10  Compact Groups

In this section we will generalize the representation of finite groups into a much broader context. While a number of difficulties arise when the cardinality of our groups become infinite, tools from analysis allow us to retain much of the original theory.

10.1 Topological Notions

A class in analysis or topology is not a prerequisite for this course, so we will not refer to the underlying machinery too rigorously—see, for example, Fulton and Harris (1991) and Rossmann (2002). We omit a formal treatment of the real numbers and assume that the reader is familiar, at least in practice, with the least upper bound property, decimal and other expansions, and so on. We summarize the necessary concepts here and focus on the underlying spirit, rather than the formality.

When dealing with infinite objects, it is often inevitable that we must consider questions of convergence: if we continue this process indefinitely, do we find ourselves arbitrary “close” to some anticipated answer? Is that the only thing we find ourselves drawn arbitrarily closely to? In other words, these limiting questions are concerned with the existence and uniqueness of solutions. Here we will consider such questions in the context of sequences, written \((x_k)_{k=1}^\infty\) and standing for an enumerated and ordered collection of objects in some larger set, to lean on intuition developed in elementary calculus. For shorthand, we abuse notation and write \[ (x_k)_{k=1}^\infty \subseteq X \] to indicate that \((x_k)_{k=1}^\infty\) is a sequence of elements all taken from \(X\).

The convergence of sequences is usually rigorized using the epsilon-delta framework—we ignore this formalism here, assuming that the reader already has some familiarity with sequences of real numbers. Interested readers can turn to standard textbooks on analysis, e.g., Rudin (2013).

Definition 10.1 A sequence \((x_k)_{k=1}^\infty \subseteq \mathbb{R}^n\) is said to be convergent if there is an \(x \in \mathbb{R}^n\) such that \(|x-x_k| \to 0\) as \(k \to \infty\); we call \(x\) the limit of the sequence and write \[ x = \lim_{k \to \infty} x_k \] or, more simply, \(x = \lim x_k.\) A (not necessarily convergent) sequence is said to admit a convergent subsequence if there is an increasing sequence \((i_k)_{k=1}^\infty\) of positive integers so that \((x_{i_k})_{k=1}^\infty\) is convergent, i.e., if we can throw out some (possibly infinitely many) of the elements in \((x_k)_{k=1}^\infty\) to obtain a convergent sequence.

Proposition 10.1 (Rudin 2013, 48) The limit of a sequence, if it exists, is unique.

Example 10.1 Consider the sequences \[ x_k = \tfrac{1}{k}, \quad y_k = \tfrac{k+1}{k} (-1)^k, \quad \text{ and } \quad z_k = k. \] The first sequence converges to \(0\); the second does not converge, but has many convergent subsequences which converge to \(1\) and \(-1\); the last has no convergent subsequences.

Definition 10.2 A set \(X \subseteq \mathbb{R}^n\) is called closed if every convergent sequence \((x_k)_{k=1}^\infty \subseteq X\) has its limit in \(X\), i.e., if \(X\) is closed under limits.

Example 10.2 The interval \([a,b] \subset \mathbb{R}\) is a closed set. More generally, cubes of the form \[ [a_1,b_1] \times \cdots \times [a_n,b_n] \subset \mathbb{R}^n \] are closed. Sets like \(\{x\} \subset \mathbb{R}\) and \(\mathbb{Z}\subset \mathbb{R}\) are also closed, though they feel quite different—these are discrete sets, to be explored as we develop the necessary topological language.

Also note that \(\mathbb{R}^n\) itself is closed (this comes from the completeness axiom of \(\mathbb{R}\)). On the other hand, \(\varnothing\) is vacuously closed.

Closed sets are the fundamental objects that allow us to speak of the topology of a space, i.e., its continuous structure.

NoteOpen sets

Many topology textbooks prefer a dual construction which we mention briefly:

Definition 10.3 A set \(X \subseteq \mathbb{R}^n\) is called open if every convergent sequence \((x_k)_{k=1}^\infty\) in \(\mathbb{R}^n\) with \(\lim(x_k) \in X\) has its tail in \(X\), i.e., there is some \(N \in \mathbb{Z}^+\) such that \(x_k \in X\) for all \(k \geq N.\) A set \(X \subseteq \mathbb{R}^n\) is open if and only if its complement \(\mathbb{R}^n \setminus X\) is closed (Rudin 2013, 34); by convention, we say that \(\varnothing \subseteq \mathbb{R}^n\) and \(\mathbb{R}^n\) itself are both simultaneously closed and open.

Intuitively, if two points are contained in many open sets then we can think of them as being nearby one another. While this sort of construction is good to be aware of, we will avoid it here.

There are fundamentally two types of convergent sequences: those that become stationary in that all but finitely many of the elements \(x_k\) are equal to the limit \(x\) and those that do not. We differentiate these situations with the following.

Definition 10.4 Given \(X \subseteq \mathbb{R}^n\), we say that \(x \in \mathbb{R}^n\) is a limit point of \(X\) if there is some sequence \((x_k)_{k=1}^\infty \subseteq X \setminus \{x\}\) converging to \(x\). In particular, every \(x \in X\) is either a limit point of \(X\) or not; when \(X\) is closed, we call the latter isolated points of \(X.\) If a closed set only contains isolated points, it is discrete.

Example 10.3 The set \(X = [0,1] \cup \mathbb{Z}\subset \mathbb{R}\) is closed. Every element \(x \in X\) with \(0 \leq x \leq 1\) is a limit point of \(X\), but every other element is an isolated point.

With these fundamentals established, we can speak of continuous functions as those that respect the underlying topological structure.

