Daniel Le, Purdue University, West Lafayette, Indiana.Bao V. Le Hung, Northwestern University, Evanston, Illinois.Stefano Morra, Universite Sorbonne Paris Nord, Villetaneuse, France.Chol Park, Ulsan National Institute of Science and Technology, Republic of Korea.Zicheng Qian, Morningside Center of Mathematics, Beijing, People's Republic of China.

• 1. Introduction2. Preliminaries3. The geometry of $\widetilde {\mathcal {F}\mathcal {L}}_\mathcal {J}$4. The invariant functions on $\widetilde {\mathcal {F}\mathcal {L}}_{\mathcal {J}}$5. Combinatorics of $\Lambda $-lifts6. Construction of invariant functions7. Invariant functions and constructible $\Lambda $-lifts8. $\widetilde {\mathcal {F}\mathcal {L}}_\mathcal {J}$, $\widetilde {Fl}_\mathcal {J}$ and Serre weights9. Canonical lifts of invariant functions10. Local-global compatibilityA. Figures