Equivariant gerbes over compact simple Lie groups

Using groupoid $S^1$-central extensions, we present, for a compact simple Lie group $G$, an infinite dimensional model of $S^1$-gerbe over the differential stack $G\mathrm{/}G$ whose Dixmier-Douady class corresponds to the canonical generator of the equivariant cohomology $H_G^3(G,Z)$.