# Joint representation of the semi-direct product of the metaplectic group and Heisenberg group

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?

• The semidirect product representation are classified by the Mackey machine. – Marc Palm Feb 25 '14 at 7:57
• 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 – m07kl Dec 29 '15 at 23:13