I am learning Borel Weil Bott theorem in rational homogeneous varieties. Working on them, I need to know two questions:
1. How to judge if a weight is singular? 
2. 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 $\alpha_i, i=1,\ldots, n$, we form a weight lattice. Let $\lambda_i,i=1,\ldots, n$ be the corresponding fundamental weight. Now I have a weight $\lambda=\sum_{i=1}^n m_i\lambda_i$, my question is 
1. How to determine if $\lambda$ is singular? Definition: a weight $\lambda$ is called singular, if there is a root $\alpha$ such that $\left<\lambda, \alpha\right>=0$.
2. If $\lambda$ is nonsingular, how to calculate the index of $\lambda$? Definition: let $w$ be a Weyl group element, the length of $w$ is the number of shortest letters of reflections. The index of $\lambda$ is the shortest length of such $w$ that takes $\lambda$ to the fundamental chamber.

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 $\lambda$?