[{\mathcal{O}}_{{\mathbf{Q}_{G}}(y)} 