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Let $R$ a root system and $\Delta$ be a simple system of roots of a Lie algebra $\mathfrak g$, $\Delta'\subset \Delta$ and $R(\Delta')=R\cap \mathbb Z(\Delta')$. Define $$p(\Delta')=\mathfrak h \bigoplus_{\alpha \in R(\Delta')} \mathfrak g_{\alpha} \bigoplus_{\alpha \in R^+ \setminus R^+(\Delta')}\mathfrak g_{\alpha}$$ the parabolic subalgebra associated to $\Delta'$.

If $\alpha$ is a simple root in $R^+(\Delta)\setminus R^+ (\Delta')$, then $\beta(h_\alpha)=0$ for all $\beta$ in $R(\Delta')$???

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If I understand your notation correctly, the answer is clearly no. Here the simple root $\alpha$ is any simple root not in $\Delta'$ and it need not be orthogonal to all roots in the subsystem. – Jim Humphreys Dec 4 '10 at 20:55
Can you give me an example where it fails? – Chris Dec 4 '10 at 21:11
Take $\mathfrak{g} =\mathfrak{sl}_3$ with $\Delta'$ containing just one simple root $\beta$ and $\alpha$ being the other simple root. – Jim Humphreys Dec 4 '10 at 22:13
This is a duplicate: – Alex B. Dec 5 '10 at 2:20
There isn't necessarily any problem, but it's good to give links between the two. – j.c. Dec 5 '10 at 22:59
up vote 0 down vote accepted

Jim gave me the answer! It is false in general!


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