## 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 be an -group and a closed normal abelian -subgroup. Then acts on by conjugation, hence acts on as well. If is a smooth representation of , then after restricting to we can construct the corresponding sheaf on , and in particular it is easy to see that is -stable. Fix a smooth character . Then the stalk carries a natural smooth action of the stabilizer : this is clear if we recall that is identified with the space of -coinvariants of .

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 is a homeomorphism under some mild technical assumptions on such as second countability, but for mostly aesthetic reasons I simply assumed this, and indeed it is obvious in the case where is the Heisenberg group.

**Theorem (Rodier) **Suppose that the -orbit is locally closed in and that the natural map is a homeomorphism. Then the functor is an equivalence of categories 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 .

*Proof.* First let us construct the quasi-inverse equivalence, which is really just a geometric form of compactly supported induction. Given a smooth representation of , the product space (where carries the discrete topology as usual) has a left action of on each factor, so we can form the quotient space . There is a natural projection , 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 . It is not hard to see that if denotes the canonical projection, it suffices to work in a neighborhood of . We already know that has a neighborhood such that there is an -equivariant homeomorphism . It is also easy to see that if is the space with the trivial -action, then is an -equivariant homeomorphism . Combining these yields an -isomorphism . If we denote by the projection map, then after descending to the quotient we get a homeomorphism as desired.

We need to finish describing the quasi-inverse functor. Write for the sheaf of continuous sections of the étale space , and note that acts on global sections of this sheaf via the formula . Clearly this restricts to an action on the compactly supported sections , and this is the desired -representation. Of course we must check that the restriction to is supported on . For this, recall that by hypothesis the -equivariant map which sends is a locally closed embedding, so we can form the pushforward with compact supports , and acts on by the same formula. Fix and , so we can write for some . Then compute

for , where we used the fact that . This shows that , whence this -representation has support .

Finally, we check that these functors are actually quasi-inverse equivalences. Since , the stalk at of the former sheaf is naturally identified with , which is in turn naturally isomorphic to because this is the fiber of the étale space over . This isomorphism is -equivariant because

.

As for the other direction, let be a smooth representation of supported on . Write for the corresponding -representation, so we need to show as -representations. This map sends to the section defined by , where we have confused with its image in the quotient . Note has compact support: viewing as a nondegenerate -module, find a compact open subset such that , where is the indicator function for . This implies that . It is completely straightforward to check that is -equivariant, and to see that this is an isomorphism we may work on the level of -representations, or equivalently sheaves on . Passing to stalks, we get the map which sends , and since these are the spaces of – and – coinvariants this is clearly an isomorphism.