## A generalized Stone-von Neumann theorem

Continuing our terminology from the last post, let be an -group, a closed normal abelian -subgroup, and a smooth character. Now acts on by conjugation, which induces an action on the Pontryagin dual .

If is a smooth representation of , we can restrict to and consider the corresponding sheaf on , which comes from the equivalence proved last time. The stalk is identified with the space of -coinvariants of , i.e. the quotient of by the subspace spanned by vectors of the form for and . Notice that for any the action of on induces an isomorphism of stalks , and in particular has a natural action by the stabilizer .

It is easy to see that the support is a -invariant subspace: in particular, if is contained in the -orbit of , then either or .

**Theorem (Rodier)** Suppose the orbit is locally closed in and that the natural map is a homeomorphism. Then the functor is an equivalence from the category of smooth representations of whose restriction to is supported on and the category of smooth representations of whose restriction to is supported at .

Let us motivate this rather general theorem by deducing the Stone-von Neumann theorem.

**Corollary (Stone-von Neumann)** Let be the Heisenberg group over a non-Archimedean local field of characteristic different from and choose a nontrivial smooth character . Then there is a unique irreducible smooth representation of with central character .

*Proof.* Recall that as a space , where is a finite-dimensional symplectic vector space over . Choose a lagrangian subspace and let , which is a closed normal abelian -subgroup of . We can inflate to a smooth character along the natural projection .

Now is a finite-dimensional vector space over , as is , which is noncanonically isomorphic to . There is a natural projection with kernel , which corresponds to a linear subspace of codimension one, and the orbit is just the affine subspace . So is actually closed in . Note that in our situation , and the map is the composition of the isomorphism , which comes from and the symplectic form, with the obvious isomorphism .

Since , the theorem says that there is an equivalence between the category of smooth representations of whose restriction to is supported on and the category of smooth representations of supported at . Obviously the latter category has a unique simple object, namely the character itself, so there is a corresponding unique irreducible smooth representation of whose restriction to is supported on .

So it remains to show that a smooth irreducible representation of has central character if and only if its restriction to is supported on . But this is just a test of eyesight: the characters of in are precisely those which restrict to on .

I think I’ll save the proof of the theorem for next time, which will make the next post a bit boring but will prevent this one from being extremely long.