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Rodier’s Mackey machine

So I’m finally getting around to the proof of that theorem of Rodier’s from several posts back. After this we’ll be done with the Stone-von Neumann theorem (keeping in mind that all this work only applies to the non-Archimedean local case).  Let’s recall the situation of the theorem and state it again for convenience.

Let G be an \ell-group and U \subset G a closed normal abelian \ell_c-subgroup. Then G acts on U by conjugation, hence acts on \widehat{U} as well. If M is a smooth representation of G, then after restricting to U we can construct the corresponding sheaf \widetilde{M} on \widehat{U}, and in particular it is easy to see that \text{Supp } \widetilde{M} is G-stable. Fix a smooth character \chi \in \widehat{U}. Then the stalk \widetilde{M}_{\chi} carries a natural smooth action of the stabilizer Z_G(\chi): this is clear if we recall that \widetilde{M}_{\chi} is identified with the space of \chi-coinvariants of M.

If you read the earlier post in which I stated this theorem, you might notice that I’ve added an extra hypothesis (although I did go back and change that post also). The map Z_G(\chi) \backslash G \to \chi^G is a homeomorphism under some mild technical assumptions on G such as second countability, but for mostly aesthetic reasons I simply assumed this, and indeed it is obvious in the case where G is the Heisenberg group.

Theorem (Rodier) Suppose that the G-orbit \chi^G is locally closed in \widehat{U} and that the natural map Z_G(\chi) \backslash G \to \chi^G is a homeomorphism. Then the functor M \mapsto \widetilde{M}_{\chi} is an equivalence of categories 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.

Proof. First let us construct the quasi-inverse equivalence, which is really just a geometric form of compactly supported induction. Given a smooth representation N of Z_G(\chi), the product space N \times G (where N carries the discrete topology as usual) has a left action of Z_G(\chi) on each factor, so we can form the quotient space E_N = Z_G(\chi) \backslash (N \times G). There is a natural projection E_N \to Z_G(\chi) \backslash G, and we will now use the result from the last post to show that this is an étale space. In fact, something a little better is true: it is a “covering space” in the usual sense, although this term is not usually applied in totally disconnected situations.

To simplify notation, write H = Z_G(\chi). It is not hard to see that if \pi : G \to H \backslash G denotes the canonical projection, it suffices to work in a neighborhood of \pi(1). We already know that \pi(1) has a neighborhood V such that there is an H-equivariant homeomorphism H \times V \to \pi^{-1}(V). It is also easy to see that if N_0 is the space N with the trivial H-action, then (n,h) \mapsto (h \cdot n,h) is an H-equivariant homeomorphism N_0 \times H \to N \times H. Combining these yields an H-isomorphism N_0 \times H \times V \to N \times \pi^{-1}(V). If we denote by \rho : E_N \to H \backslash G the projection map, then after descending to the quotient we get a homeomorphism N_0 \times V \to \rho^{-1}(V) as desired.

We need to finish describing the quasi-inverse functor. Write \mathcal{F}_N for the sheaf of continuous sections of the étale space E_N \to Z_G(\chi) \backslash G, and note that G acts on global sections s of this sheaf via the formula (g \cdot s)(Z_G(\chi)h) = s(Z_G(\chi)hg) \cdot g^{-1}. Clearly this restricts to an action on the compactly supported sections \Gamma_c(\mathcal{F}_N), and this is the desired G-representation. Of course we must check that the restriction to U is supported on \chi^G. For this, recall that by hypothesis the G-equivariant map i : Z_G(\chi) \backslash G \to \widehat{U} which sends Z_G(\chi)g \mapsto \chi^g is a locally closed embedding, so we can form the pushforward with compact supports i_!\mathcal{F}_N, and G acts on \Gamma_c(i_!\mathcal{F}_N) \cong \Gamma_c(\mathcal{F}_N) by the same formula. Fix s \in \Gamma_c(i_!\mathcal{F}_N) and \chi^g \in \chi^G, so we can write s(\chi^g) = [n,g] for some n \in N. Then compute

(u \cdot s)(\chi^g) = [n,gu^{-1}] = [n,gu^{-1}g^{-1}g] = [\chi^g(u) \cdot n,g] = \chi^g(u) \cdot s(\chi^g)

for u \in U, where we used the fact that \text{Supp } \widetilde{N} \subset \{ \chi \}. This shows that \widetilde{\Gamma_c(i_!\mathcal{F}_N)} \cong i_!\mathcal{F}_N, whence this U-representation has support \text{Supp } i_!\mathcal{F}_N \subset \chi^G.

Finally, we check that these functors are actually quasi-inverse equivalences. Since \widetilde{\Gamma_c(i_!\mathcal{F}_N)} \cong i_!\mathcal{F}_N, the stalk at \chi of the former sheaf is naturally identified with (\mathcal{F}_N)_\chi, which is in turn naturally isomorphic to N because this is the fiber of the étale space E_N over \chi. This isomorphism is Z_G(\chi)-equivariant because

[n,1] \cdot z^{-1} = [n,z^{-1}] = [z \cdot n,1].

As for the other direction, let M be a smooth representation of G supported on \chi^G. Write P = \widetilde{M}_{\chi} for the corresponding Z_G(\chi)-representation, so we need to show M \cong \Gamma_c(\mathcal{F}_P) as G-representations. This map sends m \in M to the section s_m : \chi^G \to E_P defined by s_m(\chi^g) = [g \cdot m,g], where we have confused g \cdot m with its image in the quotient M \to P. Note s_m has compact support: viewing M as a nondegenerate C^{\infty}_c(\widehat{U})-module, find a compact open subset K such that e_K \cdot m = m, where e_K is the indicator function for K. This implies that \text{Supp } s_m \subset K \cap \chi^G. It is completely straightforward to check that M \to \Gamma_c(\mathcal{F}_P) is G-equivariant, and to see that this is an isomorphism we may work on the level of U-representations, or equivalently sheaves on \widehat{U}. Passing to stalks, we get the map \widetilde{M}_{\chi^g} \to (\mathcal{F}_P)_{\chi^g}\cong \widetilde{M}_{\chi} which sends m \mapsto g \cdot m, and since these are the spaces of \chi^g– and \chi– coinvariants this is clearly an isomorphism.


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