Tannaka-Krein Duality, Pontryagin Duality

Consider finite abelian groups. We know their character theory which allows us to do harmonic analysis on these groups. The characters allow us to capture all the information about the group. In fact, we explicitly have that natural the character-group element pairing $(g, \chi) \to \chi(g)$ gives an isomorphism $G \to \hat {\hat {G}}$ . There are several ways to read this isomorphism. For instance, the Fourier inversion/duality which allows us to write functions on the group in terms of the functions of the character group and vice versa is one instance of it. An important feature is an orthogonality that is $\chi(g)$ as $\chi$ varies or $g$ varies (orthogonality of characters). Fourier transform is just one aspect of the relation. The fact that these are group isomorphisms means that the characters allow us to retrieve all of the group structure. What’s remarkable is that we can generalize this to all locally compact abelian groups. The characters (continuous group homomorphisms to the circle) form a group called the dual (Pontryagin) $\hat G$ and we have a canonical isomorphism between $G$ and $\hat {\hat {G}}$ . Character theory of finite abelian groups ( $G=\mathbb Z/n\mathbb Z, \hat G =\mathbb Z/n\mathbb Z$ ), theory of Fourier series ( $G=\mathbb R /\mathbb Z, \hat G = \mathbb Z$ ), Euclidean Fourier transforms ( $G=\mathbb R^n, \hat G=\mathbb R^{n}$ ) are all instance of the duality. We have orthogonality relations, inversion of Fourier transforms, and Plancherel formula with the appropriate choice of measure of the group and its dual.

Now, what happens when the group is not abelian? There are typically very few one-dimensional representations, and so we need all the representations to know enough about the group. Consider say, compact groups. We need all to understand all the representations in particular all the irreducible representations. The Fourier transform is now a natural transformation on the forgetful functor of the categories of representations. That is we have a map $(f, \pi) \to \int_{G} f(g) \pi(g) dg$ . That is each function maps to a family of operators acting on the representations. The Plancherel theorem, in this case (compact group), is the isometry $\displaystyle L^2(G) \to \sum_{\lambda \in \hat G} End(V_{\lambda})$ (the right side is a completion of this space of operators with properly defined inner products/measures)

We need correlations with all the matrix elements $\left \langle \pi(g) u, v \right\rangle$ to encode information about a function on the group. Class functions (constant on conjugacy classes/invariant under conjugation) can be written in terms of traces $tr(\pi(g))$ (called characters in this cases), which form an orthogonal collection parametrized by irreducible representations. (The orthogonality is reflected by Schur’s lemma which shows that between different irreducibles there are no $G$ invariant linear maps, and identity is essentially the only map on a fixed irreducible representation.) In the case of class functions, the Plancherel is an isometry from class functions to functions on the dual $\displaystyle L^2({C(G)}) \to L^2({\hat G})$ . The restriction of Fourier transform to class functions to any representation is just a scaling operator- where the scalar is given by the projection onto the trace functions $\frac{1}{\text{dim } V_{\lambda}}\int_{G} f(g) tr(\pi(g)) dg$

Ok, we have a way to get Fourier transforms and generalize harmonic analysis, but how do we retrieve the group structure? The dual (set of representations) is not a group anymore. The product structure instead is captured by tensor products. So we need to look at the “category” of representations, that is the total information about all the representations, the relations/maps between them, with the structure of direct sums and tensor products. (The Fourier transform is already an object related to this category– a natural transformation on the forget functor from this category to the category of vector spaces).

So what is the analogue of Pontryagin duality? In the abelian case, each group element gives a character on the set of characters, thus an element of the double dual. In the compact group case we think of group elements as delta functions at those elements, and consider their Fourier transform– that is they give the natural family of operators $\pi(g)$ as $\pi$ varies. This family of operators has nice properties– for instance, the action of $g$ on a space $V$ and its dual $\hat V$ are related, it acts trivially on the trivial representation, behaves well under direct sums and tensor products. Now the main insight of the Tannak duality is that these properties capture these special elements coming from delta functions. That is any family of operators which satisfy the above properties (behaving nicely with respect to direct sum, tensor product, dual constructions, acting trivially on trivial representation) form a group called the Tannaka group $T(G)$ which in the case of a compact group is isomorphic to $G$ . (The fact that this collection of the family of operators forms a group has to do with for instance the fact that the action of a group element on a space is related by inverse transpose to the action on its dual). The isomorphism in the case of compact groups needs, like in the case of abelian groups, the existence of representation that sends a non-trivial group element to a non-trivial operator. Once we have this, matrix elements/ representative functions can be used to separate points in the groups and approximate all continuous functions uniformly, then we one can show that the representative functions on the Tannaka group (potentially a subalgebra of the representative functions on the group) also form a dense closed set which separate points, approximate all continuous functions on the group, so one can then get the equivalence of categories representations on the group and its Tannak group, isomorphism of algebras of representative functions on the groups and thus isomorphism of groups. To summarize, Tannaka duality is a way to decode the group completely in terms of the category of representations– that is by capturing how the group elements act on different representations, and so just that by considering a nice subset (Tannaka group) of the collection of families of operators (all of the data is inside the category of representations), we can learn everything about the group.