바일 벡터

메모

• http://www.math.harvard.edu/~kronheim/yaft.pdf
• 각각의 \(\alpha\in \Delta^+\)에 대하여,

\[ \begin{aligned} \omega_{\alpha}(\theta^{\vee}) & = \omega_{\alpha}(\sum_{\beta \in \Delta^+} n_{\beta}^{\vee} \beta^{\vee}) \\ & = n_{\alpha}^{\vee}, \end{aligned} \] \[ \begin{aligned} \langle \omega_{\alpha},\theta \rangle &= \omega_{\alpha}(\theta^{\dagger}) \\ & = \frac{\langle \theta,\theta \rangle}{2}\omega_{\alpha}(\theta^{\vee}) \\ & = \frac{\langle \theta,\theta \rangle}{2} n_{\alpha}^{\vee} \end{aligned} \]

• 따라서

\[ \begin{aligned} \langle \rho,\theta \rangle & = \langle \sum_{\alpha\in \Delta^+} \omega_{\alpha},\theta \rangle \\ & = \sum_{\alpha\in \Delta^+} \frac{\langle \theta,\theta \rangle}{2} n_{\alpha}^{\vee} \\ & = \frac{h^{\vee}-1}{2h^{\vee}} \end{aligned} \]