Blocks of the Grothendieck ring of equivariant bundles on a finite group

If $G$ is a finite group, the Grothendieck group $\mathbf{K}\mathrm{\_}G(G)$ of the category of $G$-equivariant $\mathbb{C}$-vector bundles on $G$ (for the action of $G$ on itself by conjugation) is endowed with a structure of (commutative) ring. If $K$ is a sufficiently large extension of $\mathbb{Q}\mathrm{\_}\text{\hspace{0.17em}⁣}p$ and $\mathcal{O}$ denotes the integral closure of $\mathcal{Z}\mathrm{\_}\text{\hspace{0.17em}⁣}p$ in $K$, the $K$-algebra $K{$\mathbf{K}$}_G(G)=K \otimes_{$\mathbb{Z}$} {$\mathbf{K}$}_G(G)$ is split semisimple. The aim of this paper is to describe the $\mathcal{O}$-blocks of the $\mathcal{O}$-algebra $\mathcal{O}\mathbf{K}\mathrm{\_}G(G)$.