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Given a symplectic space $W$ over a local field $F$ and a additive character $\psi$ of $F$, we can construct the Weil representation $\omega_\psi$, which can be viewed as a representation of the semi-direct product $Mp(W)\ltimes H(W)$, where $Mp(W)$ is the metaplectic group and $H(W)$ is the Heisenberg group.

My question is: are these all of the irreducible representations of the semi-direct product $Mp(W)\ltimes H(W)$? If it is true, can anybody point out a reference? If it is false, why? Is there any easy counter-example?

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  • $\begingroup$ The semidirect product representation are classified by the Mackey machine. $\endgroup$
    – Marc Palm
    Feb 25, 2014 at 7:57
  • $\begingroup$ You can maybe find the answer in "Elements of the Representation Theory of the Jacobi Group" by Rolf Berndt,Ralf Schmidt. But I'am not sure $\endgroup$
    – m07kl
    Dec 29, 2015 at 23:13

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