Definition 10.5 If \(X \subseteq \mathbb{R}^n\) and \(Y \subseteq \mathbb{R}^m\), a function \(f: X \to Y\) is continuous if, for every convergent \((x_k)_{k=1}^\infty \subseteq X\) with \(\lim(x_k) \in X\), the sequence \((f(x_k))_{k=1}^\infty\) is convergent and \[ f\left( \lim_{k \to \infty} x_k \right) = \lim_{k \to \infty} f(x_k). \tag{10.1}\] In other words, continuous functions commute with limits. An invertible continuous map \(X \to Y\) whose inverse is also continuous is a homeomorphism; we say that \(X\) and \(Y\) are homeomorphic, written \(X \cong Y\) which means they have the same topological structure.

In other words, continuous functions send nearby points to nearby points. Practically, it can be difficult to show that a given function is continuous. We will ignore such details in this discussion and instead rely on functions we know from calculus, such as polynomials, \(\sin(x)\) and \(\cos(x)\) and \(e^x\), compositions therein, etc., that we know are continuous.

Proposition 10.2 (Rudin 2013, 87) Let \(X \subseteq \mathbb{R}^n\) be closed and suppose that \(f: X \to Y \subseteq \mathbb{R}^m\) is continuous. If \(C \subseteq \mathbb{R}^m\) is closed, then the inverse image \[ f^{-1}(C) = \{ x \in X: f(x) \in C \} \] is a closed subset of \(\mathbb{R}^n.\)

Proof. Assume, non-trivially, that \((\operatorname{im}f) \cap C\) is non-empty (otherwise \(f^{-1}(C) = \varnothing\) is closed by definition). If we build a sequence \((x_k)_{k=1}^\infty\) from elements in \(f^{-1}(C)\) converging to \(x\), then \[ f(x) = f\left( \lim_{k \to \infty} x_k \right) = \lim_{k \to \infty} f(x_k), \] where the second equality is by the definition of continuity. This limit exists and moreover belongs to \(C\), since \(C\) is closed, and hence \(x \in f^{-1}(C).\)

Since singleton sets are closed, we immediately conclude:

Corollary 10.1 Let \(X \subseteq \mathbb{R}^n\) be closed. The fibers of any continuous map \(f: X \to Y \subseteq \mathbb{R}^m\) are closed.

Even if this is not always literally true in topology (see your nearest class in point-set topology for examples of non-\(T_1\) spaces), the idea that fibers of functions should behave in a special way is a fundamental idea in math. As we learned from Noether (Theorem 2.6, Theorem 3.6), fibers are special!

Example 10.4 The function \(\mathbb{R}^{n+1} \to \mathbb{R}\) given by \[ (x_0,x_1,\dots,x_n) \mapsto x_0^2 + x_1^2 + \cdots + x_n^2 \] is continuous and so the fiber over \(1\), i.e., the unit \(n\)-sphere \(\mathbb{S}^n\), is a closed set of \(\mathbb{R}^{n+1}.\)

Example 10.5 The determinant defines a continuous function \(\operatorname{Mat}_n(\mathbb{R}) = \mathbb{R}^{n^2} \to \mathbb{R}\) and hence the fiber over \(1\), i.e., the special linear group \(\operatorname{SL}_n(\mathbb{R})\), is a closed set. By a similar argument, \(\operatorname{SL}_n(\mathbb{C})\) is a closed subset of \(\mathbb{C}^{n^2}=\mathbb{R}^{2n^2}.\) Similarly, the map \(\operatorname{Mat}_n(\mathbb{R}) \to \operatorname{Mat}_n(\mathbb{R})\) given by \[ A \mapsto A^\top A \] is continuous and so the fiber over \(\mathbb{I}\in \operatorname{Mat}_n(\mathbb{R})\), i.e., the group \(\operatorname{O}(n)\) of orthogonal matrices, is closed. The same argument shows the group \(\operatorname{U}(n)\) of unitary matrices is closed; so are the subgroups \(\operatorname{SO}(n)\) and \(\operatorname{SU}(n)\) of determinant \(1\) orthogonal and unitary matrices, respectively.

Example 10.6 On the other hand \(\operatorname{GL}_n(\mathbb{R})\) and \(\operatorname{GL}_n(\mathbb{C})\) are not closed sets. In fact, their complements (the set of singular \(n \times n\) matrices) are closed: \[ \operatorname{GL}_n(F) = \operatorname{Mat}_n(F) \setminus {\det}^{-1}(\{0\}). \]

Another key topological notion we will need is that of density, which in our context often arise as algebraically well-behaved (in particular, countable) subsets of a more complicated topological space. As we will see, dense subsets are to continuous functions as bases are to linear transformations or generating sets are to group homomorphisms.

Definition 10.6 Given a closed set \(X \subseteq \mathbb{R}^n\), a subset \(A \subseteq X\) is called dense if every element \(x \in X\) is the limit of some convergent sequence \((x_k)_{k=1}^\infty\) with each \(x_k \in A.\)

Example 10.7 The subset \(\mathbb{Q}\in \mathbb{R}\) is dense, even though the former is a countable set and the latter is uncountably infinite. Indeed, we can write any \(x \in \mathbb{R}\) as \(x = n + 0.d_1d_2d_3\dots,\) with \(n \in \mathbb{Z}\) so that \(x-n \in [0,1)\) and \(0.d_1d_2d_3\dots\) is its decimal expansion; the sequence \[ x_1 = n+\tfrac{d_1}{10}, \quad x_2 = x_1+\tfrac{d_2}{100}, \quad \cdots \quad x_k = x_{k-1} + \tfrac{d_k}{10^k}, \quad \dots \ \] is comprised entirely of rational numbers and converges to \(x.\)

