This course develops the structure theory of compact Lie groups, moving from analytic foundations to the full classification and representation-theoretic picture. It asks how compactness simplifies the geometry and algebra of Lie groups, and how one can recover global information from invariant integration, conjugacy, and the action of a group on its own [Lie algebra](/page/Lie%20Algebra). The central goal is to understand compact Lie groups through the interplay of symmetry, harmonic analysis, and combinatorial data such as roots, weights, and lattices.

The early chapters build the analytic and geometric tools: Haar measure and averaging on compact groups, maximal tori, conjugacy theorems, and the Weyl [group action](/page/Group%20Action). From there the course passes to root systems and the adjoint representation, which encode the internal geometry of the group, and then to lattices and weight data, which organize its representations. Highest weight theory, Peter-Weyl theory, and character orthogonality provide the representation-theoretic backbone, while the Weyl integration and character formulas translate abstract structure into computable formulas. The final chapter ties these ideas to the compact classical families, showing how the general theory manifests in familiar examples such as unitary, orthogonal, and symplectic groups.

# Introduction

This introductory chapter fixes the viewpoint of the course before the structural theory begins. The first course on Lie groups builds the language of manifolds, Lie algebras, homomorphisms, actions, and the exponential map. Here the emphasis shifts from local differential structure to global structure: for compact connected Lie groups, much of the group can be read from a maximal torus, provided we remember the finite symmetry group acting on it.

The main examples throughout are the compact matrix groups $T^n$, $U(n)$, $SU(n)$, $SO(n)$, and $Sp(n)$. These examples are not decorations; they are the testing ground for every general definition. The point of the theory is to explain why diagonal matrices, permutation symmetries, roots, weights, and characters organise the representation theory of compact groups.

## What Is the Structure Problem?

What information is needed to recover the main features of a compact connected Lie group $G$ without handling all of $G$ at once? A compact group has enough averaging to make representation theory finite-dimensional and unitary in nature, while connectedness prevents the group from being assembled from several unrelated components. The structure problem asks how far we can reduce questions about $G$ to questions about a torus and a finite group of symmetries.

The guiding picture is that a compact connected Lie group contains a maximal connected abelian subgroup $T$, called a maximal torus. For $U(n)$, this is the subgroup of diagonal unitary matrices; for $SU(n)$, it is the diagonal subgroup with determinant $1$. Conjugation by elements of $G$ moves tori around, and the Weyl group records the residual symmetries that preserve a chosen maximal torus.

[definition: Compact Connected Lie Group] A compact connected Lie group is a Lie group $G$ whose underlying smooth manifold is compact and connected. [/definition]

Compactness gives integration over the group, while connectedness makes the Lie algebra and the exponential map more controlling than they would be for a disconnected group. The course uses these hypotheses together: compactness supplies Haar measure and invariant inner products, and connectedness supports the conjugacy theorems for maximal tori. The first example is needed to make these hypotheses concrete inside the matrix groups used throughout the course.

[example: Diagonal Torus In Special Unitary Group] In $SU(n)$, consider \begin{align*} T=\{\operatorname{diag}(z_1,\dots,z_n): |z_i|=1,\ z_1\cdots z_n=1\}. \end{align*} The map \begin{align*} (S^1)^{n-1}\to T,\qquad (z_1,\dots,z_{n-1})\mapsto \operatorname{diag}(z_1,\dots,z_{n-1},(z_1\cdots z_{n-1})^{-1}) \end{align*} is a [group isomorphism](/page/Group%20Isomorphism), since the determinant condition forces $z_n=(z_1\cdots z_{n-1})^{-1}$ and multiplication is componentwise. Hence $T$ is a torus of dimension $n-1$. Its Lie algebra is obtained by differentiating a path \begin{align*} \gamma(t)=\operatorname{diag}(e^{ia_1t},\dots,e^{ia_nt}) \end{align*} with $a_i\in \mathbb R$. The condition $\det\gamma(t)=1$ says \begin{align*} e^{i(a_1+\cdots+a_n)t}=1\text{ for all }t, \end{align*} so $a_1+\cdots+a_n=0$. Differentiating at $t=0$ gives \begin{align*} \gamma'(0)=\operatorname{diag}(ia_1,\dots,ia_n), \end{align*} which is diagonal, skew-Hermitian, and has trace $i(a_1+\cdots+a_n)=0$. Conversely, every diagonal skew-Hermitian trace-zero matrix has this form, so \begin{align*} \mathfrak t=\{\operatorname{diag}(ia_1,\dots,ia_n):a_i\in\mathbb R,\ a_1+\cdots+a_n=0\}. \end{align*} The torus is maximal: any connected abelian subgroup of $SU(n)$ containing $T$ consists of unitary matrices commuting with every diagonal matrix in $T$, and such matrices preserve each coordinate line, so they are diagonal; the determinant-one condition then puts them back in $T$. Permutation matrices conjugate diagonal matrices by permuting the diagonal entries, and after multiplying by a diagonal unitary matrix one may choose determinant $1$. Thus the Weyl group acts on $T$ by \begin{align*} \operatorname{diag}(z_1,\dots,z_n)\mapsto \operatorname{diag}(z_{\sigma^{-1}(1)},\dots,z_{\sigma^{-1}(n)}) \end{align*} for $\sigma\in S_n$, so the finite symmetry left after choosing the torus is permutation of the diagonal entries. [/example]

This example already displays the main pattern. The continuous abelian part is the torus $T$, while the finite symmetry group is a permutation group. Chapters 4 and 5 turn this observation into roots and root data, and Chapter 9 uses it to compute characters.

