Singular supports for character sheaves on a group compactification

Let $G$ be a semisimple adjoint group over $\mathbf{C}$ and $\bar{G}$ be the De Concini-Procesi completion of $G$. In this paper, we define a Lagrangian subvariety $\mathrm{\Lambda }$ of the cotangent bundle of $\bar{G}$ such that the singular support of any character sheaf on $\bar{G}$ is contained in $\mathrm{\Lambda }$.