Example 10.8 The multiplicative group \[ \mathcal{C}_{} = \left\{ z \in \mathbb{C}^\times \mid z^k = 1 \text{ for some } k \in \mathbb{Z}^+ \right\} \] of roots of unity is a dense subset of the unit circle \(\mathbb{T}.\)

Briefly, a dense subset is one that allows us to get arbitrarily close to every element of a bigger space. Note that if \(A \subseteq X\) is dense and also closed, then we can only have \(A = X.\)

Proposition 10.3 (Rudin 2013, 99) Suppose \(X \subseteq \mathbb{R}^n\) is closed, \(Y \subseteq \mathbb{R}^m\), and \(A \subseteq X\) is dense. If \(f_1, f_2: X \to Y\) are continuous and agree on every element of \(A\), then \(f_1 = f_2.\)

Proof. Left as an exercise (Exercise H.1).

In other words, continuous functions are determined by their values on dense subsets!

The final topological notion we emphasize is known as compactness, which has many useful characterizations that we do not delve into here. Somehow, the idea is that compactness prevents situations where a particular quantity of study “runs off to infinity.”

Definition 10.7 (Rudin 2013, 40) We say \(X \subseteq \mathbb{R}^n\) is compact if every sequence in \(X\) has a convergent subsequence that converges to a point of \(X.\) Equivalently, \(X\) is closed and bounded (that is, \(X\) fits inside a sphere of some finite radius).

Example 10.9 The circle \(\mathbb{S}^1 = \operatorname{U}(1) = \mathbb{T}\) is compact. Indeed, all spheres \(\mathbb{S}^n\) are compact.

Example 10.10 The set \(\mathbb{Z}\) is closed but not compact, since the sequence \(x_k = k\) has no convergent subsequence. The open interval \(X = (0,1)\) is bounded but not compact, since the sequence \(x_k = \tfrac{1}{k+1}\) converges to \(0 \not \in X.\)

10.2 Topological Groups

Remember, the reason for developing these concepts is to think about how they interact with our existing notions of group structure. As such, we define the following:

Definition 10.8 A topological group is a group \(G \subseteq \mathbb{R}^n\) such that the binary operation \((g,h) \mapsto gh\) and the inversion map \(g \mapsto g^{-1}\) are both continuous.

In other words, if we have a convergent sequence \((g_k)_{k=1}^\infty \subseteq G\) with \(\lim g_k = g\), then \[ \lim {g_k}^{-1} = g^{-1}, \] and if we have \((g_k,h_k)_{k=1}^\infty \subseteq G \times G\) with \((g_k,h_k) \to (g,h)\), then \[ \lim g_k h_k = gh. \]

Example 10.11 The following are topological groups:

• The real line \(\mathbb{R}\) (more generally, \(\mathbb{R}^n\)) is a topological group with respect to addition.
• The integers \(\mathbb{Z}\) is a closed subgroup of \(\mathbb{R}\); more than that, \(\mathbb{Z}\) is discrete. By this we mean that \(\mathbb{Z}\) does not have an interesting topology—sequences of integers cannot converge unless they have a constant tail, i.e., points \(n \in \mathbb{Z}\) behave like open sets.
• The circle \(\mathbb{T}= \operatorname{U}(1) \subseteq \mathbb{C}\) is a topological group. More generally, unitary \(\operatorname{U}(n)\) and special unitary groups \(\operatorname{SU}(n)\) are topological groups.
• The general linear groups \(\operatorname{GL}_n(\mathbb{R}) \subset \mathbb{R}^{n^2}\) and \(\operatorname{GL}_n(\mathbb{C}) \subset \mathbb{C}^{n^2}\) are topological groups. Similarly, the associated special linear groups \(\operatorname{SL}_n(\mathbb{R})\) and \(\operatorname{SL}_n(\mathbb{C})\) are topological groups.
• Every finite group \(G\) can be thought of as a discrete subgroup of \(\operatorname{GL}_n(\mathbb{R})\), via the permutation representation of \(G\) acting on itself by (say, left) multiplication, and is therefore a topological group. Much like with \(\mathbb{Z}\), sequences in a finite \(G\) cannot converge unless they have a constant tail; this topology does not enrich the study of \(G.\)

We will only consider topological groups that are subsets of \(\mathbb{R}^N\) (or \(\mathbb{C}^N = \mathbb{R}^{2N}\)) for some \(N \in \mathbb{N}\), so topological notions are always framed in terms of sequences. For the topologists, this means we will be assuming all our topological groups are Hausdorff—in particular, singleton sets are always closed.

The interplay of topology with the symmetry inherent in groups leads to interesting phenomena in objects equipped with both structures simultaneously. For example:

Proposition 10.4 If \(G\) is a topological group and \(h \in G\) is a limit point of \(G\), then every \(g \in G\) is a limit point of \(G.\)

Proof. We will show that the identity \(e \in G\) is a limit point of \(G\) if and only if \(h \in G\) is. This follows from the continuity of the group operations. Indeed, if \((g_k)_{k=1}^\infty\) is a sequence of elements in \(G\) converging to \(e \in G\), then \(h g_k \to h\); on the other hand, if \((h_k)_{k=1}^\infty \in G\) is a sequence converging to \(h\), then \(h^{-1} h_k \to e\).

When studying topological groups, we are particularly interested in homomorphisms that also respect the underlying continuous structure. Therefore, in the context of topological groups \(G\) and \(H\), when we refer to \(\operatorname{Hom}(G,H)\) we always mean the set of all continuous homomorphisms \(G \to H.\) Accordingly:

Definition 10.9 Let \(V\) be a finite-dimensional \(\mathbb{C}\)-vector space. A representation of a topological group \(G\) is a continuous homomorphism \(\rho: G \to \operatorname{GL}(V).\) ¹

This definition might seem alarmingly technical, but we can interpret it here as ruling out badly-behaved homomorphisms.

