Past exam of the lie Groups and Differentiable Manifolds course of the University of Oxford / 2008 / A / 4

As usual here, representations are finite-dimensional and continuous. Start with any positive definite Hermitian form $h_0$ and normalized Haar measure $\mu$ on the compact group. DefineThis is Hermitian, sesquilinear, and positive definite: for $v\ne 0$ its continuous integrand $h_0(gv,gv)$ is strictly positive. Bi-invariance of Haar measure shows $h(av,aw)=h(v,w)$ for every $a\in G$.

If $U\subset V$ is invariant, then so is its orthogonal complement: for $v\perp U$ and $u\in U$, $h(av,u)=h(v,a^{-1}u)=0$. Hence $V=U\oplus U^{\perp }$ as a representation. Induct on dimension, splitting whenever a nontrivial proper invariant subspace exists. This proves every such representation is a direct sum of irreducible representations.

Schur orthogonality for irreducible complex characters of a compact group, with normalized Haar measure, isEquivalent representations clearly have equal characters, since trace is invariant under conjugation. Conversely complete reducibility gives ${\chi }_V={\sum }_{\alpha }{m}_{\alpha }{\chi }_{\alpha }$, where the nonnegative integer multiplicity is recovered by $m_{\alpha }=⟨{\chi }_V,{\chi }_{\alpha }⟩$. Equal characters thus give equal multiplicities of every irreducible constituent, hence equivalent representations.

The usual Weyl character formula requires a compact connected Lie group. Choose a maximal torus $T$, a positive root system ${\Phi }^{+}$, Weyl group $W$, and $\rho =\frac{1}{2}{\sum }_{\alpha \in {\Phi }^{+}}\alpha$. Irreducible representations are indexed by dominant integral highest weights $\lambda$ belonging to the character lattice of the actual group $T$. For $X\in L(T)$ such that $\exp X$ is regular,Weights are written as linear functionals on the complexified torus algebra; their values on the compact real torus algebra are imaginary. The denominator is $e^{\rho (X)}{\prod }_{\alpha >0}(1-e^{-\alpha (X)})$. If $\rho$ is not itself a character of $T$, the individual expressions are interpreted on its covering algebra; their quotient is single-valued and descends to $T$. The apparent singularities at nonregular elements are removable, and conjugacy into a maximal torus determines the character on the whole connected group.

The formula explicitly determines all irreducible characters from their highest weights, and its limit at the identity gives the dimension formulaCharacter multiplication and orthogonality then compute tensor-product multiplicities. For $S{U}_2$, highest weights are integers $m\ge 0$, and at $\mathrm{diag}(z,z^{-1})$,Connectedness cannot simply be omitted from this version. For a nontrivial finite group the maximal torus is trivial, yet characters can differ away from the identity. The root data on that torus alone cannot supply the characters of all disconnected compact groups.