# Compact vs. ordinary induction for the Heisenberg group

Let $G$ be a Heisenberg group over a local field. To construct an infinite-dimensional irreducible representation of $G$, we fix a central character $\psi$ and extend $\psi$ to a character $\widetilde{\psi}$ of a maximal abelian subgroup $U$, then form the induced representation $\text{Ind}_U^G \ \widetilde{\psi}$. It is well-known that, in this case, induction with compact supports coincides with ordinary induction, but the proofs that I have seen (involving some calculation with double cosets) are unsatisfying in that they make this situation seem like a coincidence.

This is especially interesting to me when compared to parabolic induction for reductive groups, where we construct representations by extending a representation of a Levi subgroup to a parabolic subgroup and then take the the induced representation. In that case, the compactly supported induction coincides with the ordinary one because parabolic subgroups are cocompact, but maximal abelian subgroups of the Heisenberg group are not cocompact and yet the analogous fact holds true. Perhaps this analogy is too tenuous, but I would hope that there is still a conceptual explanation.

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Note that some maximal abelian subgroups of the Heisenberg group (with compact center) are cocompact and used to give an alternative realization of the Schrödinger representation: ams.org/mathscinet-getitem?mr=0216825 –  Francois Ziegler Oct 29 '12 at 5:48