= Character of a Weyl-vector multiple
{title2=$\operatorname{ch}L(k\rho)$}
For a complex <semisimple Lie algebra> and an <integer> $k\ge0$, the <Weyl character formula> and <Weyl denominator formula> give
$$
\operatorname{ch}L(k\rho)=e^{k\rho}\prod_{\alpha\in R^+}(1+e^{-\alpha}+\cdots+e^{-k\alpha}),\qquad
\dim L(k\rho)=(k+1)^{|R^+|}.
$$
Apply the dilation $e^\mu\mapsto e^{(k+1)\mu}$ to the denominator identity to obtain the numerator. Cancellation leaves a finite <geometric series> for each positive <root of a root system>. Evaluating the resulting polynomial at all formal exponentials equal to one gives the <dimension>, without taking an undefined quotient at the identity.
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