Orbifold Schur index and IR formula

We discuss an orbifold version of the Schur index defined as the supersymmetric partition function in $S^3/Z_n\times S^1$. We first give a general formula for Lagrangian theories obtained by the localization technique, and then suggest a generalization of the Cordova and Shao IR formula. We confirm that the generalized IR formula gives the correct answer for systems with free hypermultiplets if we tune the background fields so that they are invariant under the orbifold action. Unfortunately, we find disagreement for theories with dynamical vector multiplets.