## Representations of totally disconnected abelian groups

So last time I promised to start talking about the Weil representation, but I decided to backtrack a bit and give a rather interesting proof of the Stone-von Neumann theorem which goes through, of all things, sheaf theory. As far as I know this is due to François Rodier, and pretty much everything I am going to say can be found in his 1977 paper *Decomposition Spectrale des Représentations Lisses*. As you can probably tell from the title, the article was written in French, so I hope it will be useful to some people to have these results in English. Also, some of my proofs will be slightly different (hopefully a bit cleaner) than Rodier’s.

Let us introduce some convenient terminology. I will call a Hausdorff, locally compact, and totally disconnected topological space an *-space*. A topological group whose underlying space is an -space is called an *-group*. For example, if is an algebraic group over a non-Archimedean local field and is the group of -points, then is an -group. Recall that a Hausdorff topological group is an -group if and only if it has a basis at the identity consisting of compact open subgroups. We also remind the reader that a complex representation of an -group is called *smooth* provided that every vector in the representation space has open stabilizer in .

I call an -group an *-group* provided that it is a filtered union of its compact open subgroups. If is an abelian -group, then is an -group if and only if the Pontryagin dual group is totally disconnected. Any finite-dimensional vector space over is an -group under addition, but the multiplicative group is not, since the units in the ring of integers are the maximal compact subgroup. A more interesting (in particular, nonabelian) example of an -group is a unipotent group over , e.g. the Heisenberg group.

Of course we hope to eventually prove something about nonabelian groups, but the purpose of this post will be to establish a connection between sheaves and representations of abelian -groups via the following theorem.

**Theorem (Rodier)** Let be an abelian -group. Then there is an equivalence between the category of smooth representations of and the category of sheaves of complex vector spaces on the Pontryagin dual .

If is any topological space, let us write for the -algebra of locally constant, compactly supported complex-valued functions under pointwise multiplication. This algebra is unital only when is compact, so it will be necessary to work with the following definition. If is a ring (not necessarily commutative or unital), we say that an -module is *nondegenerate* provided that .

**Lemma** Let be an -space. The category of sheaves of complex vector spaces on is equivalent to the the category of nondegenerate -modules.

*Proof*. We will write for the maximal ideal of consisting of functions which vanish at a fixed , and similarly denotes the ideal of functions which vanish on some compact open subset . Given a nondegenerate -module , it is natural to consider the vector space : this will be the space of sections over . It is not hard to see that the associated presheaf whose sections locally have this form (recall that an -space has a basis of compact open subsets) is a sheaf, which we call . Indeed, any compact open cover of a compact open set has a finite refinement consisting of pairwise disjoint compact open sets, so locally patching is easy. The stalk of at a point is just the space . It is not hard to check that the mapping extends to a functor from the category of nondegenerate -modules to , and we claim that this functor is an equivalence.

First, observe that has a right adjoint which sends a sheaf to its space of compactly supported global sections, with the obvious -action: the level sets of any are open. The unit morphisms are defined as follows: given , choose a compact open subset which contains , or equivalently , where denotes the indicator function for . Then pass to the quotient , from which we obtain by extending by zero outside of . This construction is easily seen to be independent of the choice of . Now pick : the counit is a morphism of sheaves, so we define it locally (over compact open sets). For any compact open subset , consider the restriction map . This vanishes on , and is clearly compatible with the restriction maps of and . We leave it to the reader to check that the unit-counit relations are satisfied.

But in fact the unit and counit are isomorphisms. It is clear enough that is surjective: if choose compact open containing and pick a representative for , then note that is sent to by checking locally. Injectivity of the unit is much harder: observe that it suffices to prove that for , we have if and only if . The nontrivial part is showing that if for all , then . First pick a compact open subset such that , which we know is possible because is nondegenerate. Then we can identify with the (unital) algebra , and the maximal ideals of all have the form for . Moreover, is a module for , and we just need to show , so by all of this we are reduced to the case that is compact. Thus we are dealing with an ordinary unital, commutative -algebra , and it is not hard to see that for each the localization map is surjective with kernel , so we have an isomorphism (actually both are canonically identified with ). But it is a well-known fact of commutative algebra that if is zero in every localization then .

As for the counit, notice first that that the canonical maps for are all surjective since has a basis of compact open subsets. This shows that the counit is an epimorphism, and it is just as clear that if goes to zero in then . Thus the counit induces an isomorphism on stalks, so it is an isomorphism.

Back to groups: for any -group we write for the Hecke algebra of , i.e. the algebra of locally constant, compactly supported complex-valued functions under convolution. It is well-known (and not so hard to see) that the category of nondegenerate -modules is isomorphic to the category of smooth representations of .

*Proof of Rodier’s theorem.* Let be an abelian -group. As we just remarked, is isomorphic to the category of nondegenerate -modules. The Fourier transform provides an isomorphism of algebras , so is in fact isomorphic to the category of nondegenerate -modules. Now the assumption on implies that is totally disconnected and hence an -space, so we can apply the lemma to see that the category of nondegenerate -modules is equivalent to as desired.