I am trying to understand the Borel-Weil theorem, but I am very confused because of the different conventions used in different sources. I am especially confused about two things: (1) the definition of the line bundle involved and the sign of the representation $\lambda$, and (2) the dual representation.
I will now try to formulate the theorem. Suppose that $G$ is a complex semisimple algebraic group with torus $T$ and Borel $B$. Let $\lambda$ be a character of the torus, it can be naturally extended to a one-dimensional representation of $B$. Then we can consider $\mathbb{C} = \mathbb{C}_\lambda$ as a $B$-module by setting $b.v = \lambda(b).v$.
Now we consider a line bundle associated to $\lambda$. We want the total space of the line bundle to be the quotient of $G \times \mathbb{C}$ by the action of $B$ given by $(g,v).b \sim (gb,(\lambda(b))^{-1}v)$. This is where I first get confused because I have come across notations $G \times^B \mathbb{C}_{- \lambda}$ and $G \times^B \mathbb{C}_{ \lambda}$ and I am not sure whether they refer to the same space. Question: what is the correct action of $B$ on the $\mathbb{C}$ component in the line bundle we are trying to define?
Supposing we have defined our line bundle, let $\mathcal{L}_\lambda$ denote its sheaf of sections. (Again, some people write $\mathcal{L}_{-\lambda}$.) The Borel-Weil theorem says that if $\lambda$ is a dominant weight then $H^0(G/B,\mathcal{L}_\lambda)$ is isomorphic to the irreducible representation $V_\lambda$ of $G$ with highest weight $\lambda$. I have come across versions which say that $H^0(G/B,\mathcal{L}_{-\lambda})$ is isomorphic to $V_\lambda$, or that $H^0(G/B,\mathcal{L}_\lambda)$ or $H^0(G/B,\mathcal{L}_{-\lambda})$ is isomorphic to the dual of $V_\lambda$.
I am very confused. Clearly all these versions say the same thing but they all use different conventions/definitions and I don't know how to interpret them. Also, what is the dual of $V_\lambda$? Is it isomorphic to $V_\lambda$ as a $G$-module? Why do some versions of the theorem mention the dual while others don't? (I noticed that the dual is mentioned mainly when the extension of the theorem called Borel-Weil-Bott theorem is also discussed, so is it the case that if $\lambda$ is dominant then the dual of $V_\lambda$ is the same as $V_\lambda$?)
I will greatly appreciate help with this.