Example 10.12 If \(f: \mathbb{R}\to \mathbb{R}\) is a homomorphism and \(f(1) = \lambda\), we can deduce that \(f(x) = \lambda x\) for any \(x \in \mathbb{Q}\) by repeatedly applying the homomorphism law (see Lemma 11.1). If we ask \(f\) to be continuous, this completely determines the map by the density of \(\mathbb{Q}\) in \(\mathbb{R}.\) Hence we have \[ \operatorname{Hom}(\mathbb{R},\mathbb{R}) = \left\{ t \mapsto \lambda t \mid \lambda \in \mathbb{R}\right\}. \] Without continuity, however, there are no other restrictions on \(f\)—they are many more discontinuous homomorphisms! For example, we could define \(f(\sqrt{2})\) to be anything; from this we will have determined the values of \(f\) on \(\mathbb{Q}(\sqrt{2}) \subset \mathbb{R}.\) We can continue in this way indefinitely, since \(\mathbb{R}\) is an infinite-dimensional \(\mathbb{Q}\)-vector space. These maps are less interesting to us at present, because they do not respect the topological structure of \(\mathbb{R}.\)

Proposition 10.5 If \(G \subseteq \mathbb{R}^n\) and \(H \subseteq \mathbb{R}^m\) are topological groups and \(\rho: G \to H\) is a homomorphism, then \(\rho\) is continuous if and only if \(\rho\) is continuous at \(e_G \in G\), i.e., if \[ \lim_{k \to \infty} \rho(g_k) = e_H \] for all sequences \((g_k)_{k=1}^\infty\) with each \(g_k \in G\) and \(\lim(g_k) = e_G.\)

Proof. We need to show the reverse implication. If \((g_k)_{k=1}^\infty\) is a sequence in \(G\) converging to \(g \in G\), then \(\lim( g^{-1} g_k ) = e\) and hence \[ \begin{aligned} \lim_{k \to \infty} \rho(g_k) & = \lim_{k \to \infty} \rho(g) \rho(g^{-1} g_k) \\ & = \rho(g) \lim_{k \to \infty} \rho(g^{-1} g_k) \\ & = \rho(g) e_H = \rho(g), \end{aligned} \] by the continuity of the group operations in \(G\) and \(H.\)

Example 10.13 Fix \(\omega \in \mathbb{R}.\) Then the function \(\rho: \mathbb{R}\to \mathbb{T}\) given by \[ \rho(t) = e^{i\omega t} \] is a continuous homomorphism.

Example 10.14 Fix \(n \in \mathbb{Z}.\) Then the function \(\rho: \mathbb{T}\to \mathbb{T}\) given by \[ \rho(z) = z^n \] is a continuous homomorphism.

In the theory of groups, we study normal subgroups because they are the subgroups that are kernels of homomorphisms. For topological groups, where fibers of continuous homomorphisms are necessarily closed, we are interested in closed normal subgroups.

If \(G\) and \(H\) are topological groups with \(G\) closed and \(\rho: G \to H\) is any continuous homomorphism, then \(\ker \rho\) is a closed normal subgroup of \(G.\)

Note

We can avoid the requirement that \(G\) need be closed if we allow the notion of induced topologies. Since this is beyond the scope of the class, and unnecessary for our purposes (we are only interested in compact groups, which are all closed), there is no loss in generality.

Lemma 10.1 Any proper non-trivial closed subgroup of \(\mathbb{R}\) has a least positive element that generates it. Similarly, any proper non-trivial closed subgroup of \(\mathbb{T}\) has an element of least positive argument that generates it.

Proof (Sketch). A proper closed subgroup cannot be dense (Exercise H.1), so in particular the identity cannot be a limit point of the subgroup. If the subgroup in question had elements not contained in the group generated by this smallest element, we could contradict it being the smallest using properties of groups.

The full details are left for Homework (Exercise H.3), as are the details of the following:

Theorem 10.1 The proper non-trivial closed subgroups of \(\mathbb{R}, \mathbb{T},\) and \(\mathbb{Z}\) are given by \[ \begin{aligned} r\mathbb{Z}, & \text{ where } r \in \mathbb{R}^+; \\ \langle e^{2\pi i/n} \rangle, & \text{ where } n \in \mathbb{Z}^+, \\ n\mathbb{Z}, & \text{ where } n \in \mathbb{Z}^+. \end{aligned} \]

Note that the closed condition for subgroups on \(\mathbb{Z}\) does not add any constraints, since every subgroup of \(\mathbb{Z}\) is already closed.

10.3 Integrals

In the representation theory of topological groups, we retain a notion of irreducibility without having to modify our definition from the context of finite groups. Moreover, both forms of Lemma 5.1 can be recovered by copying the proof for finite groups. Many of our other previous results, however, are suddenly shrouded in doubt, because a purely algebraic notion of summing over \(G\) and its irreducible representations no longer exists. With our algebraic toolkit harshly limited, we must turn to analytic approaches.

Definition 10.10 A compact group is a topological group \(G \subseteq \mathbb{R}^n\) that is also compact.

TipHeine-Borel

For our purposes, the word compact just means closed (in the sense previously described) and bounded. In other words, a closed set \(X \subset \mathbb{R}^n\) is called compact if there is some \(r \in \mathbb{R}^+\) such that \(|x| < r\) for all \(x \in X.\) The upshot of compactness is that phenomena on \(X\) that we are interested in cannot “run off to infinity,” to be explored in analysis, geometry, and topology courses.

