General form of the 4d mirror brane for T*G // H_Γ
Determine whether the 4d mirror brane to the twisted cotangent bundle T*G // H_Γ, where H_Γ is a subgroup of maximal rank in G with root set Γ, is always of the form T*{G} //_{ψ_Γ} U(Γ) after affinization when needed; explicitly identify the unipotent subgroup U(Γ) of the Langlands dual group {G} and the character ψ_Γ, and verify the consistency conditions U(Γ(∅, J, ∅)) = U_{P_J}, U(Γ(I, I, K)) = U, and U(Γ(I, ∅)) = U_{e_I}.
References
One reasonable conjecture is that the 4d mirror brane to $T*G / !!/ H_{\Gamma}$ is of the form $T*{G} / !!/{\psi}{\Gamma} {U}(\Gamma)$, (which requires to take affinization when necessary,) where $H_\Gamma$ is a subgroup of maximal rank in $G$ whose roots form the closed subset $\Gamma$, and ${U}(\Gamma)$ is a unipotent subgroup of ${G}$ that depends on $\Gamma$ with a character ${\psi}_{\Gamma}$.