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I have a question related to Mackey theory applied to discrete nilpotent groups which are not torsion-free and are infinite.

Let us suppose that $G$ is a type I infinite discrete nilpotent group with normal abelian subgroup $N$ which is regularly embedded. Also assume that the commutator subgroup of $G$ is finite, and that $N$ is two-step. In order to describe the unitary dual of $G$ using the Mackey machine, let us suppose that we fix a character in the unitary dual of the normal subgroup $N$. Let us also assume that the Little subgroup fixing the given character is finite.

  1. Do we know that the given character of $N$ must extend to a representation of the stabilizer subgroup of the fixed character as opposed to a multiplier representation?
  2. If we can extend the character to a representation (which is not a multiplier representation) the extended representation to the stabilizer subgroup inside $G$ is unique up to multiplication by a constant. But how do we construct the extended representation?

I cannot find any answer in the literature. Thank you in advance.

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