WarningBut what is compactness?

The “actual” notion of compactness will not be dwelled on here. Sometimes the term compact appears with an additional modifier, “locally”—local compactness is another topological property that is included to avoid printing outright falsehoods in these notes. All topological groups mentioned here are locally compact (the main exception is \(\mathbb{Q},\) which is nightmarish from a topological perspective) so this descriptor can be safely ignored.

Example 10.15 The circle group \(\mathbb{T}\) is compact. More generally, \(\operatorname{O}(n), \operatorname{U}(n), \operatorname{SO}(n)\) and \(\operatorname{SU}(n)\) are compact groups for all \(n \in \mathbb{Z}^+.\) There is a natural way to understand finite groups as compact, but \(\mathbb{Z}\) is not.

In what follows, we write \(C^{}(G) \coloneqq C^{}(G,\mathbb{C})\) for the vector space of continuous functions on a topological group \(G.\)

Definition 10.11 An integral on a compact group \(G\) is a nonzero linear functional \[ \int: C^{}(G) \to \mathbb{C}, \] satisfying positivity: if \(f(h) \geq 0\) for all \(h \in G\), then we must have \(\int f \geq 0.\)

Remark 10.1. We can talk about integrals more generally for locally compact groups, though we have to restrict our interest to the vector space of compactly-supported continuous functions. To see why, note that the usual Riemann integral over \(\mathbb{R}\) (which is locally compact but not compact) is not defined for many continuous functions because the integral diverges. We also note that a more general approach via measure theory is considered standard today—see, e.g., Diestel and Spalsbury (2014)—though we will quickly restrict our attention to the sort of integrals that we see in a typical undergraduate calculus class.

As we are inspired from our averaging operations over finite groups, we would like to extend the definition of integration to continuous functions from \(G\) into some inner product space \(V.\) “Continuous” here has the same meaning—that limits in \(G\) turn into limits in \(V\), under the norm induced by the inner product—though we will not worry too much about these issues here. We denote the space of all such functions as \[ C^{}(G,V) = \left\{ f : G \to V \mid f \text{ is continuous}\right\}. \] Given \(\int\) on \(C^{}(G)\), we define an extended integral on \(C^{}(G,V)\) by fixing a basis \(\{v_1,...,v_n\}\) for \(V\) and writing \(f \in C^{}(G,V)\) as a sum \(f = f_1 v_1 + \cdots + f_n v_n\) for functions \(f_i \in C^{}(G)\): \[ \int^V f \coloneqq \sum_{i=1}^n \left( \int f_i \right) v_i. \] One should immediately be suspicious—does this definition depend on the basis we picked?

Proposition 10.6 The extended integral \(\int^V\) is well-defined.

Proof. Suppose \(\{w_1,...,w_n\}\) is another basis, so that we have \(v_i = \sum_{j=1}^n \alpha_{ij} w_j\) for each \(i.\) Then we have \[ f = \sum_{i=1}^n f_i v_i = \sum_{i,j=1}^n f_i \alpha_{ij} w_j = \sum_{j=1}^n \left( \sum_{i=1}^n \alpha_{ij} f_i \right) w_j \] in the new basis. Now compute \(\int^V\) with respect to \(\{w_1,...,w_n\}\): \[ \sum_{j=1}^n \int \left( \sum_{i=1}^n \alpha_{ij} f_i \right) w_j = \sum_{i=1}^n \left( \int f_i \right) \sum_{j=1}^n \alpha_{ij} w_j = \sum_{i=1}^n \left( \int f_i \right) v_i, \] because \(\int\) is linear, so the extended integral \(\int^V\) is well-defined.

In other words, this proposition gives us permission to integrate a column vector, or even a matrix, entry-wise without anxiety. With this technicality established, we will often simply write \(\int\) when we really mean \(\int^V\) except when doing so would cause confusion.

Of particular interest is the case when we integrate operators via \(\int^{\operatorname{End}(V)}.\) Because of the way we defined the extended integral, integration commutes nicely with a variety of linear operations. We mention two key properties here.

Proposition 10.7 Let \(V\) be a vector space, let \(f: G \to \operatorname{End}(V)\) be a continuous function, and let \(A \in \operatorname{End}(V)\) and \(x \in V.\) The following identities hold: \[ \begin{aligned} A \int^{\operatorname{End}(V)} f & = \int^{\operatorname{End}(V)} (Af), \\ \left(\int^{\operatorname{End}(V)} f\right)A & = \int^{\operatorname{End}(V)} (fA), \\ \left(\int^{\operatorname{End}(V)} f\right) & x = \int^{V} (fx). \end{aligned} \]

Proof. We show the first formula; the other proofs are similar. Fix a basis \(\{v_1,\dots,v_n\} \subset V\), which gives rise to the usual basis of \(\operatorname{End}(V)\) consisting of operators \(E_{ij}\) which send \(v_j \mapsto v_i\) and all other basis elements to zero. We write \(F: G \to \operatorname{End}(V)\) and \(A\) in this basis: \[ f(g) = \sum_{i,j = 1}^n f_{ij}(g) E_{ij} \quad \text{ and } \quad A = \sum_{i,j = 1}^n \alpha_{ij} E_{ij}, \] where \(f_{ij} \in C^{}(G)\) and \(\alpha_{ij} \in \mathbb{C}.\) We can then compute directly, by linearity of \(\int\) and the definition of extended integral: \[ \begin{split} A \int^{\operatorname{End}(V)} f & = \left(\sum_{i,\ell = 1}^n \alpha_{i\ell} E_{i\ell}\right)\left( \sum_{j,k = 1}^n \left[ \int f_{jk} \right] E_{jk} \right) \\ & = \sum_{i,j,k = 1}^n \left( \alpha_{ij} \int f_{jk} \right) E_{ik} \\ & = \sum_{i,k = 1}^n \left( \int \sum_{j=1}^n \alpha_{ij} f_{jk} \right) E_{ik} \\ & = \int^{\operatorname{End}(V)} \left( \sum_{i,k = 1}^n \left[ \sum_{j=1}^n \alpha_{ij} f_{jk} \right] E_{ik} \right) \\ & = \int^{\operatorname{End}(V)} (Af). \end{split} \]

