Towards canonical representations of finite Heisenberg groups

Abstract

We consider a finite abelian group $M$ of odd exponent $n$ with a symplectic form $\omega: M\times M\to \mu_n$ and the Heisenberg extension $1\to \mu_n\to H\to M\to 1$ with the commutator $\omega$. According to the Stone - von Neumann theorem, $H$ admits an irreducible representation with the tautological central character (defined up to a non-unique isomorphism). We construct such irreducible representation of $H$ defined up to a unique isomorphism, so canonical in this sense.