## Why Compactness Changes Representation Theory

Why should compact groups have a representation theory resembling finite group representation theory? The reason is averaging. Haar measure lets us average arbitrary choices over the group, producing invariant inner products and equivariant projections.

[definition: Finite-Dimensional Unitary Representation] A finite-dimensional unitary representation of a Lie group $G$ is a homomorphism $\rho:G\to U(V)$, where $V$ is a finite-dimensional complex Hilbert space and $U(V)$ is the group of unitary linear maps $V\to V$. [/definition]

This definition identifies the representation-theoretic setting in which orthogonal complements are available. The next theorem is needed because a general finite-dimensional representation of a compact group may not initially be presented as unitary, and the course needs a mechanism that makes unitarity available without changing the underlying representation.

[quotetheorem:9712]

[citeproof:9712]

Once an invariant [inner product](/page/Inner%20Product) is available, invariant subspaces have invariant orthogonal complements. Thus the representation theory becomes semisimple in finite dimensions, just as in the finite group case. Compactness is essential here: for a noncompact group such as $(\mathbb R,+)$, the two-dimensional representation with $t$ acting by $e_1\mapsto e_1$ and $e_2\mapsto t e_1+e_2$ has no positive definite Hermitian inner product making every operator unitary, since non-identity unipotent operators cannot be unitary. The theorem also does not say that every representation is already unitary in its given coordinates; it says that the inner product can be changed without changing the underlying action. This is the analytic input behind complete reducibility, Schur orthogonality, and eventually Peter-Weyl theory. The next example is needed to connect the averaging proof with the familiar Fourier decomposition of a circle action.

[example: Averaging On The Circle] Let $G=T^1$ act on a finite-dimensional complex [vector space](/page/Vector%20Space) $V$ by a continuous representation $\rho:T^1\to GL(V)$, and start with a Hermitian inner product $(\cdot,\cdot)_0$. Define \begin{align*} (v,w)_V=\frac{1}{2\pi}\int_0^{2\pi}(\rho(e^{i\theta})v,\rho(e^{i\theta})w)_0\,d\theta. \end{align*} The integral is Hermitian and linear in the same variable as $(\cdot,\cdot)_0$, because integration preserves addition and scalar multiplication. If $v\neq 0$, then the function $\theta\mapsto (\rho(e^{i\theta})v,\rho(e^{i\theta})v)_0$ is continuous and non-negative. Its value at $\theta=0$ is $(v,v)_0>0$, so it is positive on some interval around $0$, and therefore \begin{align*} (v,v)_V=\frac{1}{2\pi}\int_0^{2\pi}(\rho(e^{i\theta})v,\rho(e^{i\theta})v)_0\,d\theta>0. \end{align*} Thus $(\cdot,\cdot)_V$ is a Hermitian inner product. For invariance, fix $e^{i\phi}\in T^1$. Using that $\rho$ is a homomorphism, \begin{align*} (\rho(e^{i\phi})v,\rho(e^{i\phi})w)_V=\frac{1}{2\pi}\int_0^{2\pi}(\rho(e^{i(\theta+\phi)})v,\rho(e^{i(\theta+\phi)})w)_0\,d\theta. \end{align*} With $u=\theta+\phi$, this becomes \begin{align*} (\rho(e^{i\phi})v,\rho(e^{i\phi})w)_V=\frac{1}{2\pi}\int_\phi^{2\pi+\phi}(\rho(e^{iu})v,\rho(e^{iu})w)_0\,du. \end{align*} The integrand is $2\pi$-periodic because $e^{i(u+2\pi)}=e^{iu}$, so the last integral equals the integral over $[0,2\pi]$. Hence \begin{align*} (\rho(e^{i\phi})v,\rho(e^{i\phi})w)_V=(v,w)_V. \end{align*} Therefore every $\rho(e^{i\phi})$ is unitary for the averaged inner product. Since $T^1$ is abelian, these unitary operators commute with one another, and the finite-dimensional spectral theorem for commuting normal operators gives a simultaneous decomposition into common eigenspaces. The eigenvalue of $z\in T^1$ on such a common eigenspace is a continuous homomorphism $T^1\to S^1$, hence has the form $z\mapsto z^m$ for a unique $m\in\mathbb Z$. Thus the averaging construction makes the circle action unitary, and the representation splits into its Fourier weight spaces. [/example]