I am reading the book. On page 80, there is a concept the inner product on $\mathfrak{h}^*$ induced by the inner product on $\mathfrak{h}$. Here $\mathfrak{h}$ is a Cartan subalgebra of a Lie algebra $g$. My questions are:
(1) how the inner product on $\mathfrak{h}$ is defined?
(2) how to define the inner product on $\mathfrak{h}^*$ using the inner product on $\mathfrak{h}$?
Thank you very much.
$\mathfrak{g}$ is always a semisimple Lie algebra in this context. This means that it has a nondegenerate Killing form. The inner product on $\mathfrak{h}$ is the restriction of the Killing form to $\mathfrak{h}$; it remains nondegenerate.
If $V$ is any vector space equipped with a nondegenerate bilinear form $B(-, -)$, then $B$ defines an isomorphism $V \cong V^{\ast}$, and you can define a bilinear form on $V^{\ast}$ using this isomorphism. There are a couple of ways to do this, but if $B$ is symmetric they all agree.