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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.

$\Box$