Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 102 / 4

Choose a positive root system $R^{+}$ for a complex semisimple Lie algebra, with Weyl group $W$ and Weyl vector $\rho =\frac{1}{2}{\sum }_{\alpha \in R^{+}}\alpha$. For a dominant integral weight $\lambda$, let $L(\lambda )$ denote its finite-dimensional irreducible representation, and use formal symbols with $e^{\mu }{e}^{\nu }=e^{\mu +\nu }$. Its formal character of a weight module is $\mathrm{ch}L(\lambda )={\sum }_{\mu }\dim L(\lambda {)}_{\mu }{e}^{\mu }$. The Weyl character formula saysHere $\det (w)=(-1{)}^{\ell (w)}$, the sign of the product of simple root reflections. The Weyl denominator formula isThe quotient is initially interpreted in the fraction field of the group algebra of the weight lattice; the theorem makes it a finite character.

Take an integer $k\ge 0$, so $k\rho$ is a dominant integral weight. Apply the dilation $e^{\mu }\mapsto e^{(k+1)\mu }$ to the Weyl denominator formula:This is exactly the numerator of the Weyl character formula for $L(k\rho )$. Divide by its denominator and use a finite geometric series in each root direction:This character of a Weyl-vector multiple also gives $\dim L(k\rho )=(k+1{)}^{|R^{+}|}$ by evaluating every formal exponential at one. This evaluation is safe in the final polynomial expression, without dividing by a zero denominator.