The Zak transform and representations induced from characters of an abelian subgroup

We consider a variant of the Zak transform for a finite group G with respect to a fixed abelian subgroup H, and demonstrate a relationship with representations of G induced from characters of H. We also show how the Zak transform can be used to study right translations by H in L²(G), and give some examples of applications for equiangular tight frames.The purpose of this note is to demonstrate some connections between the Zak transform and the theory of induced representations. We also show some applications for right shift-invariant spaces, and for equiangular tight frames that occur as orbits of induced representations. For the sake of clarity, we restrict our attention to the setting of finite groups, where we can safely ignore convergence issues. However, the results in Sections I and II should also hold (with suitable modification) on locally compact groups.