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Forsaken Fortress Strategy Free Download.Q:
Simple question about section on Lusztig-Vogan bijection for “Root Numbers”
I have a basic question about an argument in the book “Groups Acting on Semisimple Lie Algebras” by Lusztig and Vogan (especially the proof of the following theorem 3.1.2):
Let $S$ be a set of simple roots and let $A$ be the corresponding Cartan matrix. Let $V_\lambda$ denote the irreducible representation with highest weight $\lambda$. The equality $d_\lambda = \dim V_\lambda$ is equivalent to the condition that there exists a primitive vector $v_\lambda \in V_\lambda$ satisfying $\langle h, v_\lambda \rangle = \lambda(h)$ for $h \in S$ and $\langle h, v_\lambda \rangle = 0$ for $h \in A$. The Lusztig-Vogan bijection gives us a way of constructing such a vector from $v_\lambda$ by the formula $v_\lambda = \sum_{w \in W} a_w v_0$.
The authors argue that if $v_\lambda$ is such a vector then there exists $w \in W$ with $a_w
eq 0$. But why is this the case? I don’t see why this is supposed to be true. If I take the space $V_\lambda$ of highest weight $\lambda = -\alpha_1$ then there is no element $w \in W$ with $a_w = 1$. But there is $v_{ -\alpha_1} = \alpha_1$ and this satisfies $\langle h, v_{ -\alpha_1} \rangle = – \alpha_1 (h)$ for all $h \in S$ and $\langle h, v_{ -\alpha_1} \rangle = 0$ for all $h \in A$. Why can’t this be counted as a “primitive vector” as well?
A:
The authors consider the subrepresentation $\mathfrak g_\mathbb C$ of $\mathfrak g_\mathbb C$ corresponding to the fundamental weights $\lambda_i$. Then $\mathfrak
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