Proposition 10.8 If \(V\) is a vector space and \(F: G \to \operatorname{End}(V)\) is continuous, then \[ \operatorname{Tr}\int^{\operatorname{End}(V)} F = \int \operatorname{Tr}F. \]

Proof. Fix a basis \(\{v_1,\dots,v_n\} \subset V\) and the accompanying basis elements \(E_{ij}\) of \(\operatorname{End}(V)\): We have \[ \begin{split} \operatorname{Tr}\int^{\operatorname{End}(V)} F & = \operatorname{Tr}\left( \sum_{i,j=1}^n \left[ \int f_{ij} \right] E_{ij} \right) \\ & = \sum_{i=1}^n \left( \int f_{ii} \right) \\ & = \int \left( \sum_{i=1}^n f_{ii} \right) \\ & = \int \operatorname{Tr}F. \end{split} \]

Next, recall that there is a group action of \(G\) on \(C^{}(G)\) via \[ (h \cdot f)(g) \coloneqq f(h^{-1}g) \] which extends naturally to an action of on \(C^{}(G,V).\) There is also a right group action—the word “right” means that the compatibility condition corresponds to right multiplication of group elements, rather than left multiplication, which effectively reverses the order we expect actions to apply in—that we will write using \[ (h.f)(g) \coloneqq f(gh). \]

Definition 10.12 An integral \(\int\) is left-invariant if, for all \(h \in G\) and \(f \in C^{}(G)\), \[ \int h \cdot f = \int f. \] Similarly, we say that \(\int\) is right-invariant if we have \[ \int h.f = \int f \] for all \(h \in G\) and \(f \in C^{}(G).\) An integral satisfying both these conditions is simply called invariant.

Proposition 10.9 If \(\int: C^{}(G) \to \mathbb{C}\) is left-invariant (resp., right-invariant) and \(V\) is an inner product space, then the extended integral \(\int^V\) is left-invariant (resp., right-invariant).

Proof. Suppose \(\int\) is left-invariant and \(\{v_1,\dots,v_n\} \subset V\) is a basis. If \(f = f_1 v_1 + \cdots + f_n v_n\) for functions \(f_i \in C(G)\) and we fix \(h \in G\), then we have \[ \begin{aligned} (h \cdot f)(g) & = f_1(h^{-1}g) v_1 + \cdots + f_n(h^{-1}g) v_n \\ & = (h \cdot f_1)(g) v_1 + \cdots + (h \cdot f_n)(g) v_n \end{aligned} \] for all \(g \in G.\) Therefore we can see that \[ \begin{aligned} \int^V h \cdot f & = \sum_{i=1}^n \left[ \int h \cdot f_i \right] v_i \\ & = \sum_{i=1}^n \left[ \int f_i \right] v_i \\ & = \int^V f. \end{aligned} \] The proof for right invariance is similar.

Theorem 10.2 (Haar 1933) Let \(G\) be a locally compact group. Then there is a left-invariant integral \(\int\) on \(G\) which is unique up to scaling, called a Haar integral on \(G.\) If \(G\) is compact or abelian, then \(\int\) is also right-invariant. Moreover, since integrating the constant function \(1\) over a compact group gives some finite value, we can normalize so that \(\int 1 = 1\); we call this normalized invariant integral the Haar integral on \(G.\)

Example 10.16 The usual integral from calculus is a Haar integral on \(\mathbb{R}\) (recall that \(\mathbb{R}\) is locally compact, but not compact). Indeed, if \(f: \mathbb{R}\to \mathbb{C}\) is a function such that \[ \int_{- \infty}^\infty f(t) \mathop{}\!\mathrm{d}{t} < \infty \] then we know from \(u\)-substitution that \[ \int_{- \infty}^\infty f(t-s) \mathop{}\!\mathrm{d}{t} = \int_{- \infty}^\infty f(t) \mathop{}\!\mathrm{d}{t} \] for all \(s \in \mathbb{R}.\) This is called translation invariance. This integral is also right-invariant: \[ \int_{- \infty}^\infty f(t+s) \mathop{}\!\mathrm{d}{t} = \int_{- \infty}^\infty f(t) \mathop{}\!\mathrm{d}{t}. \]

Example 10.17 If we parameterize by angle, then \(f: \mathbb{T}\to \mathbb{C}\) is equivalent to a \(2\pi\)-periodic function \(f: \mathbb{R}\to \mathbb{C}\) (recall Example 3.46). With this setup, the Haar integral on \(\mathbb{T}\) is given by \[ f \mapsto \frac{1}{2\pi} \int_0^{2\pi} f(\theta) \mathop{}\!\mathrm{d}{\theta}. \] This integral is rotationally invariant because, for any \(\phi \in \mathbb{R}\), we have \[ \underbrace{\frac{1}{2\pi} \int_0^{2\pi} f(\theta - \phi) \mathop{}\!\mathrm{d}{\theta} = \frac{1}{2\pi} \int_{-\phi}^{2\pi-\phi} f(u) \mathop{}\!\mathrm{d}{u}}_{u = \theta - \phi} = \frac{1}{2\pi} \int_{0}^{2\pi} f(u) \mathop{}\!\mathrm{d}{u}, \] where the last equality follows because of the \(2\pi\)-periodicity of \(f.\)

