I am learning the Borel–Weil–Bott theorem in rational homogeneous varieties. Working on them, I need to know two questions:

- How to judge if a weight is singular?
- How to compute the index of a given weight?

To make my questions precise, let $G$ be a semisimple Lie group over $\mathbb C$. Choose a base of simple roots $\alpha_i$, $i=1,\dotsc, n$, and let $\lambda_i$, $i=1,\dotsc, n$ be the corresponding fundamental weights. Now I have a weight $\lambda=\sum_{i=1}^n m_i\lambda_i$. My questions are:

Definition: a weight $\lambda$ is called singular, if there is a root $\alpha$ such that $\langle\lambda, \alpha\rangle=0$. How to determine if $\lambda$ is singular?

Definition: let $w$ be a Weyl group element, the length of $w$ is the smallest number of reflections $s_{\alpha_i}$ occurring in a word for $w$. The index of $\lambda$ is the shortest length of such $w$ that takes $\lambda$ to the fundamental chamber. If $\lambda$ is nonsingular, how to calculate the index of $\lambda$?

For example, let $G=B_n$ and $\lambda=\lambda_{n-2}-(1+n)\lambda_{n-1}+2\lambda_n$. How can I determine whether $\lambda$ is singular? If $\lambda$ is not singular, how should I compute the index of $\lambda$?