G2 weight and root lattce

Question

The book that I'm reading tells me that G2's weight lattice coincides with its root lattice but I do not understand the concept of weight lattice. I've been able to construct G2's root diagram (the one that looks like a star) from its Dynkin Diagram but now I'm stuck. I thought that weights depend on the representation you choose so I do not know how to approach this general question.

Answer 1

This is indeed a confusing issue. I hope to find time to write a longer answer later but in short:

There are two definitions of the weight lattice (they are equivalent, but that they are equivalent is a very interesting, non-trivival fact)

1. The weight lattice is the collection of points in $\mathbb{R}^2$ (for $G_2$), or in $\mathfrak{h}^*$ (in general) that appear as a weight in some finite dimensional representation. So while you are right that weights depend on the representation and each finite dimensional representation has only finitely many of them, the weight lattice is the collection of all such weights. The surprising thing is that they form a lattice - naively you would perhaps expect that the weights of one representation had no incentive whatsoever to behave 'nice' with respect to the weights of a different representation they have no way of knowing about, but somehow they do.

2. The weight lattice is the set of point that have integer inner product with each of the points in the root lattice. Here it is not surprising that they form a lattice, but it is surprising that this definition gives the same thing as definition 1.