For readers familiar with contour integrals and the residue calculus, we can also regard elements \(z \in \mathbb{T}\) as complex numbers in their own right, i.e., the group operation is understood multiplicatively instead of additively. With this setup, the Haar integral is given by \[ f \mapsto \frac{1}{2\pi i} \oint_C \frac{f(z)}{z} \mathop{}\!\mathrm{d}{z}. \]

The group \(\mathbb{R}^+\) has the Haar integral \[ f \mapsto \int_0^\infty \frac{f(x)}{x} \mathop{}\!\mathrm{d}{x}. \] Indeed, if \(y \in \mathbb{R}^+\), then \[ \begin{aligned} \int_0^\infty \frac{(y \cdot f)(x)}{x} \mathop{}\!\mathrm{d}{x} & = \int_0^\infty \frac{f(y^{-1}x)}{x} \mathop{}\!\mathrm{d}{x} \qquad & (\text{substitute } u = y^{-1}x) \\ & = \int_0^\infty \frac{f(u)}{yu} y \mathop{}\!\mathrm{d}{u} \\ & = \int_0^\infty \frac{f(u)}{u} \mathop{}\!\mathrm{d}{u}. \end{aligned} \]

Example 10.18 Consider \(G \leq \operatorname{Aff}_{1}({\mathbb{R}})\) (c.f. Example 8.7) consisting of transformations given by \(t \mapsto \alpha t + \beta\) where \(\alpha > 0.\) Then \(G\) is locally compact (but not compact). We claim that \(G\) has a Haar integral given by \[ f \mapsto \int_{-\infty}^\infty \int_{0}^\infty f(\alpha,\beta) \frac{1}{\alpha^2} \mathop{}\!\mathrm{d}{\alpha} \mathop{}\!\mathrm{d}{\beta}. \] As we will see in Exercise H.4, this integral is left-invariant but not right-invariant.

Example 10.19 If \(G\) is a finite group, the Haar integral is given by \[ f \mapsto \frac{1}{|G|} \sum_{g \in G} f(g). \]

In all our examples (aside from finite groups), our integrals will be given in terms of integrals in the sense of calculus and, as we’ve seen from the above examples, left-invariance amounts to additional terms included in the integrand to compensate for the substitution \(u = h^{-1} g.\) For this reason, when we are studying an arbitrary compact group \(G\), we write \[ \int_G f(g) \mathop{}\!\mathrm{d}{g} \] for the Haar integral of \(f \in C^{}(G).\)

10.4 Representation Theory

Not all of our arguments in Chapter 4, Chapter 5, and Chapter 7 relied on the finiteness of the groups involved! In particular, Schur’s Lemma (Lemma 5.1) still holds by repeating the existing proof verbatim. Moreover, our alternative argument for Proposition 5.5 applies without any modification:

Proposition 10.10 Any irreducible representation of any (possibly infinite) abelian group over \(\mathbb{C}\) is \(1\)-dimensional.

The main obstruction to developing the representation theory of infinite groups, then, is Weyl’s unitary trick. Behold, the power of Haar integrals!

Lemma 10.2 (Unitary Trick) Let \(G\) be a compact group with a representation \(\rho: G \to \operatorname{GL}(V)\) over \(\mathbb{C}.\) Then there exists a Hermitian inner product \(\langle \cdot, \cdot \rangle_G\) on \(V\) such that \[ \langle g \cdot x, g \cdot y \rangle_G = \langle x, y \rangle_G \]

for all \(g \in G\) and \(x, y \in V.\)

Proof. The argument is the same as before: we construct a \(G\)-invariant inner product \(\langle \cdot, \cdot \rangle_G\) using an arbitrary inner product \(\langle \cdot, \cdot \rangle.\) Since \(G\) is compact, we use the Haar integral to define \[ \langle x, y \rangle_G \coloneqq \int_G \langle g \cdot x, g \cdot y \rangle \mathop{}\!\mathrm{d}{g}. \] For fixed \(x, y \in V\), the map \(g \mapsto \langle g \cdot x, g \cdot y \rangle\) is a continuous function \(f: G \to \mathbb{C}.\) Therefore \[ \langle h \cdot x, h \cdot y \rangle_G = \int_G \underbrace{\langle gh \cdot x, gh \cdot y \rangle}_{(h.f)(g)} \mathop{}\!\mathrm{d}{g} = \int_G f(g) \mathop{}\!\mathrm{d}{g} = \langle x, y \rangle_G, \] by right-invariance of the Haar integral. Verifying that \(\langle \cdot, \cdot \rangle_G\) is a Hermitian inner product is the same as in the proof for finite groups, so we are done!

From this we immediately deduce Maschke’s Theorem (Theorem 4.1) for continuous representations of compact groups via the same proof as before: given a subrepresentation \(W \leq_G V\), the orthogonal complement under the \(G\)-invariant inner product is also a subrepresentation and \(V = W \oplus W^\perp.\) Thus a continuous representation of a compact group is reducible if and only if it is decomposable. Moreover, because we still have Schur’s Lemma, we can apply the same argument as before to deduce Theorem 5.1:

Theorem 10.3 Let \(G\) be a compact group and \(V\) a finite-dimensional continuous representation over \(\mathbb{C}.\) Then there is a decomposition \[ V = V_1 \oplus \cdots \oplus V_r, \] where the \(V_i\) are irreducible representations. Moreover, this decomposition is unique in the sense that, if we have another decomposition \[ V = W_1 \oplus \cdots \oplus W_s, \] then \(r=s\) and, after re-ordering, we have \(V_i \cong W_i\) for each \(i.\)

