Talk about the representation theory of compact Lie groups. How do you know you have a finite-dimensional representation?
Solution
Let \(G\) be a compact Lie group. We work with continuous complex representations.
1. Every finite-dimensional representation of \(G\) is equivalent to a unitary representation.
Let \(\rho:G\to\operatorname{GL}(V)\) be finite-dimensional and choose any positive-definite Hermitian form \(\langle-,-\rangle_0\) on \(V\). Normalize Haar measure \(dg\) on \(G\) so that \(\int_G dg=1\), and define \[ \langle v,w\rangle_G =\int_G \langle\rho(g)v,\rho(g)w\rangle_0\,dg. \] This is again positive definite. By left-invariance of Haar measure, for \(h\in G\), \[ \langle\rho(h)v,\rho(h)w\rangle_G =\int_G\langle\rho(gh)v,\rho(gh)w\rangle_0\,dg =\langle v,w\rangle_G. \] Thus \(\rho\) is unitary for this averaged inner product.
2. Every finite-dimensional representation of \(G\) is completely reducible.
After 1, suppose \(W\subseteq V\) is \(G\)-invariant. Its orthogonal complement \(W^\perp\) is also invariant: if \(v\in W^\perp\), \(w\in W\), and \(g\in G\), then \[ \langle\rho(g)v,w\rangle_G =\langle v,\rho(g^{-1})w\rangle_G=0 \] because \(W\) is invariant. Hence \[ V=W\oplus W^\perp. \] Induction on \(\dim V\) decomposes \(V\) as a direct sum of irreducibles.
3. Every irreducible continuous unitary representation of a compact group is finite-dimensional.
This follows from the Peter–Weyl theorem, which decomposes the regular representation on \(L^2(G)\) into finite-dimensional irreducible unitary representations and, equivalently, implies that every irreducible continuous unitary representation of a compact group is finite-dimensional.
4. Thus the irreducible unitary dual of a compact Lie group consists of finite-dimensional representations, and arbitrary unitary representations decompose into Hilbert direct sums of these irreducibles.
3. The unitary decomposition theorem for compact groups, again part of Peter–Weyl theory, expresses a continuous unitary representation as an orthogonal direct sum of isotypic pieces built from finite-dimensional irreducibles.
5. For connected compact Lie groups, the finite-dimensional irreducibles are classified by highest weights.
Choose a maximal torus \(T\subseteq G\) and a positive system of roots. The highest-weight theorem identifies isomorphism classes of irreducible finite-dimensional complex representations with the dominant integral weights compatible with \(G\). This is the compact form of the highest-weight classification.
6. The answer to ``how do you know you have a finite-dimensional representation?’’ depends on the setting.
3. General reducible unitary representations of compact groups may themselves be infinite-dimensional; they are built as Hilbert direct sums of finite-dimensional irreducibles.