Locally compact groups

Although the above theory was probably started as a means to study the finite groups themselves, it soon became apparent that there were many applications to other subjects in mathematics and physics. This fact was a strong motivation to extend the theory of group representations to general topological groups. The representation space $V$ is allowed to be an infinite dimensional Hilbert space. A (unitary) representation is a continuous group homomorphism $\pi$ from $G$ to $U(V)$, the space of unitary operators on $V$.

In order that the results of the previous section can be generalized, one needs to define a unitary structure on the regular representation. This implies that one needs a measure $\mu$ on $G$ that is invariant both for left and right translations by elements in $G$. The existence and essential uniqueness of a left and a right invariant measure is guaranteed in the case of locally compact¹ groups by a famous result of Haar. But the requirement that the left invariant measure is itself also right invariant is a nontrivial condition. We call the group unimodular in this case. Thus, given any unimodular locally compact group $G$, three natural questions present themselves:

(1)

Describe $\hat G$, the collection of equivalence classes of irreducible unitary representations of $G$, as a topological space. Construct the irreducible representations.

(2)

Can one decompose any given unitary representation as a unique direct integral of irreducible representations?

(3)

In particular, decompose the regular representation as an explicit Hilbert integral of the spaces ${HS}_\pi$ of Hilbert-Schmidt operators in $V_\pi$, where $\pi$ varies over $\hat G$. In other words, find explicitly the measure $\nu$ on $\hat G$ such that for every $f\in L^2(G,\mu)\cap L^1(G,\mu)$

$$\int_G\vert f(g)\vert^2 d\mu(g)=\int_{\hat G} \Vert \hat f (\pi)\Vert^2 d\nu(\pi)$$

The measure $\nu$ is called the Plancherel measure of $G$.

Notice that the Plancherel measure on $\hat G$ is given by $\mu(\pi)=dim(V_\pi)$ when $G$ is a finite group.

It has to be said that even in the case of finite groups these questions are very difficult, and still open for many important groups. It is clear that the number of irreducibles is equal to the number of conjugacy classes in $G$ (by the Plancherel formula applied to central functions). However, this is a duality, not a parametrization, and there is no natural correspondence between the conjugacy classes and the irreducibles. The question of actually constructing the representations is even more difficult.

In some sense the case of the circle group $S^1$ was already treated in 1807 by Fourier. In his famous work on the heat equation with periodic boundary conditions he proved the Plancherel formula for what is now called the Fourier series of a function on a circle. But at that time there was no recognition of the fundamental role played by the group structure of $S^1$. The first real progress was made in 1934 by Pontrjagin, in the case of the general locally compact abelian group. This work represents a huge generalization of the theory of Fourier series and integrals, and is a cornerstone in much of the non-abelian theory that would be developed later.

When a group is abelian, then all its unitary irreducible representations are one dimensional. Such irreducible representations are called characters. We can multiply two characters point-wise to obtain a new character. Pontrjagin proved that this gives $\hat G$ itself the structure of a locally compact abelian group, the dual group of $G$. Moreover, he showed that the dual of $\hat G$ is $G$ itself. In 1940 Weil proved that the Plancherel measure on $\hat G$ is given by its Haar measure, suitably normalized.

The non-abelian theory started off with the study of compact groups. Schur himself considered already the representations of the groups $SO(n,{\mathbb R})$ and $SU(n)$. The general case of a compact connected Lie group was solved completely by Hermann Weyl in the 1920's. Compact Lie groups were only a small first step, in some sense still very resemblant to the case of finite groups. Serious applications in quantum theory and number theory required however the understanding of the representation theory of general locally compact groups.

A new key point arose from the work of Murray and Von Neumann in the 1930's. A von Neumann algebra is a subalgebra of the algebra of bounded operators on a Hilbert space which is closed for the weak topology and for taking adjoints. A von Neumann algebra is called a factor if its center consists of scalars only. A unitary representation $\pi$ of $G$ is called factorial if the von Neumann algebra generated by $\pi(G)$ is a factor. Examples of factorial representations are arbitrary direct sums of a single irreducible unitary representation. Factorial representations of this special kind are said to be of type I. A group is said to be of type I when all its factorial representations are of type I. The shocking discovery of Murray and Van Neumann was that not all locally compact unimodular groups are of type I. A famous counterexample is the free group $F_2$ on two elements (with the discrete topology). In general one can show that every unitary representation of a locally compact unimodular group $G$ has an essentially unique decomposition as a direct Hilbert integral of factorial representations. This does not imply the existence nor the uniqueness of a Plancherel measure on $\hat G$. However, if we further assume that $G$ is of type I, then there exists a unique Plancherel measure on $\hat G$ describing the decomposition of $L^2(G,\mu)$ as a unitary $G\times G$ representation. One can show that the ``dual'' $\hat G$ is metrizable and locally compact in this case.

Thus it became an important issue to decide whether a given unimodular locally compact group is of type I. Known examples of such groups are nilpotent Lie groups, certain (but not all) solvable Lie groups, reductive algebraic groups over local fields, and finally reductive groups over the adelic ring of a global field. Many mathematicians have studied the above program of three questions for these cases from the 1940's. It is of course too much to be discussed in detail here, and I will restrict myself to the case of reductive groups over local fields only.