Indeed, this theme of averaging allows us to construct projections onto invariant subspaces as before. Remember that, in the finite case, we could average over each operator in the image of a representation to obtain such a projection (Exercise D.5, Theorem 5.2). We obtain the same result here:

Theorem 10.4 For any compact group \(G\) and continuous representation \(\rho: G \to \operatorname{GL}(V)\), \[ \varphi \coloneqq \int_G \rho(g) \mathop{}\!\mathrm{d}{g} \] is an orthogonal projection \(V \twoheadrightarrow V^G\) with respect to the inner product \(\langle \cdot, \cdot \rangle_G.\)

Proof. For a fixed \(x \in V\), the map \(g \mapsto \rho(g) x\) is a continuous map \(G \to V.\) Thus \[ \rho(h) \varphi(x) = \int_G \rho(hg) x \mathop{}\!\mathrm{d}{g} = \int_G \rho(g) x \mathop{}\!\mathrm{d}{g} = \varphi(x) \] for any \(h \in G\), by Proposition 10.7 and left-invariance. Further, if \(y \in V^G\), then we compute \[ \varphi(y) = \int_G \rho(g) y \mathop{}\!\mathrm{d}{g} = \int_G y \mathop{}\!\mathrm{d}{g} = y. \] Hence \(\varphi(x) \in V^G\) and further \(\varphi^2(x) = \varphi( \varphi(x) ) = \varphi(x)\), so \(\varphi\) is a projection onto \(V^G.\) This is enough for our purposes—the verification of orthogonality is left as an exercise.

As before, we define the character of a continuous representation \(\rho: G \to \operatorname{GL}(V)\) as the map \(\chi_V: G \to \mathbb{C}\) given by \(\chi_V(g) = \operatorname{Tr}\rho(g).\) We denote the subspace of continuous class functions by \(\mathscr{C}({G})\) and introduce an inner product by \[ \langle f_1, f_2 \rangle \coloneqq \int_G \overline{f_1(g)} f_2(g) \mathop{}\!\mathrm{d}{g}. \tag{10.2}\] Taking the trace of the projection \(\varphi\) in the previous Theorem (by Proposition 10.8) means \[ \dim V^G = \langle 1, \chi_V \rangle. \] From here we deduce the same character orthogonality relations as before by considering the invariant subspace of \(\operatorname{Hom}(V,W) \cong V^* \otimes W\): \[ \dim \operatorname{Hom}(V,W)^G = \langle 1, \chi_{\operatorname{Hom}(V,W)} \rangle = \int_G \overline{\chi_V(g)} \chi_W(g) \mathop{}\!\mathrm{d}{g} = \langle \chi_V, \chi_W \rangle. \] Since \(\operatorname{Hom}(V,W)^G = \operatorname{Hom}_G(V,W)\), Schur’s Lemma allows us to conclude:

Theorem 10.5 (Orthonormality of characters) Let \(G\) be a compact group with \(V\) and \(W\) irreducible representations over \(\mathbb{C}\) with characters \(\chi_V\) and \(\chi_W\), respectively. Then \[ \langle \chi_V, \chi_W \rangle = \left\{ \begin{array}{ll} 1 & \text{if } V \cong W \\ 0 & \text{otherwise}. \end{array} \right. \]

From here, we deduce the majority of results from our study of finite groups: in particular, Corollary 7.2, Corollary 7.3, Corollary 7.4, Corollary 7.5. The main difference from the case of finite groups is that \(G\) need not have a finite number of conjugacy classes, so we no longer expect to have a finite list of irreducible representations. Indeed, the subspace of continuous class functions \(\mathscr{C}({G}) \leq C^{}(G)\) is infinite-dimensional, so the characters of \(G\) no longer constitute a basis in the algebraic sense.

Instead, with care taken to develop the theories of measure and convergence, one can show that appropriately “nice” class functions can be well-approximated by characters. Such a study is complicated by the fact that the space of continuous functions is not closed under limits—we can take a (pointwise, say) limit and end up with something discontinuous—and so the curious student of analysis must broaden their notion of what “nice” means. This is better left for a course in functional analysis.

Indeed, while investigating the general scenario further leads to a robust and rich theory, the technical requirements are somewhat prohibitive—functional analysis, measure theory and other pursuits become important prerequisites. We will instead focus on specific examples for the remainder of this course to get at how the theory of compact groups stands out from that of finite groups.

Diestel, J., and A. Spalsbury. 2014. The Joys of Haar Measure. Graduate Studies in Mathematics. American Mathematical Society.

Fulton, W., and J. Harris. 1991. Representation Theory: A First Course. Graduate Texts in Mathematics. Springer New York.

Haar, Alfred. 1933. “Der Massbegriff in Der Theorie Der Kontinuierlichen Gruppen.” Annals of Mathematics 34 (1): 147–69.

Rossmann, Wulf. 2002. Lie Groups: An Introduction Through Linear Groups. Oxford University Press.

Rudin, W. 2013. Principles of Mathematical Analysis. International Series in Pure and Applied Mathematics. McGraw-Hill.

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1. For the experts, \(\operatorname{GL}(V)\) has the topology induced by the operator norm. Equivalently, a group homomorphism \(\rho: G \to \operatorname{GL}(V)\) is continuous if the mapping \(G \times V \to V\) given by \((g,v) \mapsto \rho(g)v\) is continuous. Note that we can also choose a basis and realize \(\operatorname{GL}(V) \cong \operatorname{GL}_n(\mathbb{C}) \subset \mathbb{C}^{n^2}\), where the notion of limits is clear by our previous discussion.