Home > representation theory > Representations of totally disconnected abelian groups

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 \ell-space. A topological group whose underlying space is an \ell-space is called an \ell-group. For example, if \mathbb{G} is an algebraic group over a non-Archimedean local field F and G = \mathbb{G}(F) is the group of F-points, then G is an \ell-group. Recall that a Hausdorff topological group is an \ell-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 \ell-group G is called smooth provided that every vector in the representation space has open stabilizer in G.

I call an \ell-group an \ell_c-group provided that it is a filtered union of its compact open subgroups. If G is an abelian \ell-group, then G is an \ell_c-group if and only if the Pontryagin dual group \widehat{G} is totally disconnected. Any finite-dimensional vector space over F is an \ell_c-group under addition, but the multiplicative group F^{\times} is not, since the units in the ring of integers are the maximal compact subgroup.  A more interesting (in particular, nonabelian) example of an \ell_c-group is a unipotent group over F, 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 \ell_c-groups via the following theorem.

Theorem (Rodier) Let G be an abelian \ell_c-group. Then there is an equivalence between the category \mathcal{R}(G) of smooth representations of G and the category \text{Sh}(\widehat{G}) of sheaves of complex vector spaces on the Pontryagin dual \widehat{G}.

If X is any topological space, let us write C^{\infty}_c(X) for the \mathbb{C}-algebra of locally constant, compactly supported complex-valued functions under pointwise multiplication. This algebra is unital only when X is compact, so it will be necessary to work with the following definition. If A is a ring (not necessarily commutative or unital), we say that an A-module M is nondegenerate provided that A \cdot M = M.

Lemma Let X be an \ell-space. The category \text{Sh}(X) of sheaves of complex vector spaces on X is equivalent to the the category of nondegenerate C^{\infty}_c(X)-modules.

Proof. We will write \mathfrak{m}_x for the maximal ideal of C^{\infty}_c (X) consisting of functions which vanish at a fixed x \in X, and similarly \mathfrak{m}_K denotes the ideal of functions which vanish on some compact open subset K \subset X. Given a nondegenerate C^{\infty}_c (X)-module M, it is natural to consider the vector space M/\mathfrak{m}_KM: this will be the space of sections over K. It is not hard to see that the associated presheaf whose sections locally have this form (recall that an \ell-space has a basis of compact open subsets) is a sheaf, which we call \widetilde{M}. 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 \widetilde{M} at a point x \in X is just the space M/\mathfrak{m}_xM. It is not hard to check that the mapping M \mapsto \widetilde{M} extends to a functor from the category of nondegenerate C^{\infty}_c (X)-modules to \text{Sh}(X), and we claim that this functor is an equivalence.
First, observe that M \mapsto \widetilde{M} has a right adjoint which sends a sheaf \mathcal{F} \in \text{Sh}(X) to its space \Gamma_c(X,\mathcal{F}) of compactly supported global sections, with the obvious C^{\infty}_c(X)-action: the level sets of any f \in C^{\infty}_c(X) are open. The unit morphisms M \to \Gamma_c(X,\widetilde{M}) are defined as follows: given m \in M, choose a compact open subset K \subset X which contains \text{Supp } m = \{ x \in X \ | \ m \notin \mathfrak{m}_xM \}, or equivalently e_K \cdot m = m, where e_K denotes the indicator function for K. Then pass to the quotient M/ \mathfrak{m}_K M = \widetilde{M}(K), from which we obtain \widetilde{m} \in \Gamma_c(X,\widetilde{M}) by extending by zero outside of K. This construction is easily seen to be independent of the choice of K. Now pick \mathcal{F} \in \text{Sh}(X): the counit is a morphism of sheaves, so we define it locally (over compact open sets). For any compact open subset K \subset X, consider the restriction map \Gamma_c(X,\mathcal{F}) \to \mathcal{F}(K). This vanishes on \mathfrak{m}_K \Gamma_c(X,\mathcal{F}), and is clearly compatible with the restriction maps of \mathcal{F} and \widetilde{\Gamma_c(X,\mathcal{F})}. 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 M \to \Gamma_c(X,\widetilde{M}) is surjective: if s \in \Gamma_c(X,\widetilde{M}) choose compact open K \subset X containing \text{Supp} s and pick a representative m \in M for s|_K \in \widetilde{M}(K) = M/ \mathfrak{m}_K M, then note that e_K \cdot m is sent to s by checking locally. Injectivity of the unit is much harder: observe that it suffices to prove that for m \in M, we have \text{Supp } m = \varnothing if and only if m = 0. The nontrivial part is showing that if m \in \mathfrak{m}_xM for all x \in X, then m = 0. First pick a compact open subset K \subset X such that e_K \cdot m = m, which we know is possible because M is nondegenerate. Then we can identify e_K \cdot C^{\infty}_c(X) with the (unital) algebra C^{\infty}_c(K), and the maximal ideals of C^{\infty}_c(K) all have the form e_K \cdot \mathfrak{m}_x for x \in K. Moreover, e_K \cdot M is a module for C^{\infty}_c(K), and we just need to show m = e_K \cdot m = 0, so by all of this we are reduced to the case that X is compact. Thus we are dealing with an ordinary unital, commutative \mathbb{C}-algebra A = C^{\infty}_c(X), and it is not hard to see that for each x \in X the localization map A \to A_{\mathfrak{m}_x} is surjective with kernel \mathfrak{m}_x, so we have an isomorphism A/\mathfrak{m}_xA \to A_{\mathfrak{m}_x} (actually both are canonically identified with \mathbb{C}). But it is a well-known fact of commutative algebra that if m is zero in every localization M_{\mathfrak{m}_x} then m = 0.

As for the counit, notice first that that the canonical maps \Gamma_c(X,\mathcal{F}) \to \mathcal{F}_x for x \in X are all surjective since X has a basis of compact open subsets. This shows that the counit is an epimorphism, and it is just as clear that if s \in \Gamma_c(X,\mathcal{F}) goes to zero in \mathcal{F}_x then s \in \mathfrak{m}_x\Gamma_c(X,\mathcal{F}). Thus the counit induces an isomorphism on stalks, so it is an isomorphism.

\Box

Back to groups: for any \ell-group G we write \mathcal{H}(G) for the Hecke algebra of G, 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 \mathcal{H}(G)-modules is isomorphic to the category of smooth representations of G.

Proof of Rodier’s theorem. Let G be an abelian \ell_c-group. As we just remarked, \mathcal{R}(G) is isomorphic to the category of nondegenerate \mathcal{H}(G)-modules. The Fourier transform provides an isomorphism of algebras \mathcal{H}(G) \to C^{\infty}_c(\widehat{G}), so \mathcal{R}(G) is in fact isomorphic to the category of nondegenerate C^{\infty}_c(\widehat{G})-modules. Now the \ell_c assumption on G implies that \widehat{G} is totally disconnected and hence an \ell-space, so we can apply the lemma to see that the category of nondegenerate C^{\infty}_c(\widehat{G})-modules is equivalent to \text{Sh}(\widehat{G}) as desired.

\Box

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