From the group to the spaces it moves
The first four guides in this rung kept the spotlight on G itself — its Lie algebra, its exponential map, its adjoint representation. But the whole reason Lie groups matter is that they move things. A Lie group earns its keep as a group of symmetries: rotations act on the sphere, the Lorentz group acts on spacetime, SU(3) acts on the color states of a quark. This final guide changes the question from "what is G?" to "what does G do?", and the answer organizes into three layers — G acting on a manifold, G acting on a vector space, and G integrated against itself.
Recall the basic vocabulary from the Erlangen rung, now in smooth form. A smooth action is a smooth map G x M -> M, written (g, p) |-> g.p, that respects the group law: e.p = p and g.(h.p) = (gh).p. The orbit of a point p is everything you can reach, {g.p : g in G}, and its stabilizer H_p = {g : g.p = p} is the subgroup that pins p in place. These two notions — where you can go, and what fixes you — are the entire skeleton of what follows.
Homogeneous spaces: when the orbit is everything
The cleanest case is when the action is transitive — one orbit fills all of M, so any point can be carried to any other. Then M is called a homogeneous space: it looks the same from every point, because G itself shuffles the points around. Fix a basepoint p_0 with stabilizer H. The orbit-stabilizer correspondence becomes a diffeomorphism M = G/H, the space of left cosets gH, where g and g' name the same point exactly when they differ by something in H. This is the central identity of the whole guide: a homogeneous space is a quotient of a group.
Make it concrete with the two-sphere. The rotation group SO(3) acts on S^2, and you can rotate any point to any other, so the action is transitive. Pick the north pole; the rotations fixing it are exactly the spins about the vertical axis, a copy of SO(2). Hence S^2 = SO(3)/SO(2), a sphere realized as a quotient of a three-dimensional group by a one-dimensional subgroup — and indeed 3 - 1 = 2, the right dimension. The same template gives real projective space, Grassmannians, and the upper half-plane (as SL(2,R) modulo its rotation subgroup) all at once.
Representations: when the group acts linearly
Now let G act not on a curvy manifold but on a vector space V, linearly. A representation is a smooth homomorphism rho: G -> GL(V): each group element becomes an invertible matrix, and rho(gh) = rho(g) rho(h). The point is to trade the abstract group for matrices, where everything is computable. You have already met the most important example built into the group itself — the adjoint representation Ad, where G acts on its own Lie algebra. Differentiating any representation at the identity hands you a representation of the Lie algebra, dovetailing with Lie's theorems from the previous guides.
The dream is to break every representation into atoms. A representation is irreducible if V has no proper subspace preserved by all the rho(g) — no smaller invariant target to retreat into. For a compact group the dream comes true: every finite-dimensional representation splits as a direct sum of irreducibles, cleanly and uniquely. This is why compactness, the same hypothesis that made compactness precious in topology, is the dividing line of the entire theory. Compact groups behave like finite groups; non-compact groups (the Lorentz group, SL(2,R)) need infinite-dimensional representations and a far subtler analysis.
Hold the workhorse example in your hand: SU(2), one of the classical matrix groups. Its irreducible representations come exactly one for each dimension 1, 2, 3, ... — the spin-0, spin-1/2, spin-1, ... representations of quantum mechanics. The 2-dimensional one is SU(2) acting on C^2 as itself; the 3-dimensional one is the adjoint representation on its Lie algebra su(2) = R^3, which is why SU(2) double-covers SO(3). The whole list is generated by symmetric powers of the standard one. This single example, fully understood, is worth more than the general theorem stated in maximum generality.
Haar measure: the one invariant integral
To average over a group — and averaging is the engine that builds invariant inner products and splits representations — you need an integral that the group cannot shift. Haar measure is exactly that: on every Lie group there is a left-invariant measure, unique up to a positive constant, so that integrating f(x) and integrating the shifted f(gx) give the same answer. You actually already built it in guide 4. Pick any nonzero top-form at the identity and slide it everywhere by left translation; that left-invariant volume form integrates to Haar measure. The Maurer-Cartan form is the abstract bookkeeping that makes this transport canonical.
Left-invariant need not equal right-invariant. The ratio between them is a homomorphism G -> positive reals called the modular function; when it is identically 1 the group is unimodular and a single bi-invariant measure serves both sides. Good news for our story: every compact group is unimodular, and so is every semisimple or abelian group. The non-unimodular examples are characteristic of certain solvable groups — the classic being the "ax + b" group of affine maps of the line, where stretching distorts left and right averages differently. State which side you mean; sources differ on the default convention.
left-invariant: integral over G of f(x) dx = integral over G of f(g.x) dx for all g in G
build it: omega_e = nonzero n-covector at identity -> omega_g := (L_{g^{-1}})* omega_e -> dx := |omega|
modular function Delta: integral of f(x.g) dx = Delta(g) * integral of f(x) dx
Delta == 1 <=> G unimodular <= G compact, or semisimple, or abelian
compact normalization: total volume of G = 1 (so integration = averaging)Characters and Peter-Weyl: harmonic analysis on a group
With an invariant integral in hand, representations become measurable, comparable, orthogonal. The character of a representation rho is the function chi_rho(g) = trace(rho(g)). Because trace is conjugation-invariant, chi is constant on conjugacy classes — it forgets the matrices and remembers only the irreducible's fingerprint. The decisive fact, proved by averaging against Haar measure, is that the characters of distinct irreducibles are orthonormal: integral over G of chi_rho times the conjugate of chi_sigma equals 1 if the two are equivalent and 0 otherwise. A whole representation's decomposition is then read off by computing inner products of characters — exact, finite arithmetic.
The crown of the compact theory is the Peter-Weyl theorem. It says the matrix entries of all the irreducible representations, taken together, form a complete orthogonal basis for the square-integrable functions L^2(G) with respect to Haar measure. In one breath this is the vast generalization of Fourier series: on the circle G = U(1) the irreducibles are the characters e^{in theta}, and Peter-Weyl collapses to the classical statement that they span L^2 of the circle. For a general compact group the "frequencies" are the irreducible representations, and the characters are their fundamental tones.
Putting the rung together
Step back and see the architecture this rung built. A Lie group is a manifold that is also a group (guide 1); its infinitesimal shadow is a Lie algebra with a bracket (guide 2); the exponential reconnects the algebra to the group along one-parameter subgroups (guide 3); the adjoint action, Killing form, and Maurer-Cartan form let the group study itself (guide 4); and here the group finally acts outward — on homogeneous spaces, on vector spaces through representations, and on its own function space through Haar measure and characters. That last chain is where Lie theory meets physics, number theory, and geometry alike.
An honest closing about scope and difficulty. We surveyed the compact, finite-dimensional, smooth story — the part that is clean and complete. The fuller picture, the classification of irreducibles via roots and weights and the semisimple structure theory, is a full graduate course in its own right, and the representation theory of non-compact and infinite-dimensional groups (the Langlands program lives here) is open research territory. Treat this guide as a true map of the terrain, not a substitute for the climb: the theorems are stated with their hypotheses precisely so you know which trail is paved and which is still wild.