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

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.