Let $G$ a simply connected group over $k$ and $car(k)=0$.
Let $T_{+}=(T\times T)/Z_{G}$ we consider the closure $\overline{T}_{+}$ of the torus $T_{+}$ in $\prod End(V_{\omega_{i}})\times\prod\limits_{\alpha\in\Delta}A^{1}_{\alpha}$
by the map $(z,t)\mapsto (\omega_{i}(z)\rho_{\omega_{i}}(t),\alpha_{i}(z))$
where $\Delta$ is the set of simple roots, $\omega_{i}$ the fundamental weights and $\rho_{\omega_{i}}, V_{\omega_{i}}$ the irreducible representation of highest weight $\omega_{i}$.
We already know by general theorems that $\overline{T}_{+}$ is normal and Cohen-Macaulay.
Do we have that $\overline{T}_{+}$ is Gorenstein, complete intersection?
