A freely generated ring for N=1 models in class S_k

We study 4d N=1 supersymmetric theories of class S_k, obtained from flux compactifications on a Riemann surface of 6d (1,0) conformal theories describing the low energy physics on a stack of M5 branes probing a Z_k singularity. We conjecture that the protected spectrum of class S_k theories contains a freely generated ring, generalizing the Coulomb branch of the N=2 theories. We derive this by examining a limit of the supersymmetric index of 4d N=1 class S_k theories. The limit generalizes the Coulomb limit of N=2 theories, which coincides with the case of k=1 for a particular choice of flux. We conjecture a general simple formula for the index in the aforementioned limit.