Irreducible Dual Representation

For a semisimple Lie Algebra $\mathfrak{g}$ with Cartan Subalgebra $\mathfrak{t}$, let $V(\lambda)$ be the unique irreducible highest weight module with highest weight $\lambda$.

I am asked to show that the dual representation $V(\lambda)^*$ is irreducible, and to give a condition for $V(\lambda)$ to be self dual.

For the first part, my thoughts are that if I can take a basis of $V(\lambda)^*$ and show that the orbit of one of them under the action of $\mathfrak{t}$ contains all of them, then maybe I'd be done. But perhaps for this I would actually have to show it for any general basis?

For the second part I have heard that the condition is whether or not $-1$ is in the Weyl group, but as my understanding of Lie Algebras is quite weak I'm not sure why the Weyl group is important here. I would appreciate any help that you might be able to offer, thank you!

To deduces irreducibility of $V(\lambda)^*$, you don't really have to go to quotients. Given an invariant subspace $W\subset V(\lambda)^*$ consider its annihilator $U:=\{v\in V(\lambda):\forall\phi\in W:\phi(v)=0\}$. This is clearly a linear subspace in $V(\lambda)$ and a short computation shows that invariance of $W$ implies invariance of $U$. Knowing the $U=V(\lambda)$ and $U=\{0\}$ are the only possibilities for $U$, linear algebra shows that $W=\{0\}$ or $W=V(\lambda)^*$.

Concerning the highest weight of $V(\lambda)^*$ you take a basis for $V(\lambda)$ consisting of weight vectors and consider the dual basis of $V(\lambda)^*$ to conclude that the weights of $V(\lambda)^*$ are exactly the negatives of the weights of $V(\lambda)$. In particular, the highest weight of $V(\lambda)^*$ is $-\mu$, where $\mu$ is the lowest weight of $V(\lambda)$. It can be shown that $\mu=w_0(\lambda)$, where $w_0$ is the so-called "longest element" in the Weyl group. (There are cases in which $w_0=-id$ and then any $V(\lambda)$ is isomorphic to its dual, but in general it may happen that $w_0(\lambda)=-\lambda$ and hence $V(\lambda)\cong V(\lambda)^*$ without $w_0$ being $-id$).