Compact Groups, Haar Measure, and Peter–Weyl

Compactness is the single topological property that most controls a group's representation theory. Compact groups behave, representation-theoretically, exactly like finite groups — averaging replaces summation, every representation is unitary and completely reducible, and the irreducibles are finite-dimensional and organise all of $L^2(G)$. Non-compact groups (Lorentz, Poincaré) violate all of this. This page explains why, extending reps-basics.md from finite to compact groups and completing the compactness remarks of matrix-groups.md.

Compact vs. non-compact

A Lie group is compact if its underlying manifold is compact (closed and bounded, for matrix groups). From matrix-groups.md:

• Compact: $U(n),SU(n),O(n),SO(n),\mathrm{Spin}(n),\mathrm{Sp}(n)$ — the defining conditions ($M^{†}M=1$, etc.) bound the entries.
• Non-compact: $GL(n,\mathbb{F}),SL(n,\mathbb{F})$, the Lorentz/indefinite orthogonal groups $O(p,q)$ with $p,q>0$, and $Sp(2n,\mathbb{R})$ — boosts run off to infinity.

The representation-theoretic dichotomy:

This is the reason relativistic field components carry finite-dimensional non-unitary $(j_{+},j_{-})$ representations of the Lorentz group, while Hilbert-space states must carry infinite-dimensional unitary representations of the Poincaré group (Wigner's classification) — see physics-bridge.md and QFT/preliminaries.md.

Haar measure

The engine behind compact representation theory is an invariant integral. Every Lie group carries a Haar measure $d\mu (g)$ — a measure invariant under left translation, $d\mu (hg)=d\mu (g)$ (and, for compact and abelian groups, also right translation; such groups are unimodular). It is unique up to a positive scalar, and for a compact group it has finite total volume, so it can be normalised: $${\int }_{G}d\mu (g)=1.$$

This normalised Haar measure lets one average over the group exactly as one sums $\frac{1}{|G|}{\sum }_g$ over a finite group. Consequently the finite-group proofs of reps-basics.md transcribe verbatim:

• Unitarisability. Averaging any inner product, $⟨v,w{⟩}_G={\int }_G⟨\rho (g)v,\rho (g)w⟩d\mu (g)$, produces a $G$-invariant inner product — every finite-dimensional representation of a compact group is unitary.
• Complete reducibility (Maschke). The averaged projection onto an invariant subspace gives an invariant complement; representations decompose into irreps.
• Schur orthogonality. Matrix coefficients of inequivalent irreps are orthogonal in $L^2(G,d\mu )$ — the continuous analogue of the character orthogonality of character-theory.md.

For non-compact groups Haar measure has infinite total volume, the averaging integral diverges, and this whole machine breaks — which is exactly why their unitary representations must be infinite-dimensional.

The Peter–Weyl theorem

The culmination for compact groups:

Peter–Weyl theorem. For a compact group $G$, the matrix coefficients of the finite-dimensional irreducible unitary representations form a complete orthogonal basis of $L^2(G)$: $$L^2(G)\cong \widehat{\underset{\pi \in \hat{G}}{⨁}}(V_{\pi }\otimes >V_{\pi }^{\ast }),$$ the Hilbert-space direct sum over the (countable) set $\hat{G}$ of irreducibles, each appearing with multiplicity equal to its dimension.

This is the compact-group generalisation of two familiar facts: the decomposition of the regular representation of a finite group (each irrep $\pi$ occurs $\dim V_{\pi }$ times, so $|G|={\sum }_{\pi }(\dim V_{\pi }{)}^2$), and classical Fourier analysis, which is precisely Peter–Weyl for $G=U(1)$: the characters $e^{in\theta }$, $n\in \mathbb{Z}$, are a complete orthonormal basis of $L^2(S^1)$. Fourier series on the circle and spherical harmonics on $SO(3)/SO(2)=S^2$ are the two most familiar instances.

Maximal tori and the Weyl integration formula

A maximal torus $T\subset G$ is a maximal connected abelian subgroup — a product of circles $U(1{)}^r$, with $r$ the rank. Every element of a compact connected group is conjugate into $T$, so class functions (like characters) are determined by their restriction to $T$, where they become ordinary Fourier series in the torus angles. The Weyl integration formula reduces integrals over $G$ to weighted integrals over $T$, and the Weyl character formula then computes every irreducible character explicitly — the computational payoff developed in reps-lie.md and lie-classification.md.

References

• Bröcker & tom Dieck, Representations of Compact Lie Groups.
• Hall, Lie Groups, Lie Algebras, and Representations, Ch. 11–12.
• Knapp, Lie Groups Beyond an Introduction — Haar measure and Peter–Weyl in full.