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A generalized Stone-von Neumann theorem

Continuing our terminology from the last post, let G be an \ell-group, U \subset G a closed normal abelian \ell_c-subgroup, and \chi : U \to \mathbb{T} a smooth character. Now G acts on U by conjugation, which induces an action on the Pontryagin dual \widehat{U}.

If M is a smooth representation of G, we can restrict to U and consider the corresponding sheaf \widetilde{M} on \widehat{U}, which comes from the equivalence proved last time. The stalk \widetilde{M}(\chi) is identified with the space of \chi-coinvariants of M, i.e. the quotient of M by the subspace spanned by vectors of the form \chi(u) \cdot m - m for u \in U and m \in M. Notice that for any g \in G the action of G on M induces an isomorphism of stalks \widetilde{M}(\chi) \to \widetilde{M}(\chi^g), and in particular \widetilde{M}(\chi) has a natural action by the stabilizer Z_G(\chi).

It is easy to see that the support \text{Supp } \widetilde{M} \subset \widehat{U} is a G-invariant subspace: in particular, if \text{Supp } \widetilde{M} is contained in the G-orbit \chi^G of \chi, then either \text{Supp } \widetilde{M} = \chi^G or \text{Supp } \widetilde{M} = \varnothing.

Theorem (Rodier) Suppose the orbit \chi^G is locally closed in \widehat{G} and that the natural map Z_G(\chi) \backslash G \to \chi^G is a homeomorphism. Then the functor \widetilde{M} \to \widetilde{M}(\chi) is an equivalence from the category of smooth representations of G whose restriction to U is supported on \chi^G and the category of smooth representations of Z_G(\chi) whose restriction to U is supported at \chi.

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

Corollary (Stone-von Neumann) Let G be the Heisenberg group over a non-Archimedean local field F of characteristic different from 2 and choose a nontrivial smooth character \psi : F \to \mathbb{T}. Then there is a unique irreducible smooth representation of G with central character \psi.

Proof. Recall that as a space G = V \times F, where V is a finite-dimensional symplectic vector space over F. Choose a lagrangian subspace L \subset V and let U = L \times F, which is a closed normal abelian \ell_c-subgroup of G. We can inflate \psi to a smooth character \chi : U \to \mathbb{T} along the natural projection U \to F.

Now U is a finite-dimensional vector space over F, as is \widehat{U}, which is noncanonically isomorphic to U. There is a natural projection U \to L with kernel F, which corresponds to a linear subspace \widehat{L} \subset \widehat{U} of codimension one, and the orbit \chi^G is just the affine subspace \chi \cdot \widehat{L}. So \chi^G is actually closed in \widehat{U}. Note that in our situation Z_G(\chi) = U, and the map L \backslash V = U \backslash G \to \chi \cdot \widehat{L} is the composition of the isomorphism L \backslash V \to \widehat{L}, which comes from \psi and the symplectic form, with the obvious isomorphism \widehat{L} \to \chi \cdot \widehat{L}.

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

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

\Box

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.

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