Weyl character formula for compact groups

Remarks

The denominator equals $\prod_{\alpha \in \Phi^+} \left(e^{i\alpha(H)/2} - e^{-i\alpha(H)/2}\right)$, by the Weyl denominator formula. One proof multiplies $\chi_V$ by the denominator and notes that the product is $W$-alternating with integer coefficients. The Weyl integration formula and $\int_G \abs{\chi_V}^2 \, dg = 1$ then show that only the term $\lambda + \rho$ and its $W$-images survive, each with coefficient $\pm 1$. Example: for $G = SU(2)$ and $H = \operatorname{diag}(i\theta, -i\theta)$, there is one positive root with $\alpha(H) = 2\theta$, $\rho(H) = \theta$, and the representation of dimension $n + 1$ has $\lambda(H) = n\theta$. The formula gives $\chi(\exp H) = \frac{e^{i(n+1)\theta} - e^{-i(n+1)\theta}}{e^{i\theta} - e^{-i\theta}} = \frac{\sin((n+1)\theta)}{\sin\theta}$, which is $e^{in\theta} + e^{i(n-2)\theta} + \cdots + e^{-in\theta}$. Letting $H \to 0$ gives the Weyl dimension formula $\dim V = \prod_{\alpha \in \Phi^+} \frac{\ip{\lambda + \rho}{\alpha}}{\ip{\rho}{\alpha}}$ for a $W$-invariant inner product on $\mathfrak{t}^*$.

Going deeper

For a student

By conjugacy of maximal tori, a character is determined by its values on $T$, so the formula gives it everywhere. Together with the Peter–Weyl theorem it gives the theorem of the highest weight: irreducible representations of $G$ correspond to dominant integral weights. The dimension formula and Kostant's multiplicity formula follow from it, and the Borel–Weil–Bott theorem gives a geometric realisation of the same representations. For $G = U(n)$ the formula expresses irreducible characters as Schur polynomials in the eigenvalues.