λ

ch V (λ) = \frac{\sum _{w \in W} ( - 1 ) ^{ℓ (w)} e ^{w (λ + ρ)}}{\sum _{w \in W} ( - 1 ) ^{ℓ (w)} e ^{wρ}},

where

W

ℓ

ρ

\sum_{w \in W} (- 1)^{ℓ (w)} e^{wρ} = e^{ρ} \prod_{α \in Φ^{+}} (1 - e^{- α}) .

λ = k ρ

. Apply the denominator identity after replacing every formal exponential

e^{μ}

e^{(k + 1) μ}

\sum_{w} (- 1)^{ℓ (w)} e^{w ((k + 1) ρ)} = e^{(k + 1) ρ} \prod_{α \in Φ^{+}} (1 - e^{- (k + 1) α}) .

Dividing this by the ordinary denominator gives

ch V (k ρ) = e^{k ρ} α \in Φ^{+} \prod \frac{1 - e ^{- (k + 1) α}}{1 - e ^{- α}} = e^{k ρ} α \in Φ^{+} \prod (1 + e^{- α} + \dots + e^{